GB2501530A - Rheometer and rheometric method - Google Patents

Abstract

Apparatus responsive to constitutive parameters (relaxation time, power law exponent, polymeric viscosity, solvent viscosity) of the rheological properties of complex fluids comprises a flow path for the fluid comprising a shear feature that creates multiple shear patterns between adjacent elements of the flow stream, means to sense flow parameters (stress, pressure, strain, flow rate variation) of the flow along the flow path, wherein said flow parameters are sensitive to changes in rheological parameters of complex fluids, wherein said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus. Constitutive parameters of the rheological properties of complex fluids are determined by including computational means to calculate on the basis of said sensed flow parameters by inverse interpolation according to appropriate models of said rheological properties.

Description

Rheometer and rheometric method [0001] This invention relates to a rheometer and a method of determining rheometric parameters of fluids.

BACKGROUND

[0002] Fluids have many different forms and, depending on their nature, exhibit different rheological properties following different rheological relationships, some of which are complex.

It is often desirable to be able to measure the rheological properties of a fluid in order to characterise the fluid. The rheological properties of fluids are sometimes important characteristics by which a product is judged, and which may inform how a process involving the fluid is developing.

[0003] An obvious example is the manufacture of a product such as tomato ketchup. Ketchup has a complex rheological character that is very distinctive and important to the product. It would be useful to be able to measure and monitor in real time the constitutive parameters that establish that character during manufacture of the product, whereby the manufacturing process can be checked for accuracy and completeness. Clearly other processing parameters can also be important, but rheological parameters may be significant indicators of final properties of a fluid, even with products where those rheometric properties are not of paramount or significant importance to the eventual user of the product.

[0004] Another example of a product where rheometric properties are important in the final product, where the properties would be noted and expected by the final user, is emulsion paint.

[0005] Consequently, it would be desirable to have in-line rheometers in the course of a production line that monitored in real time the rheological properties of interest in a product and, perhaps, controlled the manufacturing process accordingly. However, there are many constitutive parameters that vary from fluid to fluid. Newtonian fluids are straightforward and the only parameter of interest is the viscosity, which only varies with temperature, pressure and composition and is a simple function of stress and strain.

[0006] However, few fluids are truly Newtonian and many may be described as different complex fluids. The rheological characterisation of complex fluids in chemical processes and in biotechnology is of importance for fluid control measurement and identification. Fluids containing polymers and biocolloids are typically complex fluids exhibiting non-Newtonian behaviour. It is an object of the present invention to provide an efficient microfluidic rheometer for the characterisation of complex fluids, and an associated method.

[0007] Examples of complex fluids include: (a) Power law fluids For such fluids p = K2', where: * p is the viscosity, * K is the flow consistency index (SI units Pa.s"), * j' is the shear rate or the velocity gradient perpendicular to the plane of shear (SI unit 1) and * n is the f/ow behaviour index (dimensionless).

In this model kand n are described as the constitutive parameters (k1 and k2)of the fluid in question and it may be desirable to know what they are in any given situation.

(b) Carreau fluids For such fluids models with either 3 or 4 parameters are in common usage. In the 3 parameter version, the viscosity at infinite shear rate is taken to be zero, and the viscosity is given by = [.t(l + (Afl2)(-fl'2, II where: * is the zero-shear-rate viscosity, * A is the relaxation time, * n is the power-law exponent.

In this model A, p and n are the constitutive parameters (k1, k2and k3) of the fluid exhibiting this characteristic.

(c) Viscoelastic fluid Viscoelastic materials possess both viscous and elastic characteristics when undergoing strain, and therefore exhibit time dependent stress. The Phan-Thein-Tanner (PTT) constitutive law [1,2] uses the following form for the non-dimensional constitutive equation: Were = 2p1D -ft + We[r VU + (VU)T -U Vt + flD Vt + (D. r)T]) Ill where f is usually defined using one of the following three forms: Linear model: f = 1 + Quadratic model: f = 1 + trace(r) + I [fJfE trace(r)] FsWe 1 Exponential model: f = exp [-trace(r)j, where * r=T-ZR2D, * T is the extra-stress tensor, * D is the rate of deformation tensor, * U is the velocity vector, * ji is the polymeric viscosity, * i2 is the solvent viscosity, * We is a Weissenberg number defined as We = ], where A is a relaxation time and U and L are typical velocity and length scales respectively, * s and are positive constants used to control the viscoelastic properties.

Thus the PTT constitutive law contains 5 parameters: P, A, , which correspond to constitutive parameters (k1, k2, k3, k4and k5.) of the fluid.

[0008] In Bandulasena, Zimmerman, and Sees [3], the authors described a novel methodology for calculating the rheological parameters of dilute aqueous solutions of the power law non-Newtonian fluid, xanthan gum (XG). Previous studies have verified the fidelity of finite element modelling of the Navier Stokes equations for reproducing the velocity fields of XG solutions in a microfluidic T-junction, with experimental observations obtained using micron resolution particle image velocimetry (y -Ply). As the pressure-driven fluid is forced to turn the corner of the T-junction, a range of shear rates, and therefore viscosities, is produced within the flow system. This produces a set up whereby potentially the rheological profile of XG could be established from a single experiment. An inverse method, based on finding the mapping between the statistical moments of the velocity field and the constitutive parameters of the viscosity profile, demonstrate that such a system can be used for the design of an efficient microfluidic rheometer. However, p -Ply technology is expensive and the equipment is bulky.

[0009] The advent of micromachining has enabled researchers to create microfluidic devices out of glassy and plastic materials having dimensions in the range of 1 nm to 1 mm. Microfluidic systems are being increasingly developed for many bioengineering applications such as proteomics, genomics and clinical diagnostics. Microfluidic devices offer advantages over macro scale systems in that small fluid samples can be analysed, and processing times are fast.

The surface to volume ratio for flows at micro scales is large, thus enabling precise control over reaction processes and the transfer of heat and mass.

[0010] Viscometers are designed to identify the stress response of a fluid to a single shear rate [4]. This means that in order to build the general rheological curve over a range of parameter space multiple experiments need to be performed. In order to make efficiencies in terms of cost and time, research was focused on flow regimes that enable a range of shear rates, and hence viscosities, to coexist in a single experiment. An example of such a system is a microfluidic T-junction in which a flow field can be induced by an applied pressure drop, or in the case of electrokinetic flow, by an electric field. By adopting such a geometry, a range of shear rates can be generated within a single experiment as the fluid is forced to turn the corner' of the T-junction. Bandulasena, Zimmerman and Sees [5.6] developed a procedure for assessing the total rheometric response for such a system based on the statistical moments of the velocity field analysed in conjunction with finite element modelling. The velocity field with a microchannel was obtained using micron resolution particle image velocimetry (p -Ply).

Although p. -Ply is a well-established technique for obtaining measurements of the velocity field in a microchannel geometry [7], the equipment is expensive and bulky. However, piezoelectric pressure transducers are capable of measuring the pressure field within microchannels and result in a cost-effective, portable means of obtaining statistics of the flow in the interior of the channel. Zimmerman, Sees and Craven [8] performed numerical simulations of a Carreau flow in a microchannel T-junction and demonstrated the existence of a one-to-one mapping between the first three statistical moments of the end wall pressure profile with the constitutive parameters of the viscosity profile. In order to develop a general procedure for such an analysis, a range of fluids needs to be studied. Thus rheological curves for the power law, shear thinning fluid, xanthan gum, have been obtained from knowledge of the pressure field in such a flow.

[0011] The experimental programme is outlined in section 1 below. The finite element model (the forward problem') is described in section 2. The concept of the inverse problem' is discussed in section 3 and the conclusions are summarised in section 4.

1. Outline of the Experimental Programme [0012] Xanthan gum is a natural polysaccharide of high molecular weight that is produced by the fermentation process. Its exceptional rheological properties make it a very effective stabiliser for water-based systems [9]. Consequently it is used in a broad range of applications, from the food industry to the oil industry. Dilute aqueous solutions of xanthan gum are dominated by shear thinning but possess a negligible elasticity [10,11]. Since the viscosity of a fixed concentration of a solution of xanthan gum follows a power law profile, it is an ideal fluid on which to perform studies of the rheometry of power law, shear thinning non-Newtonian fluids, as complicating effects such as viscoelasticity and extensional viscosity can be neglected.

[0013] The experimental setup is described in detail in [5]. However, for completeness, a summary is given here. The T-junction microchannel is fabricated onto a glass microchip using the processes of photolithography and hydrofluoric acid etching. The external dimensions of the chip were 15 mm X 45 mm X 1.2 mm. The photographic image of the chip is shown in Figure 1.

[0014] The channels of the T-junction extended 5 mm from the junction to the inlets and two outlets. The wet etching fabrication process gave rise to a cross-sectional shape that was in the form of a D'. The channel depth was 65 microns and its maximum width was 260 microns. The chip was deployed within a perspex holder to enable through connections to three external reservoirs of diameter 25 mm. These were connected to the inlet and outlet channels in order to suppress possible pressure imbalances arising from the effects of surface tension [12].

Reservoir diameters were made sufficiently large to negate this effect. The y -Ply system included a Nd:YAg laser (wavelength=532 nm, 100 mJ, 6ns), COD camera, synchronizer, inverted microscope, beam optics, and a fluorescence filter block, connected to a computer with supporting software.

2. The Finite Element Model -The Forward Problem [0015] In this section the equations of motion are presented for the pressure driven flow of a power law fluid within the microchannel 1-junction. The fluid density, p, is assumed to be constant. Since the channel cross-section is D-shaped, it is possible to solve for the fully three-dimensional equations, thus removing the two-dimensional approximation used in [8] for modelling the flow of a Oarreau fluid in a microchannel 1-junction.

2.1 Governing Equations [0016] The effective viscosity for a power law fluid is given by p = K ()7t_1, where K is the flow consistency index, (?) is the velocity gradient perpendicular to the plane of the shear, and n is the flow behaviour index. The parameters K and n are therefore the rheological parameters' (k1 and k2) for such a flow.

[0017] In order to nondimensionalise the equations, reference scales p, L0, u0 and Pb were set for density, length, velocity and viscosity. The length scale was set according to the channel depth. Nondimensional variables for velocity, pressure, shear rate and viscosity are given by: pL0. p =

20, (1) where we define Po = K ()Th_1, whence p = In its nondimensional form the power law for viscosity can be written as (2) For convenience of notation the asterisk is not shown hereafter. Fluid flow is governed by the Navier-Stokes momentum equation a (an. öu1 Re =----i---PI_L+ , (3) Cx', ) OX, (JXJ OX1 Ox, where Re= pU0L0 is the Reynolds number, and the continuity equation (4) OX! 2.2 Boundary Conditions [0018] At the inlet and outlets of the T-junction the tangential components of the velocity field are set to zero: = 0. (5) The pressure gradient across the microchannel was set by imposing a dimensional inlet pressure of 100 Pa together with a reference outlet pressure of 0 Pa.

2.3 Numerical Method [0019] The equations were solved on a Linux workstation using a Galerkin finite element scheme based on a tetrahedral element grid. Computations were performed with a three-dimensional domain. Use was made of asymmetrical properties of the channel system in order to reduce memory storage requirements and computational times. The commercially available software packages Comsol Multiphysics'TM and Matlab'TM were used to perform computations.

Mesh independent solutions were obtained with 31163 elements. Each simulation took around 2 cpu hours on an AMD 64-bit 3GHz processor. However, once the computational pressure fields had been established an associated inverse problem was performed using a table look-up procedure'. Consequently the time required for numerical computations is not a barrier to the adoption of online monitoring systems for determining the rheology of complex fluids.

3. Results and Discussion 3.1 Sensitivity and Factor Selection [0020] In order to determine the preferable positioning of pressure sensors in rheological experiments, it is necessary to identify which regions of the flow exhibit the greatest sensitivity to the flow behaviour index. The differences in the simulated pressure fields for flow behaviour indices n = 0.7 and n = 0.8 are shown in Figure. 2. It can be seen that the maximum pressure difference occurs in a region close to the centre of the junction. Hence a hypothetical pressure sensor is positioned in this region, as shown in Figure 3 (in which the velocity profile of the flow of pressure driven xanthan gum in a microchannel T-junction is shown, as measured using p-Ply). A hypothetical pressure sensor with diameter 100 pm is based in the region displaying maximum sensitivity to the flow behaviour index.

[0021] It should be clear from Figure 2 that there is actually only one region of high sensitivity to flow behaviour index, n, so placing a second sensor may be problematic. Zimmerman, Rees and Craven [8] proposed using the moments of the back wall pressure distribution to infer constitutive parameters of a Carreau fluid in electrokinetic flow. The discrete version of such moments can be captured by the sums and differences of sensors placed non-symmetrically along the channel walls. This idea is analogous with the usage by [13] of the raw sensor data in electrical capacitance tomographic imaging of multiphase flow regimes in pipes. Those authors use mean, fluctuation, symmetric and anti-symmetric combinations of sensor data to infer flow regime. Constructing analogous measures for a two parameter inverse methodology could take as few as two pressure sensors, if both are sensitive to constitutive parameter variation.

[0022] To find a positioning for a second pressure sensor that is sensitive to variation of the flow consistency index, K, an analogous figure to Figure 2 may be produced, but with two dissimilar values of K (not shown). The plot of sensitivity to K showed differences on the order of grid resolution noise (1 ft4). This was unexpected in that it is clear that pressure depends on viscosity, but not necessarily viscosity differences, at steady state. Since K is essentially a scaling factor, it is possible that with two different K values, K1 and K2, that the difference pressure field pi -P2, is insensitive to KI -1(2. For the sake of argument, suppose Re 0 and there are two velocity fields t41 and uf2 then, by taking the divergences of equation (3), a Lighthill equation for the pressure results: V2p1 =___H (6) ax, ox - 72p, = , (7) 0X1 CX (3u. öii, where c.. R1 -+----i5X (X1 It follows that V2 (p1 _p2)=---(4 2)) (8) ox, ox1 [0023] Bandulasena, Zimmerman and Rees [5,14] and Bandalusena, Zimmerman and Rees [6] showed for both xanthan gum and polyethylene oxide solutions that the standard deviation of the velocity is invariant with K, but the mean velocity does depend on K. For this to be true, it follows that only the magnitude of the velocity field depends on K, and thus velocity fields are self-similar with K. This is inherent in equation (3). Therefore it can be concluded that the right-hand side of equation (8) is identically zero, and thus V2(p1-p1)=0. (9) However, since bothp1 andp, have identical boundary conditions, it follows thatp1-p, have homogeneous boundary conditions, and so an unique solution to equation (9) is p1-p,=O. (10) With this analysis that the pressure field is insensitive to variation of K, it is clear that only n may be found uniquely from pressure data. Figure 4 plots the main pressure averaged over the hypothetical sensor placed as in Figure 3, against the flow behaviour index, n. The line has best fit by linear regression as P = -15.963n + 49.54 with correlation coefficient -0.9999. There is a clear linear relationship between mean pressure P and flow consistency index at any K value.

[0024] Since pressure sensors can only determine half the inverse problem, it has been proposed that a second global factor be considered, which should be sensitive to the flow consistency index K. The outflow volumetric flow rate Q, defined by Q=fh,u1dS, (11) where S is the outflow face, h1 is the unit normal to 5, and dS is the area element, should be sensitive to both K and n, since in previous studies, the mean velocity measured over a different area by p -Ply estimation, was found to be so. Clearly, there is analogous information content for the two quantities. Figure 5 shows thatlog(Q) is dependent on both the flow consistency index K and the flow behaviour index n. Since there are many techniques for estimating volumetric flow rate Q it would appear to be a good candidate factor for an inverse methodology.

3.2 Inverse Problem [0025] In many engineering applications quantities that need to be determined are often different from those that can be measured directly. For example, K and n cannot be measured directly. Thus a procedure is required that enables inference of their values from quantities that are capable of measurement, in this case from the pressure field. In order to do this, it is necessary to solve an inverse problem. The set of values that are to be reconstructed is referred to as the image, f. The quantities that are measurable form a dataset, d. The forward problem involves the mapping f -* d. In the present case, the forward problem entails numerical solution of the governing equations of motion. The mapping d -f is the corresponding inverse problem. An inverse problem is described as well posed [15] if: 1. The solution exists for any data, d, in the dataset.

2. The solution is unique in the image space.

3. The inverse mapping, d -> j, is continuous.

[0026] Since the simulation of the rheometric flow for xantham gum has already been validated, it is necessary only to find a prescription for the calculation of the inverse problem for the specified data d = (P,Q)for the image of f = (n,K). Zimmerman, Rees and Craven [8] do this for electrokinetic flow by computing the map f -* d as a lookup table, and then demonstrate the map is globally invertible by a graphical technique. Zimmerman [16] uses optimisation techniques for a higher dimensional problem, dim(d) = dim(f) = 3. For such higher dimensional problems, it is difficult to show global invertibility.

[0027] In this problem, the inverse methodology can be staged, much like back substitution in matrix equation solutions for triangular systems. This is because the Jacobean J of the map f -* d is triangular: DP 6P 3K (12) 0Q 3Q L'121 *J2H Dii 3K ap since-=O. As J can be estimated from the forward problem, invertibility of the system is guaranteed if and only if detJ=JJ=2!=O (13) DnDK 3P 30 and therefore -!=O and±!=O. 3K

[0028] Both of these conditions can be determined by inspection of Figures 4 and 5. There are no regions of zero gradient in the line P = P(n), nor in the family of n-isocurves Q = Q(n, K).

The inversion algorithm does not, therefore, require any imagination at all. The two steps involved are: (i) invert the line P = P(n) to find n = (U) using interpolation within Figure 5, read offK = Kr (Q), as these curves are all single valued.

This algorithm can readily be coded by interpolation functions for both Figures 4 and 5.

4. Conclusions

[0029] In this study, the optimal positioning of inexpensive piezo-electric sensors was considered on the surface of a microfluidic device for which rheology has been inferred from velocimetry already. Furthermore, a sufficient set of factors is selected related to pressure measurements to infer the constitutive parameters of a power law fluid, xanthan gum, for which a validated model is derived from velocimetry studies. The analysis clarified that the pressure field is insensitive to the flow consistency index K, but that a pressure sensor placed where the flow turns the corner' of the T-junction is sufficiently sensitive to determine the flow behaviour index n. As Kserves as a scaling factor, it can be shown that only global velocity measures -average velocities -can be used as factors in an inverse methodology. The volumetric flow rate was sufficiently sensitive to both K and n to design an inverse methodology that can be staged, much like back substitution in the solution of matrix equations, to identify the two power law parameters from an average pressure measurement and the volumetric flow rate measurement from a single microfluidic experiment.

[0030] However, the foregoing does not encompass the additional complexity that is associated with the determination of the constitutive parameters ofviscoelasticfluids. Inverse methods are well established protocols. The most simple method is to build a look-up table' by conducting a range of experiments varying thek1 and measuring the Q1. This calibration allows inversion by interpolation within the table. The common method now used is to conduct computational fluid dynamics simulations to predict the Q1 from fixed k, building a discrete version of the forward map or look up table.

1. Generically, a constitutive model relates effective viscous response to applied shear rates through model parameters called k1 here, where i = 1,2, ... N. 2. These k1 are usually found from the set of steady-state experiments which have either a single shear rate or a dominant shear rate by measuring the viscous response and fitting that response curve with varied shear rates.

[0031] The approach is to solve an inverse problem by posing an information-rich experiment which expresses a range of shear rates, such as when the flow turns the corner in a T-junction.

In such a flow, the forward problem is: 1. Fix k1 constitutive parameters.

2. Conduct experiments or calculate model.

3. Predict measure factor related to viscous response Q1.

The inverse method which permits rheometry runs this process in reverse'.

1. MeasureQ1.

2. Use inverse map.

3. Estimate Ic constitutive parameters.

[0032] It had been speculated that rheometry could be conducted with the Q1 being average steady flow rates through various planes or steady pressures registered at different sensors downstream. A constitutive relation comprising N parameters requires a minimum of N measured factors in order to solve the inverse problem. For example, the power law model described above needs at least 2. In this model K and n are k1 and k2. To solve the inverse problem a minimum of two dynamical measures, Q1 and Q2, are needed.

[0033] However, with Carreau fluids, A, p and n are k1, k2 and k3. To solve the inverse problem, a minimum of three dynamical measures Q1, Q2 and Q3 are needed.

[0034] For viscoelastic fluids, the PTT constitutive law contains five parameters: lit, Y2 A, 8, which correspond to k1 to k5. To solve the inverse problem for such fluids, a minimum of five dynamical measures Q1 to Q5 are needed. Numerical simulations of the PTT viscoelastic model in this geometry show a strong parametric influence on the stress tensor profile.

[0035] The present invention makes use of two observations. The first is that the constitutive parameters of the rheology of viscoelastic fluids are, of course, time-dependent and secondly that measuring steady state quantities is often problematic, for strain gauges at least, but also other pressure gauges, since the gauges themselves tend to be, or can be, susceptible to drift.

It is far from straighiforward to obtain a sensitive, invertible set of measurements from piezo-electric sensors since they do not respond uniquely at steady state with a pressure reading due to leakage of charge.

[0036] It is an object of the present invention to provide a rheometer capable of determining the rheological properties of complex fluids including viscoelastic fluids. In particular, to determine such parameters inline and in real time in connection with a process involving the use or formation of the fluid being analysed for its rheological properties. The invention includes a method of determining said constitutive parameters.

[0037] Craven, Rees and Zimmerman [19] further developed pressure sensor positioning in an electrokinetic microrheometer device. The use of computational modeling for the design of microfluidic devices is widespread. Owing to their laminar nature and precise controllability, microchannel flows lend themselves well to numerical simulation. Models for electrokinetic and pressure driven flows of Newtonian liquids in microchannels are well known and have been experimentally validated [23,33,34]. Results from accurate and reliable computational models can be used to determine quantities of interest that are not directly measurable, such as the constitutive parameters of the viscosity model, by solving an inverse problem [24,37,32].

[0038] Zimmerman, Rees and Craven [8] demonstrated in a two-dimensional numerical study that such an inverse problem was globally unique and solvable for a wide range of constitutive parameters of the more complicated Carreau viscosity model, using two-dimensional simulations of electro-osmotically driven flows in a microchannel T-junction. The mean and variance of the back wall pressure profile were found to map uniquely to the time relaxation constant and the exponential index. These concepts are expanded in Craven, Rees and Zimmerman [42] with a view to using cheap piezo electric pressure transducers for measuring the pressure field on microchannel surfaces. Bandulasena [3,5] used micron resolution particle image velocimetry (p-Ply) (as described above) to validate numerical simulations of the pressure-driven flow of power law fluids in a microchannel T-junction. This coupled experimental-computational approach was used successfully to infer the constitutive parameters of a power law fluid from the simulated velocity and pressure fields using inverse methods.

However, the equipment is expensive, not portable, and requires the microchannel to be optically transparent. If a device could use pressure only to infer constitutive parameters then it would be cheaper, more portable and could be fabricated from opaque materials if desired.

[0039] Since the viscosity of non-Newtonian fluids is a function of the shear rate, either multiple experiments at different shear rates need to be performed in order to obtain the viscous parameters [27,4] or a range of shear rates needs to be set up within a single experiment [28,36,20]. A microchip-based method for extracting viscous parameters of polymer solutions was developed by Lee and Tripathi [28]. Fluorescent intensities were used to obtain direct measurements of the channel dilution ratio. Since the pressures of the chip's reservoirs were known they were able to use the momentum balance equations to calculate the viscosities for each channel of the chip. Srivastava and Burns [36] designed a self-calibrating microfluidic capillary viscometer for obtaining power law parameters of non-Newtonian fluids. As the fluid flowed into a long capillary tube a limited range of shear rates was produced. Degre [20] used the visualization technique of particle image velocimetry to extract flow stress and shear rates from measured velocity profiles. The constitutive relations were obtained from stress and strain relationships.

[0040] The approach in Craven [19] was to use a T-junction geometry to set up a flow system exhibiting many shear rates simultaneously, hence the viscous information of the fluid viscosity function was present in a single flow [6]. This information is then extracted indirectly by measuring some of the flow characteristics, such as the pressure at different positions in the channel [8], or the average velocity vector in a given region of the flow [3]. A fully three-dimensional model of electro-osmotically driven flow of a Carreau fluid is described below. The microchannel has a curved depth profile because of the etching process used in its manufacture, causing the flow field to differ from the idealized two-dimensional case considered in [8]. The electric double layer movement at the channel walls is also present on the channel floor and ceiling, acting to constrain the fluid more tightly than in the two-dimensional case [42].

The resulting pressure profiles on the walls, floor and ceiling provide much information and their use for inferring the Carreau parameters is demonstrated. A parametric study solved the model for a wide range of values of the Carreau parameters A and n and suggested optimal positioning for piezo electric pressure transducers.

5. Electrokinetic flow in a microchannel T-junction [0041] Figure 6 shows the channel cross-section and T-junction layout considered in this work. Microchannels were etched into the chip material (usually glass or PDMS) using photolithography. When the channel had been produced, a second chip is bonded to the first in an oven to produce a ceiling for the microchannel. The channel was then completely sealed to the outside world except at the inlet and outlet openings (see also Figure 1). The etching process produced channels with curved sides, which for simplicity have been modeled as quarter circles. In reality, the etching process may produce profiles that exhibit a degree of non-uniformity along the length of a channel. However, it was assumed that the channel profile was uniform along its entire length. The microchannel considered in [19] had a depth of 80 pm and a width of 200 pm. Typical microchips may have inlet and outlet channel sections of around 10 mm, in which the flow is uniform and can therefore be approximated by analytical solutions if required.

6. Electro-osmotic flow [0042] Flows in microchannels can be pressure-driven or may be induced through electro-osmosis, where thin, charged layers of fluid close to the microchannel walls experience a driving force in the presence of an applied electric field. In the latter case, an electric double layer (EDL) forms when a fluid containing free ions is in contact with a charged surface. Trapped charges in the wall material are balanced in the fluid region adjacent to the wall by the redistribution of ions, resulting in the formation of an oppositely charged layer of fluid next to the wall. Away from the charged wall, the free ions balance out one another, so that the fluid has zero net charge. If a potential difference is applied along the channel, the charged fluid in the EDL experiences an electric body force, and begins to move along the channel. The fluid outside the EDL is brought into motion through viscous forces, and is dragged along by the motion of the fluid in the EDL resulting in an electro-osmotic flow. The EDL is very thin, with a typical width of around 10 nm for an ion concentration of 0.01 mol m3 [25, 42].

[0043] Electro-osmotic flows can be controlled precisely through the potential difference at the electrodes, while the flat plug flow profile that results allows species to be transported without the lateral mixing effects of pressure-driven flows [26]. In an electro-osmotic flow, the shear rate outside the EDL is zero across the channel, and therefore the fluid viscosity remains constant in straight channel sections. In contrast, for pressure driven flows, velocity gradients across the width of the channel produce a range of shear rates, and therefore the viscosity of such a fluid changes across the width of the channel. At microscales, inertial effects are negligible, and fluid flow is laminar and deterministic in nature. A typical electro-osmotic flow velocity for an aqueous buffer is around 1 mm s [29], which corresponds to a Reynolds number of around Re = io-in a microchannel with a characteristic length of 100 pm. In practice, the strength of the applied electric field is set to achieve the desired flow velocity. The strength of the tield required to achieve a certain velocity depends on the wall zeta potential, which is linked to the concentration of free ions (measured by the pH) of the solution. Such flows lend themselves particularly well to numerical simulation owing to their laminar nature.

7. Non-Newtonian viscosity model [0044] Polymer solutions exhibit non-Newtonian flow behaviour, where the response to shear and strain forces is nonlinear. For example, long chain polymer molecule solutions, such as xanthan gum, offer a smaller resistance to deformation as the rate of shearing is increased.

The apparent viscosity of these so-called shear-thinning liquids varies with the shear rate, leading to a generalized Newtonian model for the stress tensor. In more concentrated polymer solutions, viscoelastic behaviour is also common, where fluid elements react in an elastic fashion to the strain. Several rheological models have been employed to model shear-thinning polymeric liquids with considerable success. The Carreau viscosity model is an empirical formula that relates the apparent viscosity of a fluid element to the shear rate experienced by that element. The model falls into the generalized Newtonian class of models because it assumes a symmetric, Newtonian stress tensor, with only the scalar viscosity itself varying as a function of the shear rate. More complicated models include strain tensors whose components must satisfy additional conservation equations, and can represent both shear rate dependent and visco-elastic fluid behaviour.

[0045] The Carreau viscosity [35] of a fluid is governed by the shear rate and four model parameters; the viscosity at zero shear rate o, viscosity at infinite shear rate p, the time relaxation constant A, and the exponential index n. It is given by the expression = + ( -)[1 + (Afl2]"'2 (14) where j is the shear rate. When the shear rate is zero, the exponential term is equal to one and therefore = i'; the Carreau viscosity is equal to the viscosity at zero shear rate p0.

When the shear rate is large, the term [i+(y}1 tends to zero (with nd), and the viscosity approaches p, the viscosity at infinite shear rate. In practice, for shear-thinning fluids, the zero shear viscosities of polymer solutions obeying the Carreau model are very small, of the order of 1Op. Its main purpose in the model is to avoid numerical problems that can arise as a result of a near-zero viscosity value in Cauchy's moment equation. The model parameters p0 and p, control the minimum and maximum viscosities attained by the fluid. In order to model a shear thinning fluid, we set jz > jç. The other two parameters A and n determine the shape of the viscosity curve over a range of shear rates. In a steady microchannel flow, a finite range of shear rates is present. How the Carreau viscosity model behaves across the shear-thinning regime (i.e. not on the upper/lower limiting shear rates) is of interest.

[0046] For a shear thinning fluid, the viscosity curve falls away from p0 at ? = 0 and decays exponentially, tending to the lower limit c at high values of A. The steepness of the viscosity curve is controlled by the exponential index n, with smaller values of n giving steeper viscosity curves. The value of A serves to scale the viscosity curve along the shear rate axis.

For small values of A the viscosity curve is stretched out along the shear rate axis, resulting in a more gentle drop in viscosity over the shear range. For large A the viscosity curve is compressed along the shear rate axis, leading to a sharp drop in the viscosity curve across the shear range. When A -* 0 or n -* 1, the Carreau viscosity becomes constant, with a Newtonian viscosity Po In combination, A and n determine the shape of the viscosity curve while the range of shear rates present in the flow determine the range of viscosities. The viscous information present in a flow can be used to determine the Carreau parameters through solving the inverse problem. It is therefore important that the shape of the viscosity curve displays sensitivity to changes in A and n across the range of values of interest. It is shown below that, given a known shear rate range, there are upper and lower limits for the values of A and n beyond which the Carreau viscosity curve becomes insensitive to changes in the Carreau parameters. The parameter ranges chosen over which simulations were performed ensured that the viscosity curve was sensitive to A and n across the whole range. For a more detailed analysis of the behavior of the Carreau viscosity function, see Zimmerman, Rees and Craven [8].

8. Computational model [0047] In this section a mathematical model is described for the electro-osmotic flow of non-Newtonian fluids in microchannels, its associated boundary conditions, and the finite element methods used to solve the flow in the T-junction.

Model equations [0048] The electro-osmotic flow of a fluid driven by an applied electric field is governed by the conservation equations for applied electric potential, fluid momentum and mass. Temperature variation in the flow is assumed to have a negligible effect on the viscosity, density, and electrical conductivity of the fluid. It is also assumed that there is zero net charge on the fluid, so that electrophoresis of charged particles does not occur, and that the flow is steady state.

[0049] The equations are solved in non-dimensional form using the reference scales, po, p, E, L, and Ufor viscosity, density, electric field, length and velocity respectively. Density is taken as constant, the Carreau zero shear viscosity is used as the reference viscosity, and pressure is scaled according to viscous forces owing to the laminar nature of microchannel flows. The electric potential is scaled with the applied electric field strength to produce non-dimensional potential gradients of the order of unity in the channel. The non-dimensional variables are * X; , U; * pL0, X. n--,p = = U (15) where x7, i4, p and p are the non-dimensional length, velocity, pressure and applied electric potential respectively. In practice, the slip velocity U is set through varying the strength of the applied electric field E. The two quantities are related by the Helmholtz-Smoluchowski relation (16) where e and (are the electrical permittivity of the fluid and the zeta potential at the edge of the electric double layer. The solvent viscosity, p, is used in this expression for the velocity at the wall because the electric double layer is sufficiently thin that the macromolecules that give rise to the higher zero shear viscosity of the non-Newtonian fluid are not present in the electric double layer owing to their radius of gyration being of comparable size to the layer itself. This is known as the depletion effect, and is a valid assumption so long as the molecules do not have a tendency to attach to the wall material [40].

[0050] Dimensional analysis shows that the flow is characterized by the four dimensionless parameters: Re = = , A* = = (17) * * where Re, p, A and n are the non-dimensional Reynolds number, Carreau viscosity ratio, time shear relaxation parameter and exponential index respectively. Together, these four parameters determine the flow profile. Hereafter only non-dimensional variables are referred to, unless otherwise indicated, and the * notation is dropped, as above, for clarity. The set of non-dimensionalized governing equations for cp, u and p are VVq =0, (18) Re (u.Vu)-V.a =0, (19) Vu =0, (20) where a denotes the total stress tensor a(p,u)=-pI+2 i(v) £(u), (21) where I is the identity matrix, 11(32) is the Carreau shear rate-dependent viscosity, and E(u) is the rate of strain tensor, which is defined as E(u)=_Vu+(Vu)T. (22) The generalised shear rate [31] is the square root of the contraction of the rate of strain tensor with itself, and is an invariant of the rate of strain tensor. It is given by 32 = ,,/(u): s(u). (23) Finally, the Carreau viscosity is given in non-dimensional form by the expression 11(32) = Ito. + (Mo -Mco)[1 + (Ai2)2]o12 (24) Boundary conditions [0051] Electro-osmotic flow arises as a result of the movement of a nanometer-scale charged layer of fluid at the microchannel walls, known as the electric double layer. The computational effort required to resolve the layer directly is substantial [42]. It is common practice not to resolve the layer, instead replacing it with a locally one-dimensional approximation to the flow velocity at the edge of the layer [22,29,8]. The fluid velocity at the edge of the electric double layer is related to the electric field through the Helmholtz-Smoluchowski condition Eq. (16), which is given in the non-dimensional form by u=-Vp. (25) [0052] Under the assumptions of constant electric permittivity and zero charge on the fluid away from the walls, the applied electric potential field is independent of the velocity and pressure fields. This allows the two fields to be solved sequentially, requiring less memory space and processing time than if the equations were solved simultaneously. An electric field in the T-junction microchannel can be induced by imposing potentials at the inlet and outlet boundaries. The inlet and outlet potentials are: p 4 at inlet, (pO at outlet. (26) The inlet potential is set higher than the outlet potential by the approximate non-dimensional length of the shortest path between the inlet and the outlet; this results in the potential gradient in the inlet channel being approximately equal to -1, so that the electro-osmotic velocity is unity in the inlet channel section. The fluid pressure is set equal to zero at the inlet and outlet boundaries to achieve a zero pressure difference along the channel, so that the flow is driven solely by electro-osmosis, i.e. p=O. It is assumed that the rniciochannel wall mateiial is electrically insulating, so that at the walls there is a zero potential gradient across the boundary, Vpn=O. (27) Numerical methods [0053] The equations were discretized using the Galerkin finite element method. The microchannel 1-junction has a lineal symmetry down its center, and this was exploited to ieduce the numbei of grid elements needed in the finite element mesh. Symmetry boundaiy conditions weie used along the channel center, i.e. 7q n = 0, ufl = 0. The inlet and outlet channel sections extend two channel heights from the junction. The actual inlet and outlet channel sections are much longer, but this shortening in the model is necessary to keep the mesh to a manageable size. When inlet and outlet channels are shod, the boundary conditions at the inlet and outlet can influence the solution within the T-junction domain. In practice, a balance must be found between the number of mesh elements used and the accuracy of the approximate inlet and outlet boundary conditions. This was achieved through preliminary simulations and a trial and error approach.

[0054] The finite element mesh used covers the T-junction half-domain, split down the plane of symmetry of the junction. Half of the full T-junction is modeled to take advantage of the line of symmetly along the channel centel line, reducing the computational effort required to solve the model. The inlet and outlet channels have a length of 1.5L, where L is the channel height.

The mesh is made up of 40,010 tetrahedral elements, with the element sizes in the region of the sharp corner specified as 1% of the channel depth. The laigest flow gradients occur at the right angled coinei of the T-junction, with the higher mesh concentiation in this aiea needed to resolve the larger gradients.

[0055] The electric field and the velocity and pressure fields were solved sequentially, since the electric field is independent of the velocity and pressure fields under the model assumptions of constant electric permittivity and zero charge on the fluid. The electiic potential p was discietized using Lagrange quadratic elements, leading to a system of lineai equations with 59,115 degrees of freedom (DOF). The velocity and pressure fields were discretized using Lagrange quadratic and Lagrange linear elements respectively, resulting in a linear system with 185,494 DOF.

[0056] The electro-osmotic flow of a Carreau non-Newtonian fluid is a highly non-linear problem, which produces a system of linear equations which becomes less well-posed as A is increased and n is decreased. This increasing ill-posedness causes convergence problems for memory efficient iterative solvers such as GMRES. At higher values of A (around A>3), the memory requirements of the preconditioner algorithm become as large as the memory required to solve the system using a direct solver. This scenario compelled the use of a direct solver based on the multi frontal method and LU factorisation (SPOOLES 2.21). Computations were carried out using the Comsol Multiphysics® finite element PDE engine controlled using custom scripts in MATLAB® [41]. The hardware used was a 3 GHz machine running the Linux operating system. Peak memory usage was around 2.5 GB; memory usage increased as the condition number of the linear system worsened at higher values of A and n.

Operating parameter ranges [0057] The flow solution is dependent on four non-dimensional quantities introduced in Eq.(17); the Reynolds number Re, jt, and the non-dimensional Carreau constitutive parameters A and n. The constitutive parameters of the fluid are varied while assuming constant values for Re and p. [0058] While the zeta potential that governs the electro-osmotic slip velocity of a fluid is not known a prior/for different mixtures of polymers and macromolecules, in practice the desired slip velocity can be attained by varying the strength of the applied electric field in the microchannel. Experimental measurement of the slip velocity would require a dedicated component in the microchannel network situated before the T-junction. This however is a separate problem. In order to keep the dimensionality of the parameter space to two, we assume that Re and p. are fixed and small, in accordance with electro-osmotic and non-Newtonian flows in the literature: Re = iO, = io-(28) [0059] A characteristic of polymer solutions is their large zero shear viscosity, which is usually several orders of magnitude larger than the solvent viscosity. For example, human blood [21], mayonnaise and dilute polyethelene glycol solutions [4] and dilute linear polystyrene solutions [39] all satisfy the Carreau model and have viscosity ratios in the region 105«=p:«=102. The ranges of the Carreau parameters were chosen to include a broad range of fluids. The maximum value forA was chosen at the limit of model convergence. Convergence occurred for A=6 and n<O.5, so A=5 was taken as the upper limit for that parametric study. The value of n for various fluids always falls in the range 0cn<1, with lower values corresponding to a sharper drop in viscosity under shearing.

With the above considerations in mind, the Carreau parameters were varied over the ranges SPOOLES is a library for solving sparse real and complex linear systems of equations, written in the C language, and is available from http://www.netlib.org/linalg/spooles/.

0.1<X<5, (29) O.1<n<0.9. (30) The A range was resolved using two different step lengths. A step size of dA=0.05 was used between 0.1 «=A«=1 to examine the behaviour in this region in detail, and a larger step size of dA=1 was used for the range 1 «=A«=5 in order to cover a broader A-range whilst keeping computational requirements to a manageable level. A step size of dn=0.1 was used for all n-values. The model was run for all combinations of the parameter values expressed in the vectors =[0.10, 0.15,0.20 0.85,0.90,0.95,1.0,2.0,3.0,4.0,5.0], (31) n[0.9, 0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1]. (32) In total, 208 separate simulations were performed, requiring a computation time of approximately 400 hours.

Solution strategy [0060] The nonlinear nature of the model equations combined with the large mesh sizes used meant that for each distinct pair of Carreau parameter values, solving the model takes several hours. As the shear-thinning effect becomes more pronounced at high values of A, the number of iterations required by the Newton solver increases, and the closeness of the initial guess to the true solution starts to have a significant effect on the computation time. In extreme cases, close to the limit of model convergence, the solution can fail to converge at all if the starting guess is too far from the actual solution. It is therefore important when performing a large number of simulations to ensure that all available information is used in order to speed up computations. As an example, the time taken to solve the model for the parameter values A=3, n=0.5 was around 180 minutes when an initial solution vector of zeros was used. Using the solution at A=3, n=0.6 as an initial guess, the solution time is cut in half to 90 minutes.

[0061] Since the aim is to collect solution data at equally spaced points in parameter space, it makes sense to use the closest available already computed solution as the initial guess for the next, more non-Newtonian parameter point. By choosing to find solutions for n-values decreasing from n=1.0, which corresponds to Newtonian flow (regardless of the value of A), the previously computed solution can be used as an initial condition for the current solution. Each time the value of A is changed, the value of n is stepped starting from n1.0 again, so that the stored Newtonian solution can be used as a good initial guess to the actual solution.

[0062] Another important factor in the design of a solution strategy for a large parametric problem is the amount of available memory and data storage. The finite element node solution data for each pair of Carreau parameters is quite large at around 3 MB, and so it is impractical to store many solutions in working memory while the model is being solved. Instead, past solutions can be saved to storage for later retrieval and analysis. As the principal interest here is in the pressure profile on the walls of the T-junction microchannel, this information was extracted from the model solution at each step of the process.

Pressure data collection [0063] Zimmerman et al. [8] showed that the pressure profile on the end wall of a two-dimensional 1-junction is sensitive to the values of the Carreau parameters A and n. In order to test whether this hypothesis holds in three dimensions! it is necessary to extract the pressure profile from all of the inner surfaces of the T-junction microchannel. The extraction is complicated slightly by the curved shape of the channel walls, which necessitate the use of polar coordinates in order to obtain the pressure profile as a flat' function parameterized by two spatial variables. To unravel the curved walls of the T-junction, one method is to create a geometry "net" on which the wall pressure profiles can be plotted. The curved wall surface is a quarter-circle with radius L, as shown in Figure 7. Here, the pressure profile on the curved walls is extracted by referring to points on the wall using cylindrical polar coordinates. The wall pressure profile can be unravelled onto a flat "net" using the transformation x=LG, where x is the distance along the wall surface from the channel floor (from point A towards point B). Points on the wall with equal separation are traced out by varying 0, so that a small change in the wall angle corresponds to a small change along the wall surface; dS=L dx. Using this method the pressure can be sampled at many points on the wall and plotted on a geometry "net", as shown in Figure 10.

[0064] Owing to the irregular shape of the walls near the corners, where the inlet and outlet channel sections intersect in a curved edge, the corner was taken at the channel ceiling as the cut-off point for pressure sampling. The pressure data on the lower parts of the curved walls in the corner region (seen in Figure 8 as the square area containing the diagonal edge where the walls meet) were ignored to simplify the task of pressure sampling. Figure 8 is a top-down (xy) view showing isosurfaces of electric potential. Surfaces of equal potential appear as lines near the inlet and outlet boundaries as they are near vertical. At the junction, the curved profile of the channel walls result in the potential field having a small z-component, so that the electric field points upwards at the corner. The wall pressure points obtained from each flow profile are stored in matrices to facilitate data analysis and plotting. For example, the curved inlet wall has a pressure matrix with 23 rows and 30 columns, representing an area of 150 pm (the length of the inlet channel) by 115 pm (the length of the curved wall). The individual matrices for the curved walls, ceiling and floor sections were then assembled to produce the 1-junction geometry net seen in Figure 10. With the pressure data stored in a matrix format, post processing and analysis is made considerably easier.

9. Results [0065] Given the parametric nature of this study, a large number of flow profiles were computed. As interest resided primarily in how the pressure profile changes with different A and n values, a typical flow profile at A=3, n=0.5 was selected, which serves to illustrate the features of the flow including the velocity, viscosity and electric field profiles. These values are within the typical range of those for non-Newtonian flows, e.g. for blood, A3.313, n0.3568 [21].

Typical flow profile [0066] The applied electric potential distribution in the T-junction is shown in Figure 8. The potential falls from a value of 4 at the inlet boundary to zero at the outlet boundary, resulting in a potential gradient of vp = -1 in the inlet channel section. The potential gradient falls to around = -% in the outlet channel section as the channel volume doubles after the junction. While the isosurfaces are parallel to the vertical near the inlet and outlet boundaries, in the vicinity of the corner they are curved and have a small vertical component, indicating that the electric field exerts a body force on the electric double layer in the z-direction.

[0067] The z-component of the applied electric field is significant near the corner of the T-junction, and in the centre of the channel (along the symmetry boundary) near the end wall.

The z-cornponent is largest in magnitude at the walls immediately upstream and downstream of the corner, where the vertical force exerted by the field on the electric double layer is of the same order of magnitude as the force in the streamwise direction in the inlet channel. The EDL z-velocity is positive upstream of the corner, and negative downstream with a magnitude of 1.5 units. Fluid elements rise and fall as they turn the corner, an effect caused by the curved channel cross-section and the curved edge that results when the T-junction is formed. This significant vertical flow is a marked departure from two-dimensional simulations [8], and produces a stronger pressure on the channel floor and ceiling than on the side walls, as discussed below.

[0068] The familiar plug flow velocity field that is characteristic of electro-osmosis is observed in the inlet and outlet channels. The volumetric flow rate in the outlet channel is half of that in the inlet channel as the flow splits in two at the junction, resulting in the velocity in the outlet channel being approximately half of that in the inlet channel. There is a stagnation point at the center of the channel on the end wall, which coincides with a point of zero electric field (zero potential gradient Vp = 0) so that there is zero body force on the EDL at this point.

[0069] As witnessed in previous numerical studies [22, 29, 8], a numerical singularity occurs in the electric field at the channel corners. The magnitude of the potential gradient in the vicinity of the curved corner edge is limited only by grid density at the corner. The fixed grid used here resulted in maximum predicted velocities at the corner of about 10 units. Numerical singularities in the electric field appeared to exist, not just at the point where the corner edge joins the channel ceiling, but along the whole top part of the edge. In other words, a line of electric field singularities exist all along the corner edge, although the largest magnitude occurs at the ceiling. It was shown by doubling the mesh density at the corner from 1% to 0.5% of the channel height that the corner singularity is localized, so that the increase in mesh resolution did not increase the magnitude of the electric field outside 0.2 channel heights of the corner edge.

This numerical phenomenon has been studied in depth in [42].

[0070] The viscosity profile (isosurfaces) of a Carreau fluid with constitutive parameters A3, n=0.5 is shown in Figure 9. The viscosity falls from the zero shear viscosity p = 1 at the inlet as shearing occurs in the corner region. Many shear rates are present as the fluid is dragged around the corner by the electro-osmotic flow at the channel walls, resulting in an information-rich viscosity profile. The smallest viscosity, which corresponds to the largest shear rate, occurs at the join between the corner edge and the channel ceiling. Velocity vectors show the electro-osmotic plug flow profile in the inlet and outlet channels and a stagnation point at the center of the end wall of the T-junction. In the inlet and outlet channels, there is a zero lateral velocity gradient across the channel owing to the plug flow profile of electro-osmotic flow. The viscosity is therefore constant, taking the value p = po = 1.0. As fluid turns the corner, a range of shear rates are experienced by the fluid elements as the EDL drags them around the corner. This results in a complex viscosity profile, which is illustrated using surfaces of constant viscosity in the figure. The maximum shear rates occur at the corner of the junction, where the velocity rises sharply. As the fluid is shear-thinning, its viscosity falls to its minimum value at the corner where the shear rate is greatest.

[0071] The presence of a range of viscosities in a single tlow implies that the viscosity function has been "sampled" over a range of shear rates, so that the information contained within the flow profile can be used to infer the values of the Carreau parameters A and vi. This information is extracted by measuring the pressure profile on the T-junction walls, as described below.

Pressure field

[0072] The pressure field on the walls, floor and ceiling of the T-junction is shown for A=3, n=0.5 in Figure 10. The pressure is displayed on the geometry "net', including the channel ceiling, walls and floor. The top image shows contours of absolute pressure on the ceiling (region A) and inlet side wall of the microchannel (region B), while the bottom image shows pressure contours on the outlet inside wall (region C), channel floor (region 0) and back wall (region E). The T-junction was unraveled by taking advantage of the circular channel profile, so that curved walls can be described using polar coordinates. The pressure rises from the inlet pressure as fluid approaches the junction, resulting in an area of higher pressure in the inlet channel. As the fluid turns the corner and diverges, the pressure drops resulting in an area of fluid with lower pressure than the outlet. As the fluid approaches the outlet, the pressure balances to the outlet value. The pressure rises and falls sharply in the vicinity of the channel corner owing to the electro-osmotic forces dragging the fluid around the corner.

[0073] The non-dimensional pressure differences may be converted into physical pressure differences by inverting the original scaling from Eq.(15): p=Up* lOpp* (33) where p* is the non-dimensional pressure difference, and p0 is the zero-shear viscosity of the fluid. There is a direct relationship between the zero-shear viscosity of the fluid and the magnitude of the pressures on the microchannel walls. Typical values of the zero-shear viscosity range from 1 0 Pa s for blood, to 102 Pa s and higher for solutions of polymers such as linear polystyrene. These correspond to pressure differences in the microchannel in the range 1.p<pc1000.p* Pa.

[0074] Pressure variations along the channel walls are of the order of 2 non-dimensional units, and change with the values of the Carreau relaxation time A. As the value of A increases, the magnitude of the pressure variation on the channel walls decreases, with the pressure difference around 1 pressure unit observed forA5, n=0.5 compared with pressure differences of2 pressure units forA=1, n=0.5. These magnitudes imply that the physical pressure variation on the microchannel walls falls in the range 1-1 000 Pa, depending on the zero-shear viscosity of the particular fluid.

[0075] Very large pressures are observed along the corner edge boundary, which appears to be a line of mesh-dependent pressure singularities. The largest occurs at the point where the edge joins the channel ceiling, where the magnitude is 800 units. There is a jump in the pressure from much larger than the inlet pressure to much smaller than the inlet pressure at the corner edge, although this behaviour is restricted to an area within a radius of 0.2 units of the corner at the ceiling, and this radius reduces further moving down the edge (as z decreases), so that the pressure jump lower down the wall is much more localized. These areas are excluded from these results, based on the fact that they are artificial features of the pressure field resulting from the approximate electro-osmotic velocity boundary condition, Eq.(25).

Pressure sensitivity [0076] Figure 11 shows the total sensitivity of the pressure profile on the inner surfaces of the microchannel T-junction to the Carreau parameters (A, n) over the entire range of parameter values. The top image shows contours of the total sensitivity on the ceiling (region A) and on the inlet side wall of the microchannel (region B), while the bottom image shows sensitivity contours on the outlet inside wall (region C), channel floor (region D) and back wall (region E).

It is clear that the region of highest sensitivity is on the channel roof (region A) around one channel width upstream from the T-junction inlet, while the pressure profile along the end wall (region E) is a region of low sensitivity. Pressure sensors are best positioned in places where the pressure sensitivity is high, for example on the channel ceiling.

[0077] Pressure profiles on the inner microchannel surfaces for a range of constitutive parameters were simulated with the aim of selecting the optimal positions at which to position micropressure transducers. This data could then be used to infer the values of the Carreau parameters by solving the inverse problem. In order to be able to infer A and n accurately, it is necessary that the pressure profiles are sensitive to the changing values of A and n across the range of parameter values of interest (0.1 «=A«=5 and 0.1 «=n«=0.9). In other words, a small change in the values of the Carreau parameters must result in an appreciable change in the wall pressure profile.

[0078] Herein the spatial position on the wall pressure profile net is referenced using the coordinates (x, y). The pressure data can be described as a function with four inputs pr:p(x, y, A, n) where x, y are surface coordinates and A, n are the Carreau time relaxation constant and exponential index respectively. The sensitivities of the pressure to changes in the Carreau parameters are just the partial derivatives of pressure with respect to A and n: s,(x,y, 2, n) (34) s(x,iXn)11. (35) [0079] A measure of the total sensitivity of the pressure is desired at a particular point on the interior surface over the whole range of Carreau parameters of interest. A natural measure is therefore the sum over all A and n values of the squared parameter sensitivities. The interior surface regions with larger sensitivities summed over all constitutive values will be more sensitive to changes in A and n over the entire parameter range. For a continuous pressure profile p(x, y, A, n), the total sensitivity of the pressure to both of the Carreau parameters at a given point on the surface may be expressed as an integral of the sum of the squared sensitivities over the entire Carreau parameter range W(x,y) = f,, WAS + ws c/A c/n (36) where X[A0.1,A5]x[n0.1,n0.9] is the Carreau parameter set of interest, and wA, u',, are weighting constants that are set according to the relative magnitude or importance attached to the respective parameters. The total sensitivity P(x, y) is a measure of the sensitivity of the pressure at each point on the wall to A and n over the entire Carreau parameter range.

[0080] As described above (Pressure data collection), the pressure profile on the inner microchannel surfaces is stored as a matrix p'At) of discretely sampled pressure data with rows and columns corresponding to they and x positions on the wall. We convert this data into a piece-wise continuous function pM(x,y) by linearly interpolating between the pressure values in the matrix.

[0081] The sensitivities sjA)(x,y) and sM'v(x,y) are calculated using the central difference approximation for derivatives, and produce matrices of pressure sensitivities, with each entry in the matrix corresponding to the sensitivity of a single pressure point: 5(A,n) -p(k-1-dA.I1)_ p(A-dAn) A -2dA) = p(Afl+dflLpCA.fl-dfl) (38) where dA and dn are the constant step sizes between A and n data points respectively. The central difference derivative at a data point depends on the values at the previous and following data points, and therefore cannot be used at the first or last data points. Rather than using lower accuracy forward and backward differencing at these points, it was decided to consider only the interior points and not calculate the sensitivities at the first and last data points for both A and n. The central difference approximation is accurate to an error term that is quadratic in the step size. The error associated with the approximation is therefore small since step sizes of dAO.05, dAl and dnO.1 were used for the pressure sampling.

[0082] The integral of the total sensitivity Eq. (36) is approximated using the trapezoidal rule of integration. Let and be matrices whose elements are the squared sensitivities, so that (S2) = s. If we define a matrix of the weighted sum of squared sensitivities on the walls Q(Afl) = + (39) then the integral of squared sensitivities over the parameter set Xis given by the sum NA-2 N,-2 = dA dii i=2 j2 [Q(AtnJ) + Q(Ai+inJ) + Q(AinJ.i) + Q(Ai+itJ+i)] (40) In Eq. (40), NA and N are the number of A and n data points respectively. The limits of the sum start at 1=2 and j=2 and end at NA-2 and N-2 because the sensitivities are not defined for data points on the edge of X owing to the central difference approximation used in their calculation.

The central difference derivative at points where A=1 was computed using the Newtonian pressure solution, since a relaxation time of zero (corresponding to the data point preceding A=1 with dAl) reduces the Carreau viscosity to a constant Newtonian viscosity.

[0083] Contours of total sensitivity of the wall pressure profile to the Carreau parameters are plotted in Figure 11, with the dark areas representing low sensitivity and the light areas representing high sensitivity. The areas in which the pressure is sensitive to changes in A and n are the floor and ceiling just upstream of the junction, and the ceiling immediately upstream of the corner. The areas adjacent to the inlet and outlet show little sensitivity since the pressure is constrained to be zero at the inlet and outlet boundaries. The sensitivity grows very large at the corner owing to the singularities in the pressure field there. The area surrounding the corner edge, where W>5, are to be excluded from sensitivity results, as it is expected that the pressure predictions may be unreliable there. It is, however, likely that the largest pressure variations do occur in the region of the corner, although because the pressure jumps from high to low across the corner (see Figure 10), a discrete sensor placed at the corner would most likely average out this variation. From the sensitivity plot, it appears that the best places to position pressure sensors would be in the centre of the channel ceiling just before the junction, and on the area of sensitivity following the corner in the outlet channel.

[0084] Interestingly, the end wall pressure profile is seen to be less sensitive to the constitutive parameters than the floor and ceiling profiles. In two-dimensional simulations [8], the end wall pressure was successfully used to infer A and n. Given that there appears to be more information present in the ceiling pressure profiles than those of the end wall, this is a strong indication that inference of A and n is possible using the ceiling profile alone.

[0085] Since the microchannel is fabricated by first etching the channel layout onto a chip and then bonding a second chip to the first to create the ceiling, the simplest method of fitting micropressure sensors to a chip might be to build them into the ceiling chip before the two chips are bonded. The fact that the ceiling pressure profile is the most sensitive of the surface profiles to the Carreau parameters means that a chip can be fabricated with sensors on the channel ceiling with the expectation that the sensor readings can be used to infer the Carreau constitutive parameters A and n from a single flow experiment.

10. Discussion [0086] These results from three-dimensional simulations of non-Newtonian electro-osmotic flow in a T-junction microchannel show a significant vertical velocity component in the flow, which is of the same order of magnitude as the main channel flow at the channel walls near the corner where the electric double layer experiences a vertical body force owing to the curved microchannel walls. This component contributes to larger pressure variations over the floor and ceiling of the microchannel compared to the side walls, and especially the end wall. While the viscosity profile appears to be very similar to that of equivalent two-dimensional flows [8], the three-dimensional velocity and pressure fields render the ceiling a better place to position pressure sensors to infer the fluid constitutive parameters than the end wall.

[0087] The increased sensitivity observed here suggests that the channel ceiling is an excellent place to position micropressure transducers for pressure measurement.

[0088] The ability to determine both the A and n values ot an unknown fluid from a single experimental measurement is a key feature of a potential microrheometer design. A wide range of shear rates, and hence viscosities, are induced as the fluid is forced to turn the corners of the T-junction. In a similar flow in a T-junction, the region of detectable strain rate of the level of 200 Hz is confined to a thin region in the immediate vicinity of the corner [6]. This can be contrasted to the widespread shear field with a modal value of 200 Hz.

[0089] Another point relating to the suitability of the microchannel T-junction for inferring Carreau fluid parameters is that the magnitude of the pressure differences on the interior channel surfaces is directly related to the magnitude of the zero shear viscosity of the fluid. This relationship suggests that the microrheometer is belier suited to measuring complex fluids with high zero shear viscosities, as the resulting pressure differences will be larger and therefore easier to measure accurately with micropressure transducers.

[0090] In the study [19], the effects of temperature variation in the channel are neglected owing to the mechanisms of electrical resistance (joule) heating and viscous dissipation. It has been shown [25,38] that the dominant heating effect in a microchannel flow of the dimensions considered here is joule heating from the applied electric field. Under typical electro-osmotic flow conditions, the heating effect is counteracted by dissipation to the channel walls to the extent that the temperature increase across the T-junction itself (where the variations in viscosity occur) is small, of the order of 1 degree Kelvin. Localized heating occurs at the sharp corners in the channel, where temperature increases are much larger, causing a reduction in the fluid viscosity. As the corners are the points in the flow where the shear rate is largest, the viscosity there is already small, and the overall effect is likely to be an apparent drop in the infinite shear viscosity parameter p owing to the temperature increase at the corners. While this effect may affect the ranges over which the Carreau viscosity is sensitive to changes in its parameter values, it is unlikely to destroy the close relationship between the Carreau

parameters and the flow field.

[0091] Another assumption was that there was no electrophoresis (electric migration) of charged particles present in the flow. While this assumption can be quite valid for simple solvents, the presence of complex macromolecules such as long chain polymers or even blood cells in the test fluid may lead to aggregation of particles in the flow that in turn may acquire net charges and experience an electrophoretic body force. If polymers attain charges, their configuration may be affected by the electric field as well as the flow field, which may alter their viscous response so that they no longer follow the Carreau viscosity model.

[0092] As mentioned above, while the studies [19,3] referred to above are essential in understanding the operation of the present invention, they do not provide a solution to the problem of determining the constitutive parameters of visco-elastic fluids. Moreover, subsequent work has demonstrated that, while the theory presented in the foregoing discussions are valid, there are practical difficulties with pressure sensors that cannot reliably detect pressure and are somewhat susceptible to drift.

[0093] As stated above, it is an object of the present invention to provide a rheometer and method of determining rheometric properties of complex fluids, including visco-elastic fluids, that does not suffer from these disadvantages.

[0094] US-A-7578171 discloses a rheometer using a pulsed ultrasound Doppler based technique. Although this device eliminates the need for grab samples, its function depends on the relative movement of two surfaces. It can only analyse fluids that contain particles that reflect ultrasound, which is clearly a disadvantage when not all fluids that may require rheological monitoring include such particles.

[0095] Haward et al [43] disclose the use of a single strain rate measurement in extensional flow under oscillating conditions. The data collected is not information rich'. Using such a method involves the running of many experiments over different strain rates. In the method described, a polymer becomes trapped at a stagnation point of the device. Direct measures of birefringence with oscillations perturbing a single polymer molecule give the extensional viscosity. This is a direct method, as opposed to an inverse method.

BRIEF SUMMARY OF THE DISCLOSURE

[0096] In accordance with the present invention there is provided apparatus responsive to constitutive parameters of the rheological properties of complex fluids comprising: 1. a flow path for the fluid that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; and 2. means to sense flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; characterized in that 3. said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus.

[0097] In such a basic form, the invention has application in the fluid processing industry where all that may be required is an instrument capable of detecting changes in rheological properties where the fluid is a complex fluid. The advantage of the present invention over prior art methods is that, by pulsing the fluid, transient effects are introduced so that the effect of instrument drift over time can be negated, because the change being detected is over a relatively short period of time following a pulse, over which period instrument drift is less problematic. It is not necessary, in this scenario of application of the present invention, to know precisely what rheological parameters of the fluid might be changing. It is only necessary to know that they are changing, and that the instrument is sensitive to such changes, so that an alarm can be raised if such changes are significant.

[0098] Preferably, said sensed parameters are selected from stress, pressure, strain and flow rate variation with time. Preferably, said pressure is monitored using piezo-electric sensors disposed in regions of significant sensitivity. Preferably, strain is measured by strain gauges.

[0099] Preferably, said shear feature is a corner of a T-junction, whereby computational savings may be realized due to the symmetry of such a junction.

[00100] Preferably, said pulse or series of pulses are induced by a pulsatile pump or by actuating a pressure transducer with a pulse or series of pulses.

[00101] Other sensors, such as velocity/flowmeters, will record the transient response to the pulse disturbance.

[00102] Nevertheless, there are times when it is desired to determine certain constitutive rheological parameters of complex fluids, whether they are Carreau, power law visco-elastic or have other behavioural properties. Constitutive parameters are parameters of a model that replicates and possibly defines the rheological properties of a given fluid. Provided the model that the fluid replicates is reasonably accurate, and known, and the response of the instrument to changes in the parameters of the model are also known, it is possible by computational fluid dynamic means to deduce the parameters of the model from the measurements made by the instrument.

[00103] Thus preferably, the invention further comprises apparatus to determine constitutive parameters of the rheological properties of complex fluids comprising: 1. a tlow path for the fluid that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; 2. means to sense flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; and 3. computational means to calculate on the basis of said sensed flow parameters constitutive characteristics of the rheology of the fluid by inverse interpolation characterized in that 4. said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus and included in said calculation.

[00104] The pulse(s) transmitted through the fluid alter(s) said sensed parameters as a result of varying of individual flow rates of elements of the flow stream through the impact of the pulse on the rheological parameters of the fluid. The sensing means detect said alterations resulting from the pulse and said computational means calculates said constitutive parameters of the rheology of the fluid based on a best fit inverse computation based on the relevant model.

[00105] Preferably, said constitutive parameters in the case of Carreau fluids are A, the relaxation time, and n, the power-law exponent, in Eq (II) above.

[00106] Alternatively, said constitutive parameters in the case of visco-elastic fluids may be: the polymeric viscosity; R2' the solvent viscosity; a and being positive constants used to control the viscoelastic properties; and A, the relaxation time in Eq. (III) above.

[00107] Put more generally, this response in the detected parameters will have embedded information which is a single relaxation time, or many relaxation times, as the fluid returns to its undisturbed state. These response times are factors related to the viscous response', of the fluid and hence can be related using inverse methods to estimate the k constitutive parameters. These are not necessarily dependent on the particular relationship that the fluid rheology exhibits.

[00108] The coupling processes between embedded sensors and fluidic properties of the device are therefore crucial to inferring viscoelastic and power law fluid properties. It has been discovered that piezoelectric devices report a transient response in a steady state flow field.

Consequently, their use in the arrangements discussed above in relation to [3,19] is problematic for that reason. The present invention negates this effect by utilizing transient forcing of the fluid, which yields a fluid relaxation time constant that is sensitive to constitutive rheometric properties. These response factors can be calibrated with computational fluid dynamics (CFD) to constitutive properties of the fluid and thereby creates an effective rheometer.

[00109] CFD modelling has been used [3,19] to demonstrate that, from measurements from flow and pressure sensors in a microchannel 1-junction, it is possible to solve the mathematically proscribed inverse problem in order to obtain the constitutive viscous parameters for Newtonian, power-law and Carreau fluids. With the addition of transient effects caused by pulsing of the fluid, it is possible also to obtain the constitutive viscous parameters for visco-elastic fluids.

BRIEF DESCRIPTION OF THE DRAWINGS

[00110] Embodiments of the invention are further described hereinafter with reference to the accompanying drawings, in which: Figures 1 to 4 are taken from [3] and described above; Figures 5 to 11 are taken from [19] and described above; Figure 12 is a schematic representation showing rectangular pressure pulse-forcing profile generated by turning a syringe pump on, and then off; Figure 13 is a schematic diagram showing the transient response to the forcing depicted in Figure 12; Figure 14a, b and c are pressure responses from a strain gauge positioned at the inlet of a T-junction channel of a fluidic rheometer in accordance with the present invention for (a) Water, (b) 1 M PEO 2wt% and (c) 1 M PEO 4wt%; Figure 15 shows the sigmoidal response for PEO, polyol and xanthan; Figure 16 shows relaxation time, A, plotted against normalized flow rate for polyol (Newtonian fluid) and PEO and xanthan (non-Newtonian fluids); Figure 17 is a schematic representation of apparatus according to the invention; and Figure 18 is a view of each of the top and bottom plates of the apparatus of Figure 17 prior to assembly.

DETAILED DESCRIPTION

[00111] Referring to Figure 12, a syringe pump (not shown) is employed to deliver fluid to the inlet of a T-junction, such as described above and in [3,19], and shown in Figures 1 and 3. The force profile is a square wave shown in Figure 12. The typical sigmoidal response of the complex fluid is shown in Figures 13.

[00112] To the left-hand side of the vertical dashed line 12 in Figure 13, the pump is turned on.

A transient activation signal can be detected. (The profile of the signal could be concave or convex depending on the quantity that is measured.) To the right-hand side of the dashed line 12, the pump is turned off. A mechanical relaxation response can be measured. Atimescale for the mechanical response can be determined. This is an important parameter for determining the constitutive viscous parameters. Sigmoidal functions can be fitted to the curves on either side of the dashed line. It has been found that the parameters describing the right-hand side curve are sensitive (and hence can be mapped to constitutive viscous parameters) to different flow rates for polyol 2120, PEO 1 M 4wt% and xanthan O.5wt%. This proves the concept discussed above.

[00113] Alternative means of pulsing the flow to the T-junction include generally pulsatile-type pumping, e.g. using a peristaltic pump, or "firing" a piezo-electric element in the flow.

[00114] The following are examples of techniques that can be used for sensing/transduction of the mechanical response of the fluid to transient forcing: 1. Piezoelectric sensors -these are electromechanical devices that react to compression.

They are robust and exhibit an excellent linearity over a wide amplitude range. They have a fast response.

2. Strain gauges -these can be adhered to the walls of the flow cell. A strain gauge consists of a pattern of resistive foil which is mounted onto an insulated flexible backing material. As the fluid is perturbed, a stress is applied to the gauge and the foil is deformed, causing its electrical resistance to change. This change in resistance, which is usually measured using a Wheatstone bridge, is related to the strain by a quantity known as the gauge factor.

3. Capacitance inference of flow rate -cross-correlation effect. By recording time-series of capacitance at two or more locations across the flow cell, cross-correlation techniques can be used to infer the flow rate of the fluid.

[00115] Figures 17 and 18 show an experimental arrangement where two plates 20,22 are formed from glass. The plate 22 is etched with a T-shaped channel 24 of approximately 0.5 mm depth and approximately 90 mm length in cross arm 25 and stem 27. In the plate 20, locations 26 are etched to receive strain gauges 32 at each end of the arms 25; 34 at the junction between the arms 25,27; and 36 at the end of the arm 27. Locations 28 are also etched to receive piezoelectric transducers 38,40. A hole 42,44,46 is drilled in the plate 20 to coincide with each free end of the T-channel. The plates are secured together to enclose the channel 24 and capture the strain gauges and pressure transducers therein. Each strain gauge and transducer is wired to a power supply and interface device (not shown) that records the pressures and strains detected as changes in voltage. The strain gauges respond to a mechanical displacement caused by off-diagonal elements of the stress tensor in the fluid. The piezoelectric sensors simply respond to pressure. The positioning of the strain gauges and pressure sensors is not unique. However, numerical simulations provide insight into where would be the most sensitive parts of the channel to position them. It should also be repeated that it is not essential that the channel is a T-junction. Any geometry that induces a range of shear rates might be suitable.

(00116] Figures 14a, b and c show examples of the transient response as detected by the strain gauge 36 positioned in the inlet channel of the fluidic rheometer for two different fluids (water, and PEO). In region I in each case, the pump was turned-off. The pump was turned-on at the start of region II and turned-off at the start of region IV. The profile of the response curve (relaxation response) in region IV can be used to fit equations in order to obtain dynamic measures that can be mapped to constitutive parameters. (The small-scale oscillations are pump effects -these do not influence the statistical analysis and it has been found that these can be alleviated, in any event, by using a higher pressure pump. Also, relaxation and initial responses are inverse effects and teach essentially the same thing. It is preferred to use the relaxation effect since this is impacted less by pump effects.) This demonstrates the sensitivity of the arrangement and its suitability for the purpose of the present invention. Similar outputs are achieved by the remaining devices 32,34, 38 and 40.

(00117] In Figure 14a, water is supplied at an inlet rate of l6mltmin. In Figure 14b the PEO (at 1 M, 2 wt%) is supplied at an inlet rate of 1 6m1/min. In Figures 1 4c the PEO (at 1 M, 4wt%) is supplied at inlet rate of lml/min.

(00118] Referring to Figure 15, the sigmoidal responses for PEO (top curve: non-Newtonian), polyol (middle curve: Newtonian) and xanthan (bottom curve: non-Newtonian) are shown.

These shown can be fitted to the equation below, where b (SI units s) is the response time.

With power law and viscoelastic fluids, A is sensitive to flow rate, and hence to shear profile.

However, A is not sensitive to these quantities in the case of Newtonian fluids (e.g. polyol). The relaxation time, A, is plotted against flow rate in Figure 16 for polyol (Newtonian), and xanthan and PEO (non-Newtonian fluids).

Equation: Sigmoidal, Sigmoid, 4 Parameter a where a is a constant, V is the voltage at time t, and V0 the voltage at time to, from the piezo pressure transducers, giving the following effects and demonstrating the sensitivity of the apparatus and thus its suitability for the purposes of the present invention: Polyol 2120 (Newtonian) _________ Flow rate (mi/rn/n) b Standard ________________________ (seconds) Deviation.

0.5 24.3934 0.1400 1 24.3934 0.1389 1.5 22.0083 0.1243 2 21.5700 0.1052 2.5 21.7453 0.0606 demonstiating no change in ielaxation time with respect to flow rate.

PEC 1M 4wt% (visco-elastic) _________ Flow rate (mi/rn/n) b Standard ________________________ (seconds) Deviation 0.3 135.6844 4.6563 0.6 77.4574 1.9868 0.9* 70.3063 0.9825 1.2 54.4480 0.3626 demonstiating very substantial change in b with iespect to flow late. (*indicates the approximate position in Figure 14c).

Xanthan O.5wt% (power law) Flow rate (mi/rn/n) b Standard (seconds) Deviation 9.1573 0.7924 1.5942 0.1399 also demonstrating change in b.

[00119] Thioughout the desciiption and claims of this specification, the words "comprise" and "contain" and variations of them mean including but not limited to", and they are not intended to (and do not) exclude other moieties, additives, components, integers or steps. Throughout the description and claims of this specification, the singulai encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[00120] Features, integers, characteristics, compounds, chemical moieties or groups described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and/or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and/or steps are mutually exclusive. The invention is not restricted to the details of any foregoing embodiments. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed.

[00121] The reader's attention is directed to all papers and documents which are filed concurrently with or previous to this specification in connection with this application and which are open to public inspection with this specification, and the contents of all such papers and documents are incorporated herein by reference.

REFERENCES

[1] Phan-Thein N and Tanner RI (1977) A new constitutive equation derived from network theory. JNon-Newton/an Fluid Mech. 93 325-337.

[2] Ngamaramvaranggul V and Webster ME (2002) Simulation of pressure-tooling wire-coating flow with Fhan-ThienlTanner models. lnt. J.Num. Meth. Fluids 38 677-710.

[3] Bandulasena H.C.H., Zimmerman W.B., Rees J.M., An inverse methodology for the rheology of a power-law non-Newtonian fluid, J Mech. Eng. Sd. 222(C5) (2008) 761-768.

[4] Guillot, P., Panizza, P., Salmon, J.-B., Joanicot, M., Cohn, A., Bruneau, C.-H and Cohn T. Viscosimeter on a microfluidic chip. Langmuir, 2006, 22, 6438-6445.

[5] Bandulasena, H.C.H., Zimmerman, W.B. and Rees, J.M. Microfluidic rheometry of a polymer solution by micron resolution particle image velocirnetry: a model validation. Meas. Sci. Tech., 20, 115404, (2009).

[6] Bandulasena, H.C.H., Zimmerman, W.B. and Rees, J.M. Creeping flow analysis of an integrated microfluidic device for rheometry, J. Non-Newtonian Fluid Mech. , 2010, 165, 1302-1308.

[7] Meinhart, C.D., Wereley, 5.1. and Santiago, J.G. PIV measurements of a microchannel flow. Exp. Fluids, 1999, 27, 414-419.

[8] Zimmerman, W.B., Rees, J. M. and Craven, T.J. The rheometry of non-Newtonian electrokinetic flow in a microchannelT-junction. Microfluidics and Nanofluidics, 2006,2,481-492.

[9] Casas, J.A. Xanthan gum production under several operational conditions: molecular structure and rheological properties. Enzyme and Microbial Tech., 2000, 26(2-4), 282-291.

[10] Seevaratnam, G.K., Suo, Y. Rame, E., Walker, L.M. and Garoff, S. Dynamic welling of shear thinning fluids. Physics of Fluids, 2007, 19(1), Art. No. 012103.

[11] Song, K.W., Kim, Y.S. and Chang, G.S. Rheometry of concentrated xanthan gum solutions: Steady sheer flow behaviour. Fibers and Polymers, 2006, 7(2), 129-1 38.

[12] Maclnnes, J.M., Du, X. and Allen, R.W.K. Prediction of Electrokinetic and Pressure Flow in a Microchannel T-junction. Physics of Fluids, 2003, 15(7), 1992-2005.

[13] Jeanmeure, L.F.C., Dyakowski, T., Zimmerman, W.B.J. and Clark, W. Direct flow-pattern identification using electrical capacitance tomography. Experimental and Thermal Fluid, 2002, 26(6-7), 763-773.

[14] Bandulasena, H.C.H., Zimmerman, W.B. and Rees, J.M. Rheometry of non-Newtonian polymer solution by microchannel pressure driven flow. Applied Rheology, 2010, 20, 55608.

[15] Hadamard, J. Lectures on the Cauchy problem in linear partial differential equations.

Yale University Press, New Haven, 1923.

[16] Zimmerman, W.B. Metabolic pathways reconstruction by frequency and amplitude response to forced glycolytic oscillations in yeast. Biotechnol. Bioeng., 2005, 92(1), 91-116.

[17] [Deleted] [18] [Deleted] [19] Craven TJ, Rees JM, Zimmerman WB, Pressure sensor positioning in an electrokinetic microrheometer device: simulations of shear-thinning liquid flows. Microfluid Nanofluid, 9: 559-571 (2010) [20] Degre 0, Joseph F, Tabeling F, Lerouge 5, Cloitre M, Ajdari, A rheology of complex fluids by particle image velocimetry in microchannels. AppI Phys Lett 89: 024104 (2006) [21] Devarakonda B, Han J, Ahn CH, Banerjee RK, Bioparticle separation in non-Newtonian fluid using pulsed flow in micro-channels. Microfluid Nanofluid 3: 391-401 (2007) [22] Ermakov SV, Jacobson SC, Ramsey JM, Computer simulations of electrokinetic transport in microfabricated channel structures. Anal Chem 70:4494-4504 (1998) [23] Gao Y, Wong TN, Chai JC, Yang C, Ooi KT, Numerical simulation of two-liquid electro-osmotic flow in microchannels. Int J Heat Mass Transfer 48: 5103-5111(2005) [24] Gavrus A, Massoni F, Chenot JL, An inverse analysis using a finite element model for identification of rheological parameters. J Mater Process Technol 60(1): 447-454 (1996) [25] Horiuchi K, Dutta P, Joule heating effects in electro-osmotically driven microchannel flows. Int J Heat Mass Tran 47: 3085-3095 (2004) [26] Hunter RJ, Foundations of colloid science. Vols I-Il, Oxford University Press, Oxford (1992) [27] Kang K, Lee U, Koelling KW, High shear microfluidics and its application in rheological measurement. Exp Fluids 38: 222-232 (2005) [28] Lee J, Tripathi A, Intrinsic viscosity of polymers and biopolymers measured by microchip. Anal Chem 77(22):7137-7147 (2005) [29] Maclnnes JM, Computation of reacting electrokinetic flow in microchannel geometries.

Chem Eng Sci 57: 4539-4558 (2002) [30] Maclnnes JM, Du X, Allen RWK, Dynamics of electro-osmotic switching of reacting microfluidic flows. Chem Eng Res Des 81 (A7):773-786 (2003) [31] Owens RG, Phillips TN, Computational Rheology. Imperial College Press, London (2005) [32] Pujos C, Regnier N, Mousseau P, Defaye G, Jarny Y, Estimation of rheological law by inverse method from flow and temperature measurements with an extrusion die, MATERIALS PROCESSING AND DESIGN; Modeling, Simulation and Applications; NUMIFORM 07; Proceedings of the 9th International Conference on Numerical Methods in Industrial Forming Processes. AlP Conference Proceedings, Volume 908, pp. 1287-1294 DOI: 10.1063/1.2740987 (2007) [33] Rawool AS, Mitra SK, Numerical simulations of electro-osmotic effect in serpentine channels. Microfluid Nanofluid 2(3):261-269 (2006) [34] Rawool AS, Mitra 3K, Kandlikar SG, Numerical simulation of flow through microchannels with designed roughness. Microfluid Nanofluid 2(3):215-221 (2006) [35] Richardson JF, Chhabra RP, Non-Newtonian Flow and Applied Rheology, 2nd ed.

Butterworth-Heinemann 501 pp. Oxford (2008) [36] Srivastava N, Burns MA, Analysis of non-Newtonian liquids using a microfluidic capillary viscometer. Anal Chem 78(5): 1690-1 696 (2006) [37] Szeliga D, Gawad J, Pietrzyk M, Inverse analysis for identification of rheological and friction models in metal forming. Comput Methods AppI Mech Eng 195: 6778-6798 (2006) [38] Tang GY, Yang C, Chai CK, Gong HO, Joule heating effect on electro-osmotic flow and mass species transport in a microcapillary. Int J Heat Mass Transfer 47:215-227 (2004) [39] Tanner RI, Engineering Rheology, 2nd ed. Oxford University Press 592pp. Oxford (2002) [40] Tuinier R, Taniguchi T, Polymer depletion-induced slip near an interface. J Phys: Condens Matter 17:L9-L14 (2005) [41] Zimmerman WBJ, Multiphysics modelling with finite element methods. Series on Stability, Vibration and Conrol of Systems, Series A Vol. 18 432pp. World Scientific, Singapore (2006) [42] Craven TJ, Rees JM, Zimmerman WB, On slip flow velocity boundary conditions for electro-osmotic flow near sharp corners. Phys Fluids 20(4): 043603 (2008) [43] Haward SJ, Odell JA, Berry M, Hall T, Extensional rheology of human saliva. Rheol Acta (2011) 50:869-879

Claims (11)

1. CLAIMS1. Apparatus responsive to constitutive parameters of the rheological properties of complex fluids comprising: 1) a flow path for the fluid that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; and 2) means to sense flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; characterized in that 3) said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus.
2. 2. Apparatus to determine constitutive parameters of the rheological properties of complex fluids comprising: 1) a flow path for the fluid that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; 2) means to sense flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; and 3) computational means to calculate on the basis of said sensed flow parameters constitutive characteristics of the rheology of the fluid by inverse interpolation characterized in that 4) said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus and included in said calculation.
3. 3. Apparatus as claimed in claim 1 or 2, wherein said sensed parameters are selected from stress, pressure, strain and flow rate variation with time.
4. 4. Apparatus as claimed in claim 1, 2 or 3, wherein said pressure is monitored using piezo-electric sensors disposed in regions of significant sensitivity.
5. 5. Apparatus as claimed in any preceding claim, wherein strain is measured by strain gauges.
6. 6. Apparatus as claimed in any preceding claim, wherein said shear feature is a corner of a T-junction, whereby computational savings may be realized due to the symmetry of a 1-junction.
7. 7. Apparatus as claimed in any preceding claim, wherein said pulse or series of pulses are induced by a pulsatile pump or by actuating a pressure transducer with a pulse or series of pulses.
8. 8. Apparatus as claimed in claim 2, or any of claims 3 to 7 when dependent on claim 2, wherein said constitutive parameters in the case of Carreau fluids are A, the relaxation time, and n, the power-law exponent, in = Ro(1 + (Aj2)2)0/2, II where: * is the zero-shear-rate viscosity.
9. 9. Apparatus as claimed in claim 2, or any of claims 3 to 7 when dependent on claim 2, wherein said constitutive parameters in the case of visco-elastic fluids are: i1 the polymeric viscosity; k2 the solvent viscosity; c and being positive constants used to control the viscoelastic properties; and A, the relaxation time in Were = 2p1D -ft + We[r VU + (VU)T -U Vt + [D. Vt + (D. r)n]) Ill where f is defined using one of the following three forms: Linear model: f = 1 + Quadratic model: = 1 +!trace(r) +i[trace(r)I 2 t1 sWe Exponential model: f = exp [-trace(r)j, where * r=T-2i2D, * T is the extra-stress tensor, * D is the rate of deformation tensor, * U is the velocity vector, * We is a Weissenberg number defined as We = where U and L are typical velocity and length scales respectively.
10. 10. A method of detecting a change in constitutive parameters of the rheological properties of complex fluid comprising the steps of: 1) passing the fluid along a flow path that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; and 2) sensing flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; characterized by 3) pulsing said flow stream, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus; and 4) providing an output when said transient effects deviate more than a predetermined amount between pulses.
11. 11. A method to determine constitutive parameters of the rheological properties of complex fluids comprising: 1) passing the fluid along a flow path that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; 2) sensing flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; and 3) providing computational means to calculate on the basis of said sensed flow parameters constitutive characteristics of the rheology of the fluid by inverse interpolation characterized by 4) pulsing said flow stream, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and said transient effects are and included in said calculation.