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A high shear rate, high pressure microviscometer
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lnternationd Vol. 29. No. 7. pp. 541-557, 1996 Copyright 0 1996 Elscvier Science Ltd Printed in Great Britain, Ali rights resewed 0301-679X/96/$15.00 +O.OO
0301-679X(95)00120-4
A high shear rate, high pressure microviscometer P. L. O’Neill*
and G. W. Stachowiak
The development of a high shear rate, high pressure microviscometer is described. This viscometer was developed specifically to examine a highly loaded, dynamic contact generated in a small volume of fluid. The viscometer was used to examine the fluid film formed by the mechanism of elastohydrodynamic lubrication in the presence of an SAE 50 oil. A complete numerical analysis of the resulting elastohydrodynamic contact was performed. Copyright 0 1996 Elsevier Science Ltd Keywords:
elastohydrodynamic
lubrication,
viscometers,
Introduction The measurement of viscosity is of considerable importance in many areas of engineering and science and several different viscometers are in general use. Different viscometers are suited to different applications. For example, a viscometer may be selected on the basis of the amount of fluid available for analysis, or because the shear rate it applies to the fluid is similar to that applied to the fluid in practical usage. As non-Newtonian fluids require many observations to characterize their behaviour, the experimental facilities necessary for their study are generally more complex and costly than those developed for Newtonian fluids. The most commonly encountered viscometers are the capillary viscometer, the rotational viscometer, and the falling sphere or rolling ball viscometer. When these viscometers have not met the requirements of a particular application they have been modified. For example, capillary tube viscometers for small volumes of fluid have been developed1-3, as have high shear rate4.s, high viscosity6, and high pressure7.x rotational viscometers, and high pressure rolling ball viscometersy. New viscometers have also been developed for special applications. For example, the oscillating magnetically *To whom correspondence should be addressed: Present address: CSIRO Division of Forest Products, Private Bag IO3 Rosebank MDC, Clayton, Australia, 3169 Department of Mechanical and Materials Engineering, The University of Western Australia, Nedlands. Western Australia, 6009 Received 6 June 1994: revised 18 November 1995; accepted 11 December 1995
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viscometry
suspended sphere viscometer”‘, the band viscometer”, and the impact microviscometer’2.
High shear rate, high pressure microviscometer The design of the high shear rate, high pressure microviscometer was based on an impact microviscometer originally developed by Pauliz. The original apparatus used the phenomenon of interferometry to examine a lhtid film formed when two glass surfaces were pressed into contact with a droplet of fluid in between. The important difference between the impact microviscometer and other devices which use optical interferometry to analyse fluid contact is that the impact microviscometer used two separate beams of light at two different angles of incidence to illuminate the contact. If only one beam of light was used then the optical film thickness (n x h) would be found at the point of illumination. When two beams of light were used it was possible to evaluate the refractive index (n) and the real height (!z) of the fluid film independently. In the viscometer developed by Pauli the viscosity of very small volumes of fluid could be measured; however, the shear rates which could be applied were extremely low. In a first design the shear rate resulted purely from the squeezing out of the fluid from the contact region and was generally less than 1 ss’. In a modified designi the ball could be rotated after an entrapment was formed, but this could achieve a maximum shear rate of only 100 SC’. For this study the original impact microviscometer was modified so International
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A high shear rate microviscometer:
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that high rates of shear, of the order of 100000 s-r, which is typical of elastohydrodynamic lubrication, could be applied to the droplet of fluid. A schematic diagram of the high shear rate, high pressure microviscometer is shown in Fig 1. The high shear rate, high pressure microviscometer consists of a glass ball which is held in a steel shaft. The shaft has an air turbine and a collar machined in one end, and is supported by an air bearing. The weight of the collar applies a significant load to the contact region. Air supplied to the apparatus has three functions: (1) (2) (3)
to lubricate the air bearing, to raise and lower the shaft assembly, and to drive the air turbine, resulting in the rotational motion of the ball.
Below the shaft a glass hemisphere is mounted, and a droplet of fluid is placed on it. When the air is turned on, the glass ball is raised off the surface of the glass hemisphere, and the shaft begins to rotate. The rotational speed of the glass ball in the shaft assembly is measured via an optical pick up. When the turbine is rotating at the required velocity it is dropped onto the glass hemisphere, and so applies a high rate of shear to the fluid. The tilt applied to the shaft as shown in Fig 1 increases the radius of rotation
of the sphere at the point of contact between the sphere and the glass hemisphere, thereby providing the required high shear rate. The mirrors and lenses are arranged beneath the hemisphere to project the laser light into the contact between the ball and hemisphere, and also to reflect the resulting interferometric images into the camera. The laser used was a HeNe laser rated at 3.5 mW power. The wavelength of the light from this laser is 632.8 nm. The optical interferometry technique described in References 12 and 13 is used in the high shear rate, high pressure microviscometer to obtain two images of the contact region so that the refractive index (n) and the real height (h) of the fluid within the contact region could be determined independently. Examples of interferometric images taken in the apparatus are shown in Fig 2. These are two sets of interference fringes taken at two different angles of illumination, one at an angle of 7’, and the other at an angle of 28’. From the images the characteristic horseshoe shape of an elastohydrodynamic contact is evident. From these two images the fluid film thickness (/z) and the refractive index (n) for all points in the contact can be found. The refractive index at the point (~,y) is determined from the equation: n* (N sir?&
ng =
- ~zsinzt3i) (11
CM-W
and the fluid film thickness at the point determined from the equation:
2
(~,y)
is
(nz - &in*e,)
where Cl1and Cl2are the angles of illumination of the incident laser light, n3 is the refractive index of the fluid at the point (x, y), n2 is the refractive index of the glass and A is the wavelength of the incident light. This nomenclature’ is illustrated in Fig 3. !Vr and N2 are the fringe orders found from the.images shown in Fig 2 at the position (x, y). The derivation of these two equations is described in Appendix I.
Method of analysing experimental
data
The viscosity distribution across the contact is found from the Reynolds equation. The full Reynolds equation for compressible or incompressible flow is of the form:
=6
Fig 1 Schematic diagram pressure microviscometer 548
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& CP~(U,+ &I) + ; (NV, + Vz)l+ 4~~~~
1
where p is pressure in the lubricating film (Pa), x is distance in the direction of sliding (m), y is distance in the direction normal to sliding in the plane of the oil film (m), q is fluid viscosity (Pa.s), h is hydrodynamic film thickness (m), p is density of the fluid (kg mp3), t is t ime (s), U,, U2 are roof (z = h) and floor (z = 0) surface velocities respectively, x direction 7 1996
A high shear rate microviscometer:
Fig 2 im !age.s obtained from incidence : angle
P. L. O’Neili and G. W. Stachowiak
the high shear rate, high pressure microviscometer:
(m s-‘), and VI, Vz are roof (z = h) and floor (z = 0) surfac :e velocities respectively, y direction (m SC’). In the contact formed in the high shear rate, high pressure microviscometer, the velocities VI and Vz are taken to be zero and the squeeze film term (2d(ph)/ dt) is COI rsidered to be negligible. The assumption that the sque eze film term is negligible was based on a comparis #on between consecutive images of the contact region tatken with the apparatus in operation. It was found th [at there was no difference between images captured during an experiment lasting for five seconds. Tribology
(a) T incidence angle, (b) 2r!P
This length of time is greater than the duration of the actual experiment. The position and number of the fringes in the image denote a particular value of opt Lical film thickness (n X h). However, the refractive in .dex n is dependent only on the density p of the fl uid. Therefore, if the product (n x h) is constant at aI-lY position with time, then the product (p x h) nNIst also be constant with time, and its first derivative \uith respect to time (d(ph)/dt) must be zero. Thereff ore, for the purposes of analysing the data from the high shear rate, high pressure microviscometer, the Reynolds equation reduces to the form: International
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A high shear rate microviscometer:
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(l), and the density can then be found from the Lorentz-Lorenz equation: G-11 --EK n*+lp
(3)
(61
where n is the refractive index of the fluid, p is its density in kg me3 and K is a constant for any given fluid measured in m3 kg-l. The accuracy of this equation has been confirmed for pressures up to 1.4 GPa for paraffin oil and 0.7 GPa for glycerine14. The pressure variation with x and y. This can be found from the elasticity equation. The expression for fluid film thickness in an elastohydrodynamic contact can be written as follows?
where ho is a constant equal to the centre line film thickness at any instant measured from the base of the undeformed sphere to the undeformed plane (m); S(x,y) is the separation due to the geometry of the undeformed solids (m), and d(x,y) is the elastic deformation (m).
Fig 3 Description of nomenclature used in the equations for the caLculation of refractive index and fluid film thickness
These three components are illustrated in Fig 4.
The separation of the surfaces due to the geometry of the undeformed solids in a contact between a sphere and a flat surface is given by:
where U = U1 + UZ. This equation can be non-dimensionalized following conversions:
of fluid film thickness
using the
..2
L
.*2
where R is the radius of the sphere (m) and x,y the distance of the point of interest from the centre of the contact region in the x and y directions respectively (m).
The Reynolds equation then becomes:
The separation due to elastic deformation can be determined by solving the potential equation for distortion given by16:
where
Although Equation (5) cannot be solved by analytical methods, a solution can be found using numerical methods. In this study the equation is solved using a finite difference scheme to approximate the partial derivatives. The finite difference scheme used is described in Appendix II. The following information must be obtained before Equation (5) can be solved by finite difference methods: (1) (2)
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The variation in the height of the fluid film across the contact. This is found from Equation w The density variation across the contact. This can be found from the Lorentz-Lorenz equation which is an accurate approximation for the relationship between density and refractive index. The refractive index can be found from Equation Tribology
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Fig 4 Components ness 7 1996
of elastohydrodynamic
film
thick-
A high shear rate microviscometer:
The section chosen is that marked by the horizontal line along the image in Fig 2, and Equations (1) and (2) were used for the calculations. The longitudinal section of an elastohydrodynamic fluid film is characterized by a region of constant film thickness at the centre of the contact and a constriction at the exit. The film thickness over the central region is called the central film thickness, while the film thickness at the constriction is the minimum film thickness. From Fig 5 the constant film thickness in the centre of the contact is evident; however, the constriction at the exit is very small.
where Er is the reduced Youngs modulus (Pa), p(~,y) the non-uniform pressure distribution (Pa), and A the contact area (mz). Equation 9 gives the distortion at the point (x’, y’) due to the non-uniform pressure distribution over the area A. This equation is solved using an iteration process. The pressure is distributed across small rectangular blocks, and it is assumed that the pressure across each block is uniform. The mesh used in the solution of this equation is shown in Appendix III.
The constriction at the exit of elastohydrodynamic contacts gives interferometric images a characteristic ‘horseshoe’ shape, as can be seen in the images shown in Fig 2. The order of the fringes in an interferometric image is due to the product of the refractive index and the film thickness. Therefore a graph of the product of these two quantities should show the presence of the constriction which is seen on the images. The graph of the product of refractive index and film thickness for the section indicated by the line along Fig 2 is shown in Fig 6. From this graph the constriction at the exit of the elastohydrodynamic contact is clear.
Now, with the information obtained on the film thickness, density and pressure distributions across the contact, the Reynolds equation can be solved for viscosity using finite difference methods.
Experimental
results
The high shear rate, high pressure microviscometer was tested by analysing SAE 50 standard oil. The interferometric images obtained under a shear rate of 0.62 ms-i and a load of 5 N are shown in Fig 2. Tiie information obtained from these images was used to determine: (1) (2) (3)
There are two points to note about the density values shown in Fig 5. The first is that at the exit or cavitation region the density is assumed to be zero. In the cavitation region of an elastohydrodynamic contact the pressure is negative, and the solution of the Lorentz-Lorenz equation for density will also be negative. Rather than retain these negative values, they are set to zero. The second point is that some of the density values are very large. They are as much as twice that measured at atmospheric pressure, and much greater than that which would be predicted from established compressibility values. This difference may be attributed in part to rounding off errors in the
the fluid film thickness and density across the contact, the pressure distribution across the contact, and the viscosity across the contact.
Density and film thickness variation elastohydrodynamic contact
in
The variation in density and film thickness across a longitudinal section of the contact is shown in Fig 5.
20
40
position
Fig 5 Film thickness and density variation rate of 0.62 rns-l and a load of 5 N
along
60
x axis
P. L. O’Neill and G. W. Stachowiak
60
100
(non-dimensional)
calculated along a longitudinal Tribology
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A high shear rate microviscometer:
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0 0
20
40
position
along
60
x axis
60
Fig 6 Product of .film thickness and refractive index calculated along a longitudinal to-a shear rate &f-O.62 rns-l and a load of 5 N calculation; however, the experimental apparatus developed here is unique in that it is assessing directly a highly dynamic contact. The apparatus is operating at shear rates of the order of 100000 s-l. In comparison, other densitometers are static devices, which rarely apply any shear rate other than the squeezing out of the fluid induced by the high applied loads. It is known that dynamic contacts differ significantly from static contacts and that one cannot predict the behaviour of fluids in dynamic contacts from results obtained on static contacts. Therefore, the density of a fluid in a highly loaded, dynamic contact may, in fact, be higher than is generally found in static or quasi-static densitometers. The density and film thickness variation across the transverse section of the contact marked in Fig 2 is shown in Fig 7. From the film thickness distribution it can be seen that there is a very slight constriction at the centre of the contact. This is also a characteristic feature of elastohydrodynamic contacts. Pressure distribution contact
in elastohydrodynamic
Based on the calculated values of film thickness, the pressure distribution over the contact was determined. The computed pressure distribution across the longitudinal and transverse sections marked in Fig 2 are given in Figs 8 and 9 respectively. The pressure distribution along a longitudinal section of an elastohydrodynamic fluid film is characterized by a pressure spike. The pressure spike occurs just before the constriction at the exit of the contact. In this analysis it was found that the pressure spike was located after the constriction, i.e. at the point where the film thickness started to increase. In the case analysed in this study the constriction in the thickness of the fluid film at the exit is only very small and this could 552
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(non-dimensional)
section of a contact subjected
explain the location of the pressure spike. Other irregularities in the pressure distribution of Figs 8 and 9 can be attributed to numerical errors in the calculations. The position of the pressure spike can also be explained by comparing the actual film thickness found from the experiments by fringe counting to the film thickness which would result under the calculated pressure. This comparison is shown in Fig 10 for the longitudinal section and Fig 11 for the transverse section. The discrepancy between the actual and calculated height is due to the fact that negative pressures were set to zero in the calculation. It can be seen from Fig 10 that at the point where the actual film thickness starts to increase, the film thickness calculated from the pressure distribution suddenly decreases, so the calculated pressure distribution, with all negative pressures set to zero, predicts a constriction which the fringe counting technique does not show to be present. The actual and calculated heights found for the transverse section are virtually equal. Figure 9 shows the pressure distribution across the transverse section of the contact. This pressure distribution is very irregular at the edges of the contact. This is due to the fact that the methods used to determine the pressure distribution are approximate and numerical errors may be introduced to the solution. However, the computed pressure distribution does show the general trends found in elastohydrodynamic contacts. The highest pressure is found at the centre of the contact, and the pressure decreases towards the edges. Viscosity contact
distribution
in the elastohydrodynamic
The viscosity in the contact was found by solving the Reynolds equation using finite difference methods. 7 1996
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-10
-a
-6
-4
-2
position
along
Fig 7 Density and film thickness variation 0.62 rns-l and a loid of 5 N
0
y axis
2
4
P, L. O’Neill
a
6
and G. W. Stachowiak
10
(non-dimensional)
across a transverse section of a contact subjected to a shear rate of
0 20
40
position
along
Fig 8 Pressure and fdm thickness distribution shear rate of 0.62 ms-’ and a load of 5 N
60
x axis
(2)
100
(non-dimensional)
calculated for the longitudinal
The following problems were encountered in evaluating the viscosity in the contact: (1)
80
It can be seen from Fig 8 and especially from Fig 9 that the pressure distribution found does not follow a smooth curve. Solving the Reynolds equation requires the evaluation of the first and second partial derivatives of pressure using finite difference methods. If the pressure distribution is irregular then the partial derivatives evaluated from the pressure values are also irregular. If the pressure is constant in the contact then all the information on the left-hand side of the Tribology
section of the contact subjected to a
Reynolds equation (5) is zero and the information contained in the Reynolds equation is lost. That is, a singularity exists if the pressure is constant. In this analysis at some points within the contact the pressure distribution was approximately constant and the viscosity calculated by finite difference methods was found to be identically equal to zero, even at very high pressures, In order to overcome these two problems the following conditions were placed on the values of viscosity obtained: (1) The dimensionless viscosity q’ could not be less than 1. That is, the viscosity could not be less International
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0.6
,
-10
’
-a
,
.
-6
I
’
-2
position
along
Fig 9 Pressure and film thickness distribution shear rate of 0.62 ms-r and a load of 5 N
I
0
-4
’
2
y axis
4
a
6
10
(non-dimensional)
calculated for the transverse section
v
actual
height
calculated
4
qf the contact subjected to a
height
0 0
40
20
position
along
60
x axis
a0
(non-dimensional)
Fig 10 Comparison between the actual film thickness determined from the experiments and the film thickness which would theoretically result from the applied pressure (longitudinal section)
(2)
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than that which the fluid would have under an equivalent shear rate at atmospheric pressure. Any values of viscosity found to be less than 1 were set equal to 1. If the viscosity was less than the viscosity of the neighbouring point, but the pressure was greater than at the neighbouring point, then the viscosity was set equal to the higher viscosity. That is, a singularity has occurred at that point, so the viscosity takes on the value it had before entering the singularity. Tribology
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The values calculated for viscosity along the longitudinal and transverse sections of the contact are shown in Figs 12 and 13 respectively. The irregularity in the pressure distributions shown in Figs 7 and 8 is reflected in the irregularity of the viscosity distributions. Conclusion In this study the method of optical interferometry was used to evaluate the pressure, film thickness, density and viscosity distributions found in a highly loaded 7 1996
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-6
-0
-4
position
-2
-
actual
,-
calculated
height
4
6
0
along
y axis
P. L. O’Neill and G. W. Stachowiak
2
height
0
10
(non-dimensional)
Fig 11 Comparison between the actual film thickness determined from the experiments and the film thickness which would theoretically result from the applied pressure (transverse section)
60
20
40
position
Fig 12 Viscosiq and pressure distribution
along
60
x axis
across the longitudinal
dynamic elastohydrodynamic contact. Such a complete evaluation of a contact of this type has not been done before using experimental methods. The information obtained can be used to gain a better understanding of elastohydrodynamic contacts and high pressure/high shear rate behaviour of lubricants. There is a drawback in the method of optical interferometry which affects its performance. This limitation is the lack of detail generated in the regions of constant film thickness. For example, the images shown in Fig 2 can be divided into three Tribology
60
100
(non-dimensional)
section of the contact
areas. The first area is the entrance to the contact zone. Here there is a lot of information in the form of interference fringes. However, most of this information is not required in the calculations because in this area pressure and viscosity change slowly. The second area is the area of contact. In this area the two surfaces are virtually parallel, so very little information is available from the interferometric images. However, in this area significant changes are occurring in pressure and viscosity. Therefore, the scarce amount of information is used to generate a significant amount of data. International
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position
along y axis (non-dimensional)
Fig 13 Viscosity and pressure distribution across the transverse section of the contact
The third area is the exit of the contact. From Fig 2 it is evident that in this area the fringes are indistinct and could easily be incorrectly measured. Considering the abundance of information obtained by interferometry in the region where it is not needed, and the paucity of information in the region where it is needed, the method of measuring the fluid film thickness and density is not in accord with the way in which the data is analysed. The ideal would be the situation where the method of measuring variations in film thickness and density is more accurate over areas of constant or approximately constant film thickness. Despite this limitation, the optical interferometry method has been used successfully to completely evaluate a highly loaded, dynamic lubricated contact using only a very small volume of fluid.
10. Leyh C. and Ritter RX. New viscosity measurement: the oscillating magnetically suspended sphere. Rev. Sci. Instrum. 1984, 55, 4, 570 11. Hull H.H. The band viscometer. J. Coff. Sci. 1952, 7, 316 12. Paul G.R. Optical determination of the high pressure refractive index and viscosity of liquids entrapped in point contacts, Ph.D. Thesis, University of London, 1972 13. Paul G.R. and Cameron A. The ultimate shear stress of fluids at high pressures measured by a modified impact microviscometer. Proc. Roy. Sot., Lond. Ser. A, 1975, 346, 227-244 14. Poulter T.C., Rotchey C. and Benz C.A. The effect of pressure on the index of refraction of paraffin oil and glycerine. Phys. Rev. 1932, 41, 366-367 15. Hamruck B.J. and Dowson D. Ball Bearing Lubrication: the Elastohydrodynamics of Elliptical Contact, Wiley, New York, 1981 16. Ranger A-P., Ettles C.M. and Cameron A. The solution of the point contact elastohydrodynamic problem, Proc. Roy. Sot. Lond. Ser A, 1975, 346, 227-244
References 1. Geist J.M. and Cannon M.R. Ind. Eng. Chem. (Anal) 1946, 18, 611 2. Lillard, J.G. Anal. Chem. 1952, 24, 1042 3. Lidstone, F.M. A microviscometer of improved design. Chem. Ind. 1952, 873-874 4. Asbeck W.K., Laidermann D.D. and Van Lw M. A high shear method of rating brushability of paints. J. CON. Sci. 1952, 7, 306 5. Merrill E.W. A coaxial cylinder viscometer for the study of fluids under high velocity gradients. J. Coil. Sci. 1954, 9, 7 6. Markovitz H., Elyash L.J., Pudden Jr. F.J. and Dewitt T.W. A cone-and-plate viscometer. J. Coil. Sci. 1955, 10, 165 7. Bair S. and Winer W.O. The high shear stress rheology of liquid lubricants at pressures of 2 to 200 MPa. J. Tribol. 1990, 112, 246-253 8. Cohen A. and Richon D. New apparatus for simultaneous determination of phase equilibria and rheological properties of fluids at high pressures: its application to coal paste studies up to 773K and 30MPa. Rev. Sci. Instrum. 1986, 57, 1192-1195 9. Sawamura S., Takeuchi N., Kitamura K. and Taniguchi Y. High pressure rolling-ball viscometer of a corrosion resistant type. Rev. Sci. Instrum. 1990, 61, 871-873
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Appendix I. Derivation of equations for film thickness and refractive index in an interferometric contact Figure Al shows the contact formed in the high shear rate, high pressure microviscometer. A beam of light is directed into the fluid-filled contact between the glass hemisphere and the glass sphere at an angle 8. The incident beam of light is reflected off the flat surface of the hemisphere, generating the first reflected beam, beam 1. The rest of the light leaves the hemisphere with an angle of refraction 4, passes through the fluid and is reflected off the sphere, generating the second reflected beam, beam 2. The single incident beam of light has been split into two mutually coherent beams which will recombine. The intensity of the recombined beam will depend on the optical path difference between the two separate beams and the phase change which occurs on reflection from the two surfaces. The optical path difference is (2 n3 h cos4) where n3 is the refractive index of the
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A high shear rate microviscometer:
P. L. O’Neill
and G. W. Stachowiak
DYl (x,, Yi-,J Fig A2 Nomenclature
used to define partial derivatives
Appendix II. Finite difference method \ Fig Al Principles of optical interferometry experimental apparatus
bean
1
used in the
fluid and h is the fluid film thickness. As long as the hemisphere and sphere are made of the same material the phase change on reflection can be ignored as it will be the same from both surfaces. The two reflected beams will interfere constructively and produce bright fringes if: NA = 2n3 h cos+
= N1 A
(AlI
where Nr and N2 are the fringe orders found from the two images produced and +r and & are the angles of refraction of the two beams. A third relationship can be found from Snell’s law: sin g2 n3 _ sin0, (A3) sin +2 n2 sin+, where n2 is the refractive index of the glass hemisphere. These three equations, (Al)-(A3), are solved simultaneously to give the equations for determining the refractive index of the fluid and the fluid film thickness from the interferometric images: (N: ~-. sin*B, - %sin*&) ..nf = n$ --..-~ (A4) (N:-Iv2,) h = __-.-~-~ NIA
2 (n$ - n$s.in20,)
af(xk) ax f(xk + ,)‘Dxl’
- f(xk)‘Dx12*Dx2* - f(xk - ,)‘Dm2 Dxl.m2.(DXl+ DX2) (A@
a*fhd ,
where N is the fringe order which can take on any integer value and X is the wavelength of the incident light. The reflected beams will interfere destructively and produce dark fringes if: (N + 1/2)h = 2n3 h cos+ Consider the situation occurring in the experimental rig where two beams of incident light are used to illuminate the contact, one at angle +r and the other at angle &. The equations for bright fringes will be: 2n3hcos+,
Given a function f(x) for which the values f(xk _ r), and f(xk + r) are known, xk _ ,, xk and xk + r being three consecutive non-equally spaced points, and given that DXI is the distance between xk - , and xk and DX2 iS the distance between xk and xk + ,, as shown in Fig A2, the equations for the first and second partial derivative from finite difference methods are:
f(xk)
ax'
f(x k + I).DXl - f(x/&(DXl
+ Dx2) + f(Xk - 1)*DX2 Dxl.DxL!.(DXl+ Dx2) (A7)
Appendix III. Finite difference mesh Figure A3 shows the finite difference mesh used in the analysis. This mesh was chosen as it keeps the number of mesh points low which will make the computing faster, it has a large area, particularly in the entrance zone, and the mesh is fine at the exit where the pressure in an elastohydrodynamic contact changes rapidly.
(A3 Fig A3 Finite difference mesh used in the analysis Tribology
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lnternationd Vol. 29. No. 7. pp. 541-557, 1996 Copyright 0 1996 Elscvier Science Ltd Printed in Great Britain, Ali rights resewed 0301-679X/96/$15.00 +O.OO
0301-679X(95)00120-4
A high shear rate, high pressure microviscometer P. L. O’Neill*
and G. W. Stachowiak
The development of a high shear rate, high pressure microviscometer is described. This viscometer was developed specifically to examine a highly loaded, dynamic contact generated in a small volume of fluid. The viscometer was used to examine the fluid film formed by the mechanism of elastohydrodynamic lubrication in the presence of an SAE 50 oil. A complete numerical analysis of the resulting elastohydrodynamic contact was performed. Copyright 0 1996 Elsevier Science Ltd Keywords:
elastohydrodynamic
lubrication,
viscometers,
Introduction The measurement of viscosity is of considerable importance in many areas of engineering and science and several different viscometers are in general use. Different viscometers are suited to different applications. For example, a viscometer may be selected on the basis of the amount of fluid available for analysis, or because the shear rate it applies to the fluid is similar to that applied to the fluid in practical usage. As non-Newtonian fluids require many observations to characterize their behaviour, the experimental facilities necessary for their study are generally more complex and costly than those developed for Newtonian fluids. The most commonly encountered viscometers are the capillary viscometer, the rotational viscometer, and the falling sphere or rolling ball viscometer. When these viscometers have not met the requirements of a particular application they have been modified. For example, capillary tube viscometers for small volumes of fluid have been developed1-3, as have high shear rate4.s, high viscosity6, and high pressure7.x rotational viscometers, and high pressure rolling ball viscometersy. New viscometers have also been developed for special applications. For example, the oscillating magnetically *To whom correspondence should be addressed: Present address: CSIRO Division of Forest Products, Private Bag IO3 Rosebank MDC, Clayton, Australia, 3169 Department of Mechanical and Materials Engineering, The University of Western Australia, Nedlands. Western Australia, 6009 Received 6 June 1994: revised 18 November 1995; accepted 11 December 1995
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viscometry
suspended sphere viscometer”‘, the band viscometer”, and the impact microviscometer’2.
High shear rate, high pressure microviscometer The design of the high shear rate, high pressure microviscometer was based on an impact microviscometer originally developed by Pauliz. The original apparatus used the phenomenon of interferometry to examine a lhtid film formed when two glass surfaces were pressed into contact with a droplet of fluid in between. The important difference between the impact microviscometer and other devices which use optical interferometry to analyse fluid contact is that the impact microviscometer used two separate beams of light at two different angles of incidence to illuminate the contact. If only one beam of light was used then the optical film thickness (n x h) would be found at the point of illumination. When two beams of light were used it was possible to evaluate the refractive index (n) and the real height (!z) of the fluid film independently. In the viscometer developed by Pauli the viscosity of very small volumes of fluid could be measured; however, the shear rates which could be applied were extremely low. In a first design the shear rate resulted purely from the squeezing out of the fluid from the contact region and was generally less than 1 ss’. In a modified designi the ball could be rotated after an entrapment was formed, but this could achieve a maximum shear rate of only 100 SC’. For this study the original impact microviscometer was modified so International
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A high shear rate microviscometer:
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that high rates of shear, of the order of 100000 s-r, which is typical of elastohydrodynamic lubrication, could be applied to the droplet of fluid. A schematic diagram of the high shear rate, high pressure microviscometer is shown in Fig 1. The high shear rate, high pressure microviscometer consists of a glass ball which is held in a steel shaft. The shaft has an air turbine and a collar machined in one end, and is supported by an air bearing. The weight of the collar applies a significant load to the contact region. Air supplied to the apparatus has three functions: (1) (2) (3)
to lubricate the air bearing, to raise and lower the shaft assembly, and to drive the air turbine, resulting in the rotational motion of the ball.
Below the shaft a glass hemisphere is mounted, and a droplet of fluid is placed on it. When the air is turned on, the glass ball is raised off the surface of the glass hemisphere, and the shaft begins to rotate. The rotational speed of the glass ball in the shaft assembly is measured via an optical pick up. When the turbine is rotating at the required velocity it is dropped onto the glass hemisphere, and so applies a high rate of shear to the fluid. The tilt applied to the shaft as shown in Fig 1 increases the radius of rotation
of the sphere at the point of contact between the sphere and the glass hemisphere, thereby providing the required high shear rate. The mirrors and lenses are arranged beneath the hemisphere to project the laser light into the contact between the ball and hemisphere, and also to reflect the resulting interferometric images into the camera. The laser used was a HeNe laser rated at 3.5 mW power. The wavelength of the light from this laser is 632.8 nm. The optical interferometry technique described in References 12 and 13 is used in the high shear rate, high pressure microviscometer to obtain two images of the contact region so that the refractive index (n) and the real height (h) of the fluid within the contact region could be determined independently. Examples of interferometric images taken in the apparatus are shown in Fig 2. These are two sets of interference fringes taken at two different angles of illumination, one at an angle of 7’, and the other at an angle of 28’. From the images the characteristic horseshoe shape of an elastohydrodynamic contact is evident. From these two images the fluid film thickness (/z) and the refractive index (n) for all points in the contact can be found. The refractive index at the point (~,y) is determined from the equation: n* (N sir?&
ng =
- ~zsinzt3i) (11
CM-W
and the fluid film thickness at the point determined from the equation:
2
(~,y)
is
(nz - &in*e,)
where Cl1and Cl2are the angles of illumination of the incident laser light, n3 is the refractive index of the fluid at the point (x, y), n2 is the refractive index of the glass and A is the wavelength of the incident light. This nomenclature’ is illustrated in Fig 3. !Vr and N2 are the fringe orders found from the.images shown in Fig 2 at the position (x, y). The derivation of these two equations is described in Appendix I.
Method of analysing experimental
data
The viscosity distribution across the contact is found from the Reynolds equation. The full Reynolds equation for compressible or incompressible flow is of the form:
=6
Fig 1 Schematic diagram pressure microviscometer 548
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& CP~(U,+ &I) + ; (NV, + Vz)l+ 4~~~~
1
where p is pressure in the lubricating film (Pa), x is distance in the direction of sliding (m), y is distance in the direction normal to sliding in the plane of the oil film (m), q is fluid viscosity (Pa.s), h is hydrodynamic film thickness (m), p is density of the fluid (kg mp3), t is t ime (s), U,, U2 are roof (z = h) and floor (z = 0) surface velocities respectively, x direction 7 1996
A high shear rate microviscometer:
Fig 2 im !age.s obtained from incidence : angle
P. L. O’Neili and G. W. Stachowiak
the high shear rate, high pressure microviscometer:
(m s-‘), and VI, Vz are roof (z = h) and floor (z = 0) surfac :e velocities respectively, y direction (m SC’). In the contact formed in the high shear rate, high pressure microviscometer, the velocities VI and Vz are taken to be zero and the squeeze film term (2d(ph)/ dt) is COI rsidered to be negligible. The assumption that the sque eze film term is negligible was based on a comparis #on between consecutive images of the contact region tatken with the apparatus in operation. It was found th [at there was no difference between images captured during an experiment lasting for five seconds. Tribology
(a) T incidence angle, (b) 2r!P
This length of time is greater than the duration of the actual experiment. The position and number of the fringes in the image denote a particular value of opt Lical film thickness (n X h). However, the refractive in .dex n is dependent only on the density p of the fl uid. Therefore, if the product (n x h) is constant at aI-lY position with time, then the product (p x h) nNIst also be constant with time, and its first derivative \uith respect to time (d(ph)/dt) must be zero. Thereff ore, for the purposes of analysing the data from the high shear rate, high pressure microviscometer, the Reynolds equation reduces to the form: International
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(l), and the density can then be found from the Lorentz-Lorenz equation: G-11 --EK n*+lp
(3)
(61
where n is the refractive index of the fluid, p is its density in kg me3 and K is a constant for any given fluid measured in m3 kg-l. The accuracy of this equation has been confirmed for pressures up to 1.4 GPa for paraffin oil and 0.7 GPa for glycerine14. The pressure variation with x and y. This can be found from the elasticity equation. The expression for fluid film thickness in an elastohydrodynamic contact can be written as follows?
where ho is a constant equal to the centre line film thickness at any instant measured from the base of the undeformed sphere to the undeformed plane (m); S(x,y) is the separation due to the geometry of the undeformed solids (m), and d(x,y) is the elastic deformation (m).
Fig 3 Description of nomenclature used in the equations for the caLculation of refractive index and fluid film thickness
These three components are illustrated in Fig 4.
The separation of the surfaces due to the geometry of the undeformed solids in a contact between a sphere and a flat surface is given by:
where U = U1 + UZ. This equation can be non-dimensionalized following conversions:
of fluid film thickness
using the
..2
L
.*2
where R is the radius of the sphere (m) and x,y the distance of the point of interest from the centre of the contact region in the x and y directions respectively (m).
The Reynolds equation then becomes:
The separation due to elastic deformation can be determined by solving the potential equation for distortion given by16:
where
Although Equation (5) cannot be solved by analytical methods, a solution can be found using numerical methods. In this study the equation is solved using a finite difference scheme to approximate the partial derivatives. The finite difference scheme used is described in Appendix II. The following information must be obtained before Equation (5) can be solved by finite difference methods: (1) (2)
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The variation in the height of the fluid film across the contact. This is found from Equation w The density variation across the contact. This can be found from the Lorentz-Lorenz equation which is an accurate approximation for the relationship between density and refractive index. The refractive index can be found from Equation Tribology
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Fig 4 Components ness 7 1996
of elastohydrodynamic
film
thick-
A high shear rate microviscometer:
The section chosen is that marked by the horizontal line along the image in Fig 2, and Equations (1) and (2) were used for the calculations. The longitudinal section of an elastohydrodynamic fluid film is characterized by a region of constant film thickness at the centre of the contact and a constriction at the exit. The film thickness over the central region is called the central film thickness, while the film thickness at the constriction is the minimum film thickness. From Fig 5 the constant film thickness in the centre of the contact is evident; however, the constriction at the exit is very small.
where Er is the reduced Youngs modulus (Pa), p(~,y) the non-uniform pressure distribution (Pa), and A the contact area (mz). Equation 9 gives the distortion at the point (x’, y’) due to the non-uniform pressure distribution over the area A. This equation is solved using an iteration process. The pressure is distributed across small rectangular blocks, and it is assumed that the pressure across each block is uniform. The mesh used in the solution of this equation is shown in Appendix III.
The constriction at the exit of elastohydrodynamic contacts gives interferometric images a characteristic ‘horseshoe’ shape, as can be seen in the images shown in Fig 2. The order of the fringes in an interferometric image is due to the product of the refractive index and the film thickness. Therefore a graph of the product of these two quantities should show the presence of the constriction which is seen on the images. The graph of the product of refractive index and film thickness for the section indicated by the line along Fig 2 is shown in Fig 6. From this graph the constriction at the exit of the elastohydrodynamic contact is clear.
Now, with the information obtained on the film thickness, density and pressure distributions across the contact, the Reynolds equation can be solved for viscosity using finite difference methods.
Experimental
results
The high shear rate, high pressure microviscometer was tested by analysing SAE 50 standard oil. The interferometric images obtained under a shear rate of 0.62 ms-i and a load of 5 N are shown in Fig 2. Tiie information obtained from these images was used to determine: (1) (2) (3)
There are two points to note about the density values shown in Fig 5. The first is that at the exit or cavitation region the density is assumed to be zero. In the cavitation region of an elastohydrodynamic contact the pressure is negative, and the solution of the Lorentz-Lorenz equation for density will also be negative. Rather than retain these negative values, they are set to zero. The second point is that some of the density values are very large. They are as much as twice that measured at atmospheric pressure, and much greater than that which would be predicted from established compressibility values. This difference may be attributed in part to rounding off errors in the
the fluid film thickness and density across the contact, the pressure distribution across the contact, and the viscosity across the contact.
Density and film thickness variation elastohydrodynamic contact
in
The variation in density and film thickness across a longitudinal section of the contact is shown in Fig 5.
20
40
position
Fig 5 Film thickness and density variation rate of 0.62 rns-l and a load of 5 N
along
60
x axis
P. L. O’Neill and G. W. Stachowiak
60
100
(non-dimensional)
calculated along a longitudinal Tribology
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0 0
20
40
position
along
60
x axis
60
Fig 6 Product of .film thickness and refractive index calculated along a longitudinal to-a shear rate &f-O.62 rns-l and a load of 5 N calculation; however, the experimental apparatus developed here is unique in that it is assessing directly a highly dynamic contact. The apparatus is operating at shear rates of the order of 100000 s-l. In comparison, other densitometers are static devices, which rarely apply any shear rate other than the squeezing out of the fluid induced by the high applied loads. It is known that dynamic contacts differ significantly from static contacts and that one cannot predict the behaviour of fluids in dynamic contacts from results obtained on static contacts. Therefore, the density of a fluid in a highly loaded, dynamic contact may, in fact, be higher than is generally found in static or quasi-static densitometers. The density and film thickness variation across the transverse section of the contact marked in Fig 2 is shown in Fig 7. From the film thickness distribution it can be seen that there is a very slight constriction at the centre of the contact. This is also a characteristic feature of elastohydrodynamic contacts. Pressure distribution contact
in elastohydrodynamic
Based on the calculated values of film thickness, the pressure distribution over the contact was determined. The computed pressure distribution across the longitudinal and transverse sections marked in Fig 2 are given in Figs 8 and 9 respectively. The pressure distribution along a longitudinal section of an elastohydrodynamic fluid film is characterized by a pressure spike. The pressure spike occurs just before the constriction at the exit of the contact. In this analysis it was found that the pressure spike was located after the constriction, i.e. at the point where the film thickness started to increase. In the case analysed in this study the constriction in the thickness of the fluid film at the exit is only very small and this could 552
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(non-dimensional)
section of a contact subjected
explain the location of the pressure spike. Other irregularities in the pressure distribution of Figs 8 and 9 can be attributed to numerical errors in the calculations. The position of the pressure spike can also be explained by comparing the actual film thickness found from the experiments by fringe counting to the film thickness which would result under the calculated pressure. This comparison is shown in Fig 10 for the longitudinal section and Fig 11 for the transverse section. The discrepancy between the actual and calculated height is due to the fact that negative pressures were set to zero in the calculation. It can be seen from Fig 10 that at the point where the actual film thickness starts to increase, the film thickness calculated from the pressure distribution suddenly decreases, so the calculated pressure distribution, with all negative pressures set to zero, predicts a constriction which the fringe counting technique does not show to be present. The actual and calculated heights found for the transverse section are virtually equal. Figure 9 shows the pressure distribution across the transverse section of the contact. This pressure distribution is very irregular at the edges of the contact. This is due to the fact that the methods used to determine the pressure distribution are approximate and numerical errors may be introduced to the solution. However, the computed pressure distribution does show the general trends found in elastohydrodynamic contacts. The highest pressure is found at the centre of the contact, and the pressure decreases towards the edges. Viscosity contact
distribution
in the elastohydrodynamic
The viscosity in the contact was found by solving the Reynolds equation using finite difference methods. 7 1996
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-10
-a
-6
-4
-2
position
along
Fig 7 Density and film thickness variation 0.62 rns-l and a loid of 5 N
0
y axis
2
4
P, L. O’Neill
a
6
and G. W. Stachowiak
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(non-dimensional)
across a transverse section of a contact subjected to a shear rate of
0 20
40
position
along
Fig 8 Pressure and fdm thickness distribution shear rate of 0.62 ms-’ and a load of 5 N
60
x axis
(2)
100
(non-dimensional)
calculated for the longitudinal
The following problems were encountered in evaluating the viscosity in the contact: (1)
80
It can be seen from Fig 8 and especially from Fig 9 that the pressure distribution found does not follow a smooth curve. Solving the Reynolds equation requires the evaluation of the first and second partial derivatives of pressure using finite difference methods. If the pressure distribution is irregular then the partial derivatives evaluated from the pressure values are also irregular. If the pressure is constant in the contact then all the information on the left-hand side of the Tribology
section of the contact subjected to a
Reynolds equation (5) is zero and the information contained in the Reynolds equation is lost. That is, a singularity exists if the pressure is constant. In this analysis at some points within the contact the pressure distribution was approximately constant and the viscosity calculated by finite difference methods was found to be identically equal to zero, even at very high pressures, In order to overcome these two problems the following conditions were placed on the values of viscosity obtained: (1) The dimensionless viscosity q’ could not be less than 1. That is, the viscosity could not be less International
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0.6
,
-10
’
-a
,
.
-6
I
’
-2
position
along
Fig 9 Pressure and film thickness distribution shear rate of 0.62 ms-r and a load of 5 N
I
0
-4
’
2
y axis
4
a
6
10
(non-dimensional)
calculated for the transverse section
v
actual
height
calculated
4
qf the contact subjected to a
height
0 0
40
20
position
along
60
x axis
a0
(non-dimensional)
Fig 10 Comparison between the actual film thickness determined from the experiments and the film thickness which would theoretically result from the applied pressure (longitudinal section)
(2)
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than that which the fluid would have under an equivalent shear rate at atmospheric pressure. Any values of viscosity found to be less than 1 were set equal to 1. If the viscosity was less than the viscosity of the neighbouring point, but the pressure was greater than at the neighbouring point, then the viscosity was set equal to the higher viscosity. That is, a singularity has occurred at that point, so the viscosity takes on the value it had before entering the singularity. Tribology
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The values calculated for viscosity along the longitudinal and transverse sections of the contact are shown in Figs 12 and 13 respectively. The irregularity in the pressure distributions shown in Figs 7 and 8 is reflected in the irregularity of the viscosity distributions. Conclusion In this study the method of optical interferometry was used to evaluate the pressure, film thickness, density and viscosity distributions found in a highly loaded 7 1996
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-6
-0
-4
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-2
-
actual
,-
calculated
height
4
6
0
along
y axis
P. L. O’Neill and G. W. Stachowiak
2
height
0
10
(non-dimensional)
Fig 11 Comparison between the actual film thickness determined from the experiments and the film thickness which would theoretically result from the applied pressure (transverse section)
60
20
40
position
Fig 12 Viscosiq and pressure distribution
along
60
x axis
across the longitudinal
dynamic elastohydrodynamic contact. Such a complete evaluation of a contact of this type has not been done before using experimental methods. The information obtained can be used to gain a better understanding of elastohydrodynamic contacts and high pressure/high shear rate behaviour of lubricants. There is a drawback in the method of optical interferometry which affects its performance. This limitation is the lack of detail generated in the regions of constant film thickness. For example, the images shown in Fig 2 can be divided into three Tribology
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(non-dimensional)
section of the contact
areas. The first area is the entrance to the contact zone. Here there is a lot of information in the form of interference fringes. However, most of this information is not required in the calculations because in this area pressure and viscosity change slowly. The second area is the area of contact. In this area the two surfaces are virtually parallel, so very little information is available from the interferometric images. However, in this area significant changes are occurring in pressure and viscosity. Therefore, the scarce amount of information is used to generate a significant amount of data. International
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position
along y axis (non-dimensional)
Fig 13 Viscosity and pressure distribution across the transverse section of the contact
The third area is the exit of the contact. From Fig 2 it is evident that in this area the fringes are indistinct and could easily be incorrectly measured. Considering the abundance of information obtained by interferometry in the region where it is not needed, and the paucity of information in the region where it is needed, the method of measuring the fluid film thickness and density is not in accord with the way in which the data is analysed. The ideal would be the situation where the method of measuring variations in film thickness and density is more accurate over areas of constant or approximately constant film thickness. Despite this limitation, the optical interferometry method has been used successfully to completely evaluate a highly loaded, dynamic lubricated contact using only a very small volume of fluid.
10. Leyh C. and Ritter RX. New viscosity measurement: the oscillating magnetically suspended sphere. Rev. Sci. Instrum. 1984, 55, 4, 570 11. Hull H.H. The band viscometer. J. Coff. Sci. 1952, 7, 316 12. Paul G.R. Optical determination of the high pressure refractive index and viscosity of liquids entrapped in point contacts, Ph.D. Thesis, University of London, 1972 13. Paul G.R. and Cameron A. The ultimate shear stress of fluids at high pressures measured by a modified impact microviscometer. Proc. Roy. Sot., Lond. Ser. A, 1975, 346, 227-244 14. Poulter T.C., Rotchey C. and Benz C.A. The effect of pressure on the index of refraction of paraffin oil and glycerine. Phys. Rev. 1932, 41, 366-367 15. Hamruck B.J. and Dowson D. Ball Bearing Lubrication: the Elastohydrodynamics of Elliptical Contact, Wiley, New York, 1981 16. Ranger A-P., Ettles C.M. and Cameron A. The solution of the point contact elastohydrodynamic problem, Proc. Roy. Sot. Lond. Ser A, 1975, 346, 227-244
References 1. Geist J.M. and Cannon M.R. Ind. Eng. Chem. (Anal) 1946, 18, 611 2. Lillard, J.G. Anal. Chem. 1952, 24, 1042 3. Lidstone, F.M. A microviscometer of improved design. Chem. Ind. 1952, 873-874 4. Asbeck W.K., Laidermann D.D. and Van Lw M. A high shear method of rating brushability of paints. J. CON. Sci. 1952, 7, 306 5. Merrill E.W. A coaxial cylinder viscometer for the study of fluids under high velocity gradients. J. Coil. Sci. 1954, 9, 7 6. Markovitz H., Elyash L.J., Pudden Jr. F.J. and Dewitt T.W. A cone-and-plate viscometer. J. Coil. Sci. 1955, 10, 165 7. Bair S. and Winer W.O. The high shear stress rheology of liquid lubricants at pressures of 2 to 200 MPa. J. Tribol. 1990, 112, 246-253 8. Cohen A. and Richon D. New apparatus for simultaneous determination of phase equilibria and rheological properties of fluids at high pressures: its application to coal paste studies up to 773K and 30MPa. Rev. Sci. Instrum. 1986, 57, 1192-1195 9. Sawamura S., Takeuchi N., Kitamura K. and Taniguchi Y. High pressure rolling-ball viscometer of a corrosion resistant type. Rev. Sci. Instrum. 1990, 61, 871-873
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Appendix I. Derivation of equations for film thickness and refractive index in an interferometric contact Figure Al shows the contact formed in the high shear rate, high pressure microviscometer. A beam of light is directed into the fluid-filled contact between the glass hemisphere and the glass sphere at an angle 8. The incident beam of light is reflected off the flat surface of the hemisphere, generating the first reflected beam, beam 1. The rest of the light leaves the hemisphere with an angle of refraction 4, passes through the fluid and is reflected off the sphere, generating the second reflected beam, beam 2. The single incident beam of light has been split into two mutually coherent beams which will recombine. The intensity of the recombined beam will depend on the optical path difference between the two separate beams and the phase change which occurs on reflection from the two surfaces. The optical path difference is (2 n3 h cos4) where n3 is the refractive index of the
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A high shear rate microviscometer:
P. L. O’Neill
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DYl (x,, Yi-,J Fig A2 Nomenclature
used to define partial derivatives
Appendix II. Finite difference method \ Fig Al Principles of optical interferometry experimental apparatus
bean
1
used in the
fluid and h is the fluid film thickness. As long as the hemisphere and sphere are made of the same material the phase change on reflection can be ignored as it will be the same from both surfaces. The two reflected beams will interfere constructively and produce bright fringes if: NA = 2n3 h cos+
= N1 A
(AlI
where Nr and N2 are the fringe orders found from the two images produced and +r and & are the angles of refraction of the two beams. A third relationship can be found from Snell’s law: sin g2 n3 _ sin0, (A3) sin +2 n2 sin+, where n2 is the refractive index of the glass hemisphere. These three equations, (Al)-(A3), are solved simultaneously to give the equations for determining the refractive index of the fluid and the fluid film thickness from the interferometric images: (N: ~-. sin*B, - %sin*&) ..nf = n$ --..-~ (A4) (N:-Iv2,) h = __-.-~-~ NIA
2 (n$ - n$s.in20,)
af(xk) ax f(xk + ,)‘Dxl’
- f(xk)‘Dx12*Dx2* - f(xk - ,)‘Dm2 Dxl.m2.(DXl+ DX2) (A@
a*fhd ,
where N is the fringe order which can take on any integer value and X is the wavelength of the incident light. The reflected beams will interfere destructively and produce dark fringes if: (N + 1/2)h = 2n3 h cos+ Consider the situation occurring in the experimental rig where two beams of incident light are used to illuminate the contact, one at angle +r and the other at angle &. The equations for bright fringes will be: 2n3hcos+,
Given a function f(x) for which the values f(xk _ r), and f(xk + r) are known, xk _ ,, xk and xk + r being three consecutive non-equally spaced points, and given that DXI is the distance between xk - , and xk and DX2 iS the distance between xk and xk + ,, as shown in Fig A2, the equations for the first and second partial derivative from finite difference methods are:
f(xk)
ax'
f(x k + I).DXl - f(x/&(DXl
+ Dx2) + f(Xk - 1)*DX2 Dxl.DxL!.(DXl+ Dx2) (A7)
Appendix III. Finite difference mesh Figure A3 shows the finite difference mesh used in the analysis. This mesh was chosen as it keeps the number of mesh points low which will make the computing faster, it has a large area, particularly in the entrance zone, and the mesh is fine at the exit where the pressure in an elastohydrodynamic contact changes rapidly.
(A3 Fig A3 Finite difference mesh used in the analysis Tribology
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