GB2456540A - Rheometry Method and System for Spectral Analysis of Squeeze Films - Google Patents

Abstract

A method for analysing the dynamic response of sample fluids and semi-solids in a squeeze film using a rheometer with a first plate having a surface of substantially spherical form (convex) and a second plate having a surface of a substantially flat form is described. The method includes a data measurement step, a pre-processing step, and a spectral analysis step. In the data measurement step a squeeze film of a test material sample is supported between the plates, a force is applied and an input signal and output signal measured. In the pre-processing step dynamic equations for the geometric arrangement of the plates of the system are derived, and used to determine a non-linear spectral analysis algorithm describing the nonlinear relationship between the input and output. In the spectral analysis step non-linear spectral analysis of the measured data is performed using the non-linear spectral analysis algorithm to obtain material property result data. A rheometer system is also described.

Description

1

2456540

Method and system for spectral analysis of squeeze films

Field

5 The invention relates to a method for analysing the dynamic response of sample fluids and semi-solids in a squeeze film and a system for use in the method.

Background

10 A squeeze film is formed when two surfaces in close proximity and separated by an interstitial fluid provided in the gap between the surfaces, are moved relative to each other in a manner that is generally normal to the plane of the surfaces. As the fluid moves in or out of the gap between the surfaces depending on the direction of motion, a force is experienced by the bounding surfaces of the film. The force is highly non-linear being 15 inversely proportional to the film thickness, Stefan [ 1 ]. Using squeeze film systems (for example the Micro Fourier Rheometer), it is possible to analyse material properties based on measurements of motion of the bounding surfaces and the force on those surfaces, Phan Thien [2],

20 Often squeeze films are formed in a system having two substantially flat plates which need to be maintained parallel to each other if the standard mathematical formulations being used are to correctly model the dynamics.

In one approach a Micro Fourier Rheometer (MFR) has been used (Field [3], for 25 example, or systems such as those produced by GBC Scientific of Australia. The device applies a broadband displacement to the upper plate in an axisymmetric squeeze film. The lower plate is stationary and is supported by a piezoelectric force transducer which generates the output force signal. Substantially flat plates are used which prior to testing are aligned using a locking universal coupling in order to create a parallel geometry. The 30 plates are moved relative to each other, and experimental data including the force and upper plate displacement are collected and a spectral analysis is used to determine test

2

material characteristics. Referring to the prior art Figure A, an example of a squeeze film rheometer system for the analysis of squeeze films is shown, in which a squeeze film 1 is provided between two support plates 2 and 3. The upper plate 2 vibrates in a direction perpendicular to its plane. The motion is generated by an actuator, in this case a terfenol 5 actuator 4. This motion is detected by a displacement transducer 5. The lower plate 3 remains stationary and is supported by a force transducer 6 which measures the force developed in the squeeze film due to motion of the upper plate 2. Parallelism between the plates is adjusted using a universal coupling 7 below the lower plate. The overall system is supported using a frame 8.

10

In practise, there is a problem with rheometer systems in that it is very difficult to ensure that the support plates are maintained and remain parallel. Any skew or misalignment leads to a reduction in the force between the plates. Typically this will cause a large variation in the material characteristics identified from the squeeze film dynamics 15 particularly for small film thicknesses.

In one approach (Sakai [4]) aimed to address the problems associated with misalignment of substantially flat plates by testing a standard material (such as a Newtonian fluid of known viscosity), and using the data obtained to generate a correction factor which is 20 applied in further tests. However, this technique is cumbersome and means that any variation in the test set-up such as a change in gap height (very common in squeeze film analysis) necessitates a recalibration of the system using the known material. Also due to incorrect formulation of the spectral analysis, testing was confined to relatively large gaps and low amplitude oscillation which limited the application of the squeeze film technique 25 for analysis purposes.

Problems of plate skew and misalignment, and the resulting lack of plate parallelism which reduces squeeze film force has prompted the using of an arrangement in which one of the squeeze plates has a spherical surface topography. The spherical surface ensures 30 the minimum film height is still more or less in the centre of the film so that the system dynamics remain relatively insensitive to any small variations in plate inclination. The

3

spherical geometry is often modelled as a parabolic profile, (Adams[6], Meeten [7], Narumi [8]). The parabolic profile is a good approximation to the spherical topography but references 6,7 and 8 only performed gradual motion in a compression mode. As such the time histories of the displacement and force were recorded and matched to predicted 5 data. This technique does not lend itself to incorporation in a squeeze film rheometer and does not provide information over an extended dynamic frequency range.

Non-linear spectral analysis of squeeze films has been performed using both water and a water/glycol mixture in a flat plate axisymmetric configuration [Esmonde [9,10]]. The 10 spectral analysis procedures were shown to be capable of identifying multiple non-linear phenomena associated with squeeze film dynamics and that it was possible to distinguish the relative importance of the different phenomena involved. While some variation in results was noted at different film thicknesses, it was not understood at the time that the underlying problem was due to lack of parallelism between the plates.

15

The need remains for a method and system that is easy to use and that effectively addresses the problems resulting from plate skew and misalignment. Moreover, there is a need for a system which is robust and easy to use and provides more accurate and easily repeatable results.

20

There are therefore a number of problems with the conventional squeeze film analysis methods and systems that need to be addressed.

Summary

25 These needs and others are addressed by a method in accordance with the teachings of the invention for analysing the dynamic response of sample fluids and semi-solids in a squeeze film using a rheometer with a first plate having a surface of substantially spherical form and a second plate having a surface of a substantially flat form, the method including:

30

4

a. in a data measurement step supporting a squeeze film of a test material sample between the plates, applying a force and measuring an input signal and output signal;

5 b. in a pre-processing step deriving dynamic equations for the geometric arrangement of the plates of the system, and using these dynamic equations to determine a non-linear spectral analysis algorithm describing the nonlinear relationship between the input and output; and

10 c. in a spectral analysis step performing non-linear spectral analysis of said measured data using the non-linear spectral analysis algorithm to obtain material properly result data.

The invention thus provides a method in accordance with claim 1, with further

15 advantageous features provided in the claims dependent thereto.

According to a further aspect, the invention provides a system in accordance with claim

11.

20 Brief Description Of The Drawings

The present invention will now be described with reference to the accompanying drawings in which:

Figure 1 is schematic view of a squeeze film according to the invention;

25

Figure 2 is a flow diagram showing steps of a method according to the invention;

Figure 3 is a schematic view of the experimental instrumentation

30 Figure 4 is an input/output block diagram of the squeeze film dynamics;

5

Figure 5 is a graph showing the spectral analysis obtained with flat plates that were nominally aligned parallel to each other. The test fluid is lPas silicone oil, the upper plate diameter is 25mm and the average film thickness is 30 microns. Results are presented in terms of coherence, gain and phase between 0 and 25Hz.

5

Figure 6 is a graph of force ratio for flat plates. The force obtained using flat plates in a nominally parallel configuration and with a 0.25° tilt is given as a ratio of the force that would have been obtained with ideally parallel plates. The theoretically predicted ratio for a 0.25° tilt is also presented.

10

Figure 7 is a graph showing the spectral analysis obtained with a spherical upper plate and a flat lower plate aligned tangentially to the upper plate. The test fluid is lPas silicone oil, the upper plate has a lower surface with a spherical radius of curvature of 0.3m and a plate diameter of 25mm. The minimum film thickness at the centre of the 15 plates is 30 microns. Results are presented in terms of coherence, gain and phase between 0 and 25Hz.

Figure 8 is a graph of force ratio using a spherical upper plate. The force ratio in this case is the force that was obtained as a ratio of the force that would be obtained for 20 perfectly aligned plates. Perfect alignment is achieved when the lower plate is tangentially orientated to the spherical upper plate. Experimental results with the lower plate nominally aligned tangentially (termed "parallel") and with a 0.25° tilt are presented. The theoretically predicted ratio for a 0.25° tilt at the lower plate is also presented.

25

Detailed Description Of The Drawings

The method of the invention and the corresponding system are based on and/or may be incorporated into a squeeze film rheometer such as for example, the Micro Fourier 30 Rheometer as manufactured by GBC Scientific described with reference to Figure A above. It will be appreciated that this exemplary embodiment of implementation in the

6

context of a Micro Fourier Rheometer is provided to assist in an understanding of the teaching of the invention and it is not intended to limit the invention to such an arrangement

5 In the specification focus is on the particular steps and features of a method according to the present invention and an example geometrical arrangement of a system 20 (Figure 1) according to the invention. Further details of example parameters for operation of the system are provided under comparative results presented in a later section.

10 Referring to Figure 1 a geometrical arrangement of a pair of support plates of the system 20 is described. The system 20 comprises a first upper plate 21 having a surface of substantially spherical form and a second lower plate 22 having a surface of substantially flat form arranged in a squeeze film geometry. In each case, the surface is that surface which contacts the film 23. The system 20 is configured to and is operable to analyse the 15 dynamic response of a sample of a fluid or semi-solid material provided in a film 23 supported between the plates 21,22. The same functionality could be achieved with a spherical lower plate and a flat upper plate.

Referring to Figure 2 steps of a method of analysing a squeeze film 23 according to the 20 invention are described.

In step 100 dynamic equations 110 are formulated to describe the relationship between plate motion and plate force derived for the system 20 having a spherical support surface in the theoretical analysis.

25

In the step 100, which is essentially a pre-processing step, squeeze film dynamic equations 110 are derived for the particular geometrical arrangement of the plates of the system 20 being used. In this case, the system includes an upper plate 21 having a spherical surface and a lower plate 22 having a substantially flat surface. The spectral 30 formulation is required for considering squeeze film dynamics. For squeeze films with a spherical topography, as in this case, the appropriate dynamic equations 110 are used to

7

construct a transfer function describing the appropriate non-linear input output relationship between plate velocity and plate force for development of a non-linear spectral analysis algorithm 115. With this approach a time invariant transfer function is obtained permitting direct calculation of viscosity.

5

In a data measurement a step 101, the system 20 is operated, and input signal and output response data 120 are measured and captured. The data 120 includes for example, plate displacement/velocity as input data and plate force as output response. Referring to Figure 3, in which a squeeze film 23 is provided between plates 21 and 22, plate 10 displacement is measured by means of the displacement transducer 34 and plate velocity values which are required for the analysis may be determined using a forward difference approximation. Plate force may be measured by means of the force transducer 36. Average minimum film thickness ho, plate radius R and spherical plate radius R, are also measured.

15

The system 20 as shown also includes an actuator 33, and a signal conditioner and controller 32, and a data processor 31.

In a step 102, the measured data 120 is processed using the non-linear spectral analysis 20 algorithm 115 to obtain the required result data.

Thus, in effect the method of the invention provides use of two plates one of which has a surface of spherical form and the other which has a surface of substantially flat form, and the reformulation of the dynamic equations to construct the non-linear input output 25 relationship between plate velocity or displacement and plate force, which are then incorporated into a non-linear spectral analysis algorithm for processing the measured input and output response data 120.

The results data that may be obtained by the method of the invention include constitutive 30 properties of material such as for example viscosity or elasticity.

8

The above method according to the invention advantageously provides formulations for the geometrical arrangement of the squeeze films 23 which facilitate improved and direct calculation of material properties such as viscosity.

5 Further details of the steps 100, 101, and 102 of the method are as follows:

Pre-processing Step 100

The pre-processing step 100 is described with reference to Figure 4.

To analyse the data appropriately the system dynamics must be formulated correctly. Figure 4 shows a single input/single output model for the system where the output is taken as the plate force and the input X is chosen so that the transfer function becomes the measured property of the material in the squeeze film. For a Newtonian fluid the 15 measured property is the viscosity r]. For the case of a Newtonian fluid the dynamics of the squeeze film system are evaluated for the spherical geometry to determine the appropriate expression for X.

Determining the form of X is referred to here as the pre-processing step 100. For a 20 spherical upper plate with spherical radius Rs the film height h at a distance r from the central axis is

25 where h() is the minimum film thickness at the centre of the plates. Assuming inertia effects are negligible, the pressure at the edge of the plates (r=R) is zero, there is no slip between the fluid and the plates, the plate force Fp for an upper plate velocity v;, is determined as

10

(1)

30

P l\p-Jc-R2) ^ -Vc-/?2) (6 -Vc)

fp =n

9

(2)

5 where b=h0+Rs c = Rs~

10

This equation can be rewritten as

F=r\X

(3)

Thus X the input to the system is the composite expression in the brackets above Alternatively a simpler approximate expression is obtained by modelling the spherical 15 geometry as a parabola so that h = h0 +

2Rr

(4)

Making the same assumptions as above (for the full spherical solution) the plate force 20 becomes

371/?V

f

2h0

ha +

R

2 \2

2R.

(5)

25

Again the input X becomes the composite expression in brackets above. This simplification achieved using a parabolic approximation to the spherical sur&ce leads to minimal errors when estimating the material properties such as viscosity.

10

It is further noted that the above preprocessing step may be adapted to include analysis of inertia effects of the system.

5

Sample testing and data Measurement Step 101

Input signal and output response data 120 is captured. The step 101 is carried out largely in accordance with the methods of operation of the test set up as shown in Figure 3.

10 Relative motion between the plates of the squeeze film is generated using the actuator 33. In this case the upper spherical plate is moved thereby compressing or extending the material between the plates in the axial direction. The motion of the upper plate is determined using the displacement transducer 34 and the plate force is determined using the force transducer 36.

15

The analogue force and displacement signals are conditioned appropriately using the signal conditioner and controller 32 to generate their digital equivalent representation. The digitised data is then processed using the data processor 31 as described in step 102 during a data processing step using an appropriate software.

20

Plate displacement may be measured directly and velocity values which are required for the analysis method of the invention may be determined using a forward difference approximation.

25 Spectral Analysis Step 102

Systems are typically defined in terms of their transfer function, determined from the ratio of the output response to the input signal. This calculation is often done using broadband signals transformed into the frequency domain and carried out with repeated 30 measurements to give an averaged response. Firstly the input and the output to the system are defined. In the case of squeeze films consisting of flat plates and a Newtonian

11

fluid, the input has often been chosen as the plate velocity with the plate force considered as the output. If the system is modelled using the lubrication approximation, then from standard analysis we have

5 _ 3a/?V„

" 2 A3

(6)

Forming the ratio between the output and the input one obtains 10 F/'_ 3?ri?4r|

v, 2h3 (7)

However on close inspection it can be seen that the right hand side of equation 7 is not invariant due to the inclusion of h3 in the denominator. The film thickness h is a dynamic 15 quantity and will thus corrupt the estimate of the transfer function as defined in equation 7 over the course of the averaging process. Ideally the transfer function should remain constant and thus the formulation in equation 8 may be used.

fp _ -3ttR4X\

vp 2

20 p- (8)

or to give the material property directly, in this case the viscosity fp

25

~nr (9)

12

velocity and the instantaneous film thickness to produce a new composite term which is then Fourier transformed to create the input term in the spectral analysis.

As has been shown earlier the squeeze film system 20 has been designed and configured to be relatively impervious to misalignment effects when a spherical plate is used. Thus the appropriate equation derived for the spherical plate/tangential flat plate of Figure 1 is used to evaluate this type of squeeze film. As stated previously the parabolic approximation may be used in which case

The spectral analysis is performed so as to directly produce an estimate of the viscosity and thus the transfer function for the system is formulated as

(10)

-1

(11)

3 7tR4v p

r

2h0 h0 +

V

The input X and output Y are thus

Y(S) = 3{Fp(0) X(f) = 3

3nR4vp

13

(12)

5

Here the symbol 3 signifies the Fourier transform, (t) indicates temporal dependence and (0 indicates frequency dependence. Using this input and output, the spectral analysis is then performed using ensemble averages of cross and auto spectral quantities in the usual manner.

10

Example 1

Comparative Results

15 Tests were performed with a Micro Fourier Rheometer as described with reference to Figure A, Figure 1 and Figure 3.

Sample test data was captured and analysed using a method taking account of the nonlinear relationships between the input and output data and was analysed in the frequency 20 domain.

Tests were initially carried out with a flat plate geometry in the rheometer as shown in Figure A. The tests were then repeated by substituting the system 20 (Figure 1) with spherical upper plate into the rheometer of Figure A thereby replacing the flat plates.

25

Geometric arrangement: substantially flat plates

Spectral results using a pair of flat, nominally parallel plates are shown in Figure 5. The 30 test was performed using 1 Pas silicone oil between substantially flat plates having a 25mm diameter at 30|im static film thickness. Broadband upper plate 2 displacement

14

between 0 and 20Hz was used with an amplitude up to 20fim. The data is presented in terms of the coherence, gain and phase. The results are presented on the basis of 40 ensemble averages. A coherence value close to unity indicates that the transfer function (given here by the gain and phase) is a reliable representation of the relationship between 5 the input and output signals.

The coherence remains relatively high between 3 and 20Hz. The drop off below 3Hz is partially due to the low frequency cut-off imposed by the piezoelectric transducer while the drop above 20Hz simply reflects the input spectrum employed during the testing.

10

For 1 Pas silicone oil the gain should be unity and the phase should be 0° corresponding to a constant as described by equation 9. This is not the case from the results and it is apparent that the estimate for the viscosity is not constant and well below the value of unity at all frequencies. These tests were repeated over a range of film thicknesses and 15 the viscosity estimates are presented in Table 1. Note the value for viscosity shown was taken from the gain at 10Hz.

Film Height [microns]

Viscosity [Pas]

30

0.04

50

0.10

100

0.52

200

1.01

400

1.06

600

1.18

800

1.22

1000

1.25

Table 1

20 Viscosity Estimates (Flat Plates)

15

From the smallest gap of 30|am up to 1mm the viscosity is determined at between 4% and 125% of the expected value. This is clearly unacceptable if the viscometer is to be relied on to accurately measure material characteristics. It is believed this variation is due mainly to geometric effects associated with plate misalignment. Despite great care being 5 taken when aligning the plates it was not possible to ensure that they were absolutely parallel and as seen in Figure 6 even small deviations can lead to large variations in plate force and thereafter measured viscosity.

Next, the lower plate was tilted at 0.25° to the upper plate and viscosity measurements 10 were repeated. The experimental results are also shown in Figure 6 along with the corresponding values derived theoretically (see appendix 1).

The measured values at 0.25° are in relatively close agreement with those obtained theoretically. The difference at small film thicknesses can be attributed to the 15 axisymmetric approximation used in the theoretical analysis. The difference at larger film thickness may arise as a result of edge effects which become relatively more apparent as the gap increases and were not included in the analysis.

Geometrical arrangement: spherical upper plate/flat lower plate

20

Tests were repeated using lPas silicone oil with a spherical shaped plate 21. The spherical radius of curvature was 0.3m to counteract misalignment effects without excessively reducing the plate force.

25 Figure 7 shows the spectral results using a spherical upper plate with an average minimum film thickness h0 of 30(im. From the graph it can be seen that the viscosity estimate has improved from that obtained with the flat plates. As before tests were repeated at a series of film thicknesses up to 1mm, the results of which are given in table 2.

30

16

Film Height [microns]

Viscosity [Pas]

30

0.76

50

1.01

100

1.14

200

1.06

400

1.11

600

1.15

800

1.16

1000

1.15

Table 2

Viscosity Estimates (Spherical Plates)

5

The viscosity estimates now range from 76 to 116% of the expected value. These tests were done with the spherical upper plate 21 and the lower plate 22 aligned tangentially insofar as could be achieved experimentally. The effects of misalignment was examined by purposely imposing a 0.25° tilt on the lower plate corresponding with the analysis 10 used to derive the plate force equation for this system given in the appendix 2. The results are shown in figure 8 along the experimental data for the nominally tangential alignment. While there is some small variation in experimental data when the lower plate is tilted, the variation is minimal and should be compared with the flat plate results in figure 6.

15

The theoretically predicted values are somewhat less than the experimental values at the lowest film thicknesses. This is most likely the result of the axisymmetric approximation in the theoretical analysis. The disparity at larger film thickness has a distinct nature exhibited by both sets of experimental data. It is believed that the disparity is due to edge 20 effects and will be investigated further at a later date.

17

Despite the differences between the experimental and theoretical data for the spherical plate tests, the main points of interest are the surprising improvement in viscosity estimate over the flat plate analysis and the intransient nature of the estimates for varying degrees of plate inclination. This shows that the system of the invention can now be used 5 to reliably predict viscosity at different film thicknesses without recourse to knowledge of the plate alignment.

The method and system according to the invention provides a number of advantages over the use of prior art systems, including the following:

10

1) Use of a plate having a spherical support surface has the advantage of reducing susceptibility to support plate skew or misalignment such as has been found to occur during normal use of the prior art systems.

2) Reformulation of squeeze film dynamic equations to account for the spherical

15 geometry has the advantage of providing the correct mathematical description of the squeeze film dynamics.

3) Non-linear spectral analysis of the reformulated equations which has been shown to directly provide more accurate and robust estimates of the constitutive properties of the material under test.

20

Taking the above three points the invention advantageously provides a method and system that may be operated to obtain direct and reliable measurements of material characteristics from squeeze film tests.

25 The system is advantageously much more robust than prior art systems being relatively unaffected by lack of parallelism or misalignment that may occur between the plates.

Thus in effect the method according to the invention effectively addresses and deals with the gap dependence which was a feature and limitation in prior art squeeze film rheometry.

30

18

With the system of the invention, new applications of squeeze film rheometry are opened up for example, new commercial applications such as quantitative measurement of constitutive properties of raw materials or finished product as required in the manufacturing sector. In this sense the technology described here moves squeeze film 5 rheometry from the realms of specialist laboratory use to a more general application such as quality control. Squeeze film rheometry has in the prior art been considered a highly specialised field. Such systems normally need highly skilled operators to calibrate, operate and maintain them and detailed interpretation of results obtained. Results from this type of analysis in the past have been gap dependent and hence rather unreliable due 10 as we have noted due to problems plate skewness.

The method of the invention provides a more robust method and system which is more user friendly and provides useable output results directly without the need for further analysis. For example, the formulations for the squeeze films developed in the method of 15 the invention facilitate direct calculation of viscosity.

The method of the invention also addresses calibration issues by providing a method and system robust to skew.

20 In comparison with prior art methods where a material of known properties was required to recalibrate the system when any changes were introduced such as a change in film thickness the method of the invention obviates the need for such additional steps and measures.

25 The method also means that transfer function is a constant and therefore the system performance is not amplitude dependent so that reliable estimates of material properties are obtained irrespective of the level of input excitation.

While in the specification the term spherical plate has also been used to refer to a plate 30 have a substantially spherical support surface, being the surface which in use contacts the film, and the term flat plate has been used to refer to a plate having a substantially flat

19

support surface. The term spherical geometry has been used to describe a system having one spherical plate and one flat plate, while the term flat geometry has been used to describe a system having two flat plates. It will be appreciated that these different terms are well understood by the skilled person in the field to describe these components and it 5 is not intended to limit the scope of the invention to any specific wording as the functionality of the plates used in accordance with the teaching of the invention will be apparent irrespective of the wording applied to same.

The words comprises/comprising when used in this specification are to specify the 10 presence of stated features, integers, steps or components but does not preclude the presence or addition of one or more other features, integers, steps, components or groups thereof.

20

APPENDIX 1 5 FLAT INCLINED PLATE

Theoretical analysis

10

15

20

777//////////////////////////////!

25 Figure A1.1

For axisymmetric squeeze films where the radial dimension of the plates is large in comparison to the film thickness and the viscosity of the liquid between the plates is relatively large (>10mPas) the fluid dynamics described by the momentum equations 30 reduce to the simple Stefan solution

(All)

or dz~

35

40

subject to the boundary conditions v,. =0 z = 0,h where P is the pressure, r is the radial coordinate, T| is the viscosity, v, is radial velocity, z is the axial coordinate and h is the instantaneous film thickness at radius r which is given by h = /20 +/?.tan«-r.tana.sin<9

(A 1.2)

= /?0 +{7?-r.sin(^.tana

45

21

Using equations A 1.1 and A 1.2 one obtains v, = —^[z2 ~(h0 +(R-r.sin6).tana)z[ 2rj dr

(A 1.3)

The inclination of the upper plate leads to a loss of symmetry so that the azimuthal angle 10 must also be included when considering continuity

15

lit h0~(R-i sin 0) tan «

J?" Jv, dzdO = -n.r~vp o o

Substituting for v,. form equation A1.3

2/r h0-IR-r sin B) tan a

(A 1.4)

20 o

J" J

— -r—[z2 -(hQ + (R-r.sh\0).\zxia)^dzd8 = 2rj o/-

-71 J' Vr

(A1.5)

The pressure gradient term will be a function of 0 but to simplify matters it is taken outside the integral operators on the assumption that the gradient in the radial direction is 25 much more important than that in the azimuthal direction. This assumption is reasonable for small inclination angles.

2jt h^tR-r sin Rj tana

— IV f —[z2-(hQ+(R-r.sixid).\an.a)z\dzd9 = -n.r2\

30 dr o o 2n

(A 1.6)

Thus an expression for the pressure gradient is obtained dP_ 12r|rvp

35 dr ~ (hn + Rtana)[2(h(] + R tan a)2 + 3(r tana,)2]

As before this can be integrated twice over the area of the plates to obtain

(A 1-7)

40

fp=-

4/rr|v

6 (h0 + R tan o.) tan o.

2(h0 + Rxana)' In

2(h0 + R tm a)2

2(h0 +Rtana)2 + 3(7? tan a/

+ 3R" tan" a

J

(A1.8)

45

5

10

15

20

25

30

35

40

45

22

APPENDIX 2

SPHERICAL PLATE AND FLAT INCLINED PLATE

Theoretical analysis

Plate force equation for spherical upper plate and flat inclined lower plate is determined in a similar iashion to that for inclined flat plates in appendix 1. For the spherical upper plate the film thickness is described by h = h0 +RS- tJrs2 -r~ + r tana sin0 The plate force is then determined as

Fp = 2k [term^ + term ^ + tenn^ + term^ + term^ 4- term^j term, = 2"Kb\s In s-s

L

term 2 = kb

I

y2-4P5 2P5+Y-A/Y2-4P5 ^

In 1 +2s

2P 2Pi+Y+)/Y -4f35

V

2P

(A2.1)

y ^ / 2 \

Hi lnty3s +YS+5J

'R

term3 = X

term a = —

4 2

2

5 2i

5 In s

2

*R

,;to(p,^+s)-,;+g-ln(p5:+/+8)

2(1

+

^ 2p5+Y --JY2 -4(38

:ln f -> A

H-28

R

2(32^/y2 -4(35 2(3S+Y+-JY2-4(38

term^ =

§b V

vj/.s In f \

\|/5+(0

yl|/S+^2 j

+■ to ln(y.s+co)-,Q ln(yj+^)

'R

'0

—(co-&) ¥

4-

f 2 A 2 «

5 2 V J

ln(\(/5+(o)-

( ~ 2 ^

2 Q

5 ~2 ¥ y

In

'R

5R

term^ = k s—

23

15

20

A- = —X. In

\ f \

V)fsr + CO

PV +JSR +5

+ ()> In

a = ■

ah

2d(c-b )

P = 2-d y = 2db 8 = d(c-b2)

<j> =

ac

10 (c-b2)-y/4d2b2 -4(2-d)d(c-b2)

(0 = 2db - yj4d2b2 - 4(2 - d)d(c - b2)

= 2db + -j4d2b2 -4(2-d)d(c-b2)

\if=2{2-d)

SR-h 0 + S0_h()

a = 12^ b = h0+Rs c-Rs2 d = 3tan2a

25

30

35

24

References

I. Stefan J., "Versuche uber die scheinbare adhasion" Sitz. Kais. Akad. Wiss. Math. Nat. Wien. (1874) v69(2) pp713-735

5 2. Phan Thien N., "Small strain oscillatory squeezing of simple fluids" J. Aust Math Soc B (1980) v32 pp22-27

3. Field J.S., Swain M.V., Phan Thien N., "An experimental investigation of the use of random squeezing to determine the complex modulus of viscoelastic fluids" Journal of Non-Newtonian Fluid Mechanics (1996) v65 pp 177-194 10 4. Sakai S., "Improvements of an oscillatory squeezing flow rheometer for small elasticity measurements of liquids" (2004) v44 pp 16-28 5. Bell D,, Binding D.M., Walters K., "The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility", Rheol. Acta (2006) v46 pp 111-121

15 6. Adams M.J., Edmondson B., CaugheyD.G., Yahya R., "An experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic fluids" Journal of Non Newtonian Fluid Mechanics (1994) v51 pp61 -78

7. Meeten G.H. "Squeeze flow between plane and spherical surfaces", Rheol. Acta. (2001) v40, pp279-278

20 8. Narumi t., Hosokawa Y., Hasagawa T. "Experiments on the reverse squeezing flow of dilute polymer solutions" JSME Int. Jour. (1990) v33(2) ppl93-199 9. H. Esmonde, J.A. Fitzpatrick, H.J. Rice and F. Axisa- 'Modelling and Identification of Non-Linear Squeeze Film Dynamics' Journal of Fluids and Structures, (1992) v6, pp223-248

25 10. H.Esmonde, J.A. Fitzpatrick, H.J. Rice and F. Axisa - "Reduced order modelling of non-linear squeeze film dynamics" Proc Instn Mech Engrs, Part C J. of Mech Eng Sci Vol 206pp225-238.

II. Rice, H. J., Fitzpatrick, J. A., "A generalised technique for spectral analysis of nonlinear systems," Mechanical Systems and Signal Processing, (1988)Vol. 2, pi95-207.

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25

Claims (1)

1. A method for analysing the dynamic response of sample fluids and semi solids in a squeeze film using a rheometer with a first plate having a surface of substantially spherical form and a second plate having a surface of a substantially flat form, the method including:

a. in a data measurement step supporting a squeeze film of a test material sample between the plates, applying a force and measuring an input signal and output signal;

b. in a pre-processing step deriving dynamic equations for the geometric arrangement of the plates of the system, and using these dynamic equations to determine a non-linear spectral analysis algorithm describing the nonlinear relationship between the input and output; and c. in a spectral analysis step performing non-linear spectral analysis of said measured data using the non-linear spectral analysis algorithm to obtain material property result data.

2. The method as claimed in claim 1 wherein the measured data includes plate displacement or velocity as input and plate force as output.

3. The method as claimed in claims 1 or 2 wherein the non-linear spectral analysis algorithm defines the relationship between the input plate velocity or displacement and the output plate force.

4.

The method as claimed in any preceding claim wherein the non-linear spectral analysis algorithm comprises a time invariant transfer function useable for

26

direct calculation of the material property, of the test material under examination.

The method as claimed in claim 4 wherein the material property is viscosity or elasticity.

10

7.

The method as claimed in claims 1 to 5 wherein non-linear spectral analysis algorithm is developed for a geometrical arrangement in which the plates are arranged at an angle to each other, or substantially parallel to each other.

The method as claimed in any of claims 1 to 6 wherein the plate force is described by:

6kvpb

15 F =

R2

b "

P=T1

l\p-ylc-R2) ip-ylc-R2) ~yfc) 3b.lnw - yjc - R2

.... + 6 7TV,

(b ~>fc)

+ Vc - R2 -4c

S7rvp R~

20

where b = ha + R, c - R^~

25 8. The method as claimed in claim 7 wherein the transfer function can be simplified using a parabolic approximation to

FP ="H

371tfV

2h,

h0 +

R

2 r

2 R.

/ -i

27

to give the material property of viscosity rj directly wherein the quantity

3KR\,

2K

h +

V

- is formed in the time domain from the

2 R,

experimental data comprising the time histories of the plate velocity and instantaneous film thickness to produce a composite term which is Fourier transformed to create the input term in the spectral analysis.

The method as claimed in any preceding claim where the dynamic equations for the geometric arrangement of the plates are determined and reformulated to provide a constant transfer function and the non-linear spectral analysis algorithm is then Fourier transformed for spectral analysis purposes.

The method as claimed in any preceding claim wherein the preprocessing step includes analysis of inertia effects of the system.

The method as hereinbefore described with reference to the accompanying drawings.

A rheometer system for analysing the dynamic response of sample fluids and semi-solids materials in a squeeze film the system having a first plate having a surface of substantially spherical form and a second plate having a surface of a substantially flat form, the system further including:

- data measurement means having an actuator for providing a relative movement of the plates supporting a sample squeeze film, and transducer means operable to measure an input signal and output signal;

28

- a pre-processing module operable to derive dynamic equations for the geometric arrangement of the plates of the system, and using these dynamic equations to determine a non-linear spectral analysis algorithm describing the nonlinear relationship between the input and output; and

- a spectral analysis processor for performing non-linear spectral analysis of said measured data using the non-linear spectral analysis algorithm to obtain material property result data.

A system as hereinbefore described with reference to the drawings.