=
K
= viscosity ratio of dispense phase to continuum phase
p Y
n p
u 7
+b
= = = = = = =
Horton, T. J., Fritsch, T. R., Kintner, R. C., Can. J . Chem. Eng. 43, 143 (1965). Hu, S., Kintner, R. C., A.I.Ch.E. J . 1, 42 (1955). Hughes, R. R., Gilliland, E. R., Chem. Eng. Progr. 48, 497 (1952). Klee, A. J., Treybal, R. E., A.Z.Ch.E.J. 2, 444 (1956). Lane, W. R., Green, H. L., in “Surveys in Mechanics,” G. K. Batchelor and R. M. Davies, eds., Cambridge University Press, London, 1956. Licht, W., Narasimhamurty, G.S.R., A.Z.Ch.E. J . 1, 336 (1955). Savic, P., National Research Council of Canada, Rept. M T 2 2 (19.53). Spells, K .E., PTOC.Phys. SOC. B 6 5 , 541 (1952). Taylor, T. D., Acrivos, A., J . Fluid Mech. 18, 466 (1964). Warshay, M., Boqusz, E., Johnson. M., Kintner, R. C., Can. J . Chem. Eng. 37, 29 (1959). Wellek, R. M., Agrawal, .4.K., Skelland, A. H. P., A.I.Ch.E. J . 12, 854 (1966).
angular coordinate
0
cos 0 kinematic viscosity, sq. cm. sec.-l constant of integration, Equation 27 density, g. ~ m . - ~ surface tension, dynes cm.-l stress stream function
SUBSCRIPTS AND SUPERSCRTPTS = interior of bubble or drop = radial direction 8 = angular direction Literature Cited
Batchelor, G. K., J . Fluid Mech. 1, 177 (1956). Brenner, H., Cox, R. G., J . Fluid Mech. 17, 561 (1963). Davis. R. E.. Acrivos. A.. Chem. ERP.Sci. 21. 681 (1966). Habeiman, i V . L., Morton, R. KT, D. W.‘Taylor Model Basin Rept. 802 (1953).
RECEIVED for review July 27, 1967 ACCEPTEDDecember 20, 1967 Work financially supported by the National Science Foundation and by the Petroleum Rcsearch Fund administered by the .4merican Chemical Society.
VISCOUS HEATING IN PLANE AND CIRCULAR FLOW BETWEEN MOVING SURFACES JEROME
GAVlS AND
ROBERT L.
LAURENCE
Department of Chemical E~gineering,The Johns Hopkins Uniuersity, Baltimore, M d .
21218
The equations of motion and energy for the nonisothermal flow of a Newtonian liquid with exponential dependence of viscosity on temperature between moving parallel surfaces with no pressure gradient are presented and solved. The equations are nonlinear but admit solutions in terms of known elementary functions. Solutions are presented for plane and circular flow between isothermal surfaces and between an isothermal and an adiabatic surface. An interesting feature i s that two solutions exist for each value of the applied shear stress. Plots are presented for the temperature and velocity profiles for plane flow and for circular flow with a ratio of outer to inner cylinder radius of 1.6.
heating is an important problem during flow of liquids in such applications as viscometry, extrusion, and lubrication. I t is especially serious in viscometry; small changes of temperature in the liquid can lead to large errors in measurement because the viscosity of most liquids is highly temperature dependent. This had led to several attempts, reported in the literature (Nahme, 1940; Philippoff, 1942; Hausenblas, 1950; Kearsley, 1962; Aslanov, 1963; Turian and Bird, 1963), to solve the equations of motion and energy for simple, viscometric flows of a Newtonian liquid with temperature dependent viscosity to obtain temperature and velocity profiles. Most viscous liquids exhibit exponential temperature dependence, at least approximately, over relatively wide ranges of temperature. Use of such dependence leads to highly nonlinear equations. Believing these insoluble, Philippoff sought approximate solutions for laminar flow in a long tube with an isothermal wall, whereas Hausenblas linearized the problem by assuming a hyperbolic dependence of viscosity on temperature. Surprisingly, the equations of motion and energy for onedimensional flows with exponential dependence of viscosity on temperature are soluble in closed form in terms of known VISCOUS
232
l&EC FUNDAMENTALS
elementary functions. The ease of solution is deceptive, however. Nahme, Aslanov, and Turian and Bird were led into presenting incomplete solutions; they failed to observe that there are two solutions for one applied shear stress, and Aslanov erroneously concluded that viscous dissipation leads to an instability in the flow field as a result. Nor did Philippoff or Hausenblas obtain double solutions, in Hausenblas’ case probably because of the linearization. Only Kearsley, who solved the equations for flow in an infinitely long tube with an isothermal wall, obtained a complete solution of the problem with double valued dependence of the temperature and velocity fields on shear stress. Kaganov (1965) recognized that there should be two solutions for steady flow between rotating concentric cylinders, although he did not attempt to find the solution. The fact that there are two solutions is the most interesting feature of the viscous heating problem. Not only is this another example of the surprises which greet the investigator in the so far too little studied realm of nonlinear boundary value problems, but knowledge of it may be useful in understanding phenomena observed in practical applications where viscous heating is severe, such as in plastics extrusion. The purpose of this paper is to present complete solutions for
flow between infinite parallel surfaces one of which is moving with constant velocity. This includes flow between plane walls and flow between concentric cylinders. Although the equations to be solved are superficially similar to those solved by Kearsley for tube flow, the method he used for solution (which he did not describe) cannot be applied here. For this reason, and because the procedure leading to the complete solution is not immediately evident, the procedure will be described in some detail. Bratu (1914) first posed and solved the mathematical problem formally equivalent to the boundary value problem for the temperature field in plane flow. The solutions for circular flow between isothermal walls, and for the case, both in plane and circular flow, in which one wall is adiabatic will be obtained by an extension ofBratu’s method of solution.
0=
P(T
- To)/To
and
r~
z/h
with p expressed by Equation 1, Equation 8 and the boundary conditions of Equation 4 become the following boundary value problem for the temperature field (9)
BrlE
liquid Properties
P0VO2 ~
= Brinkman number
k To
The thermal conductivity and density of the liquid are assumed to be independent of temperature. Temperature variations of these properties are so small compared with variations of viscosity for viscous liquids that this should be a reasonable approximation. The viscosity will be assumed to have the temperature dependence IJ =
I n terms of the dimensionless variables
IJO
exp[--P(T
-
To)/ToI
Plane Flow-Isothermal Walls
Equations. The equation of motion for nonisothermal flow of a Newtonian liquid between infinite parallel, plane walls, one of which is moving with a constant velocity, is
5
(. 2)
The velocity field is
(1)
in which 1.10 is the measured viscosity at TOand p is an empirically determined constant. Nahme (1940) has shown that such an expression is able to approximate closely the true variation of viscosity of several viscous liquids over at least a 50’ C. temperature range. Therefore, the results to be obtained will be applicable to real systems.
dz
This is equivalent to Bratu’s problem. then given by
(12)
= 0
Solution. Multiplication of Equation 9 by 2(dO/d{) and integration yields
where rn is an integration constant. The 0({) curve has a maximum at In rn. To the left of the maximum d8/d{ is positive; the plus sign must then be used for integration of Equation 14 between 0 and In rn. T o the right of the maximum d0/dr is negative and the negative sign is to be used between In rn and 1. Although there is a singularity at In m, the integrals may be shown to exist. Equation 14 may be integrated subject to the boundary conditions of Equation 10 to give
The energy equation is
(3)
or when the integrations are performed and the resulting expressions are rearranged and simplified
The boundary conditions are
T(0) = T(h) = To v ( 0 ) = 0;
v(h) = vo
(4)
(5)
T h e viscosity, P O , will be assumed known at the wall temperature. Integration of Equation 2 subject to the conditions of Equation 5 gives
(7) Equation 3 then becomes
6 = 1r1 {m sech2
[(
(2
- I)]}
(16)
with m given as the solution of
Direct computation shows that there are two values of rn for each A 1 when A 1 < 3.5138, one value at XI = 3.5138, and no real values of rn when hl > 3.5138. This is illustrated in Figure 1, where rn is plotted against XI. There are thus two different temperature profiles for each value of XI for XI < 3.5138. The profile is unique only when XI = 3.5138. There are no solutions to the problem for XI > 3.5138. The temperature profiles are illustrated in Figures 2 and 3 for several values of XI. Figure 2 shows the profiles for the lower values of rn at each XI, whereas Figure 3 shows them for the higher values of rn.
V O L 7 NO. 2 M A Y 1 9 6 8
233
"
I
I
I
0
05
I O
A,
I 5 20 25 (m,liolhermol WoIIsI
30
35
02
01
0
Figure 1 . Plot of rn and n as functions of X I
03
05
04
Figure 3. flow-high
0
07
06
5 [Isothermal
OB
09
I O
Wollil
Temperature profiles for plane values of rn and n
02
0 4
06
01
02
03
08
I O
04
05
I O
09
o a 2 0 7
-
P a 0 6 g o 5
-
z 0 4
03 0 2
0 1 0 0
06
( 1 I I O l h e r m O l WoI
Figure 2. Temperature profiles for plane flow-low values of rn and n
Figure 4.
The velocity profiles may be determined when the temperature profile, Equation 16, is inserted into Equation 13. After integration, rearrangement, and simplification there results
07
09
09
I O
31
Velocity profiles for plane flow
values and fixes one of the allowable solutions for that XI. Thus the temperature and velocity profiles are unique in the Brinkman number. Figure 5 shows a plot of pBr1 against AI. Plane Flow-One Adiabatic Wall
There are, of course, two different velocity profiles for each XI when X I < 3.5138, and one when X l = 3.5138. These are plotted in Figure 4 for several values of XI. There is a maximum shear stress which can be applied. This is, from Equation 11) (
~
0
= ) 1.8745 ~ ~ (kT~po/fih*)~~'
(19)
Furthermore, for each shear stress below this value there are two possible temperature and velocity profiles. Physically, however, this presents no problem, since the ambiguity is resolved when the Brinkman number is fixed through the magnitude of Vo. By Equation 11 every value of the Brinkman number determines a value of hl in the range of allowable 234
l&EC FUNDAMENTALS
Equations. For simplicity, the adiabatic wall will be taken to be the stationary one. The equations are identical to those for isothermal walls except for the boundary conditions on T or 0. The boundary value problem is then given by Equations 9,11, 12) 13, and the conditions
Solution. The first integration, Equation 14, gives with the adiabatic boundary condition, the first of the conditions of Equation 20 n = :e
(21)
where 0 0 is the temperature at the adiabatic wall, yet to be determined. The maximum temperature is 00; the slope is,
Circular Flow-Isothermal Walls
Equations.
The equation of motion now is written
The energy equation is r dr
The boundary conditions on T are
T(Ri) = T(R2) = 0
(29)
Either the inner or the outer cylinder may rotate. T o accommodate both situations the boundary conditions on w will be written for both cylinders rotating with the constant angular velocities w l and w2
w(R1)
wiR1; ~ ( R Z=) w z R z
(30)
The solution of Equation 27 subject to the conditions of Equation 30 is therefore, always negati.ve. The integral of Equation 14, corresponding to Equation 15, now is written
which upon integration, rearrangement, and simplification becomes 0 = In ( n sech2
{I}
[(,)"'
The energy equation then becomes
(23)
where n is given by the solution of
I n dimensionless form, with
(%)
liZ
n = cosh2
p
E
r/Rl;
R2/Rl
K
Equations 1,29, and 32 become the boundary value problem Again, n is double valued in XI, and a maximum value of X I occurs, beyond which no solution exists. A given value of X i gives the same values of n in Equation 24 as 4 XI gives for m in Equation 16. Figure I , therefore, serves to illustrate n as a function of X I if the scale of the abscissa is altered to reflect the factor of 4. The temperature profiles for isothermal walls and for one adiabatic wall are also related to each other. The profiles for one adiabatic wall are given by the right halves of the profiles for isothermal wails. Figures 2 and 3, therefore, with the scales of the abscissae altered, serve to illustrate the temperature profiles for the present case. The velocity profiles are given by
The velocity profiles are also related to those for isothermal walls and are illustrated in Figure 4 with suitably altered scales for the abscissa and ordinate. The maximum shear st:ress which may be applied is
X z r
dpI2
Pa
1
with the velocity field given by
Solution.
The substitutions
-2t
t z l n p ; s=O transform Equation 33 into d2S -
This is just one half of th.e maximum shear stress for the isothermal walls case. The values of PBrl are shown in Figure 5 with abscissa and ordinate scales for this case.
sK
[
PBr2
dt2
+ XZe* = 0
and the boundary conditions of Equation 34 into s(0) = 0;
s(1n
K)
VOL 7
= -2 In NO. 2
K
MAY 1968
(40) 235
,
0.3
0.2
,
,
,
,
,
n
,
,
,
or, after integration, rearrangement, simplification, and transformation back into (e, p ) coordinates
e = 21n
+ A sinh (a1In p ) ] P
cosh (a1 In p )
‘iu
(43)
where
:-0 - l o5o
01
03
02
(44)
0 4
(45)
t 'bpi
and p is determined as the solution of
Figure 6. Temperature profiles for circular flow in (s,t) coordinates
K
I
I
cosh (a1 In
K)
+ A sinh
(a1 In K )
(46)
The constant p can only be determined over a limited range of X Z from Equation 46, depending upon K. When, for example, K = 1.6, direct computation shows that p has a minimum value of unity at X p = 9.9242 and increases without bound as X p decreases toward zero. Values of p determined from Equation 46 are plotted against corresponding values of X in Figure 7 for K = 1.6. When the initial slope is positive
I 0’
\
=
I
from which 6 = 21n
P
cosh (a,In p )
-A
sinh
(01
In p )
withp determined as the solution of I
0
10
5
15
20
K
25
= cosh (a1 In
K)
- A sinh (a1 In K )
(49)
h*
Figure 7. Xp. K
Plot of p as a function of
= 1.6
The problem is, thus, formally similar to that of plane flow except that the boundary conditions are no longer homogeneous. The equivalent of Equation 14 is
ds
-
= f(2
Xp)l/Z(p
dt
I n the
(e,
- es)l/s
(41)
p ) coordinate system, the initial slope-i.e., always positive. I n the (s, t ) coordinate system, the axis has been rotated through an angle (- tanh-’ 2) relative to the axis. The initial slope, dsldtlo, therefore, may have negative values as low as -2. When the initial slope is negative, the slope must be negative over the entire range. Then there is no maximum in the s ( t ) curve [although, of course, there is a maximum in the O ( p ) curve], and the negative sign must be used in the integration of Equation 41. This is illustrated in Figure 6. O n the other hand, when the initial slope is positive, a maximum must occur in the s ( t ) curve since s must eventually reach - 2 In K from an initial value of 0 (Figure 6). The maximum occurs when s = In p. Therefore, the positive sign must be used in the integration from 0 to In p and the negative sign from In p to --In 2 K , when the initial slope is positive. When the initial slope is negative d O / d p 1-is
236
I&EC FUNDAMENTALS
Nowp is single valued for Xz 5 9.9242 ( K = 1.6) with different (higher) values than those given by Equation 46. For X Z > 9.9242, however, p is double valued up to Xz = 24.8433 when only onep occurs. There are no solutions for X Z > 24.8433. The values of p calculated from Equation 49 are plotted against X 2 in Figure 7 for K = 1.6. The curve joins continuously with that determined from Equation 46 at X Z = 9.9242, the value of X p at which the solution changes from that given by Equation 42 to that given by Equation 47. AS before, p is a double valued function of X Z for all allowable values of X p except hp = 24.8433, its maximum. The corresponding temperature profiles are plotted in Figures 8 and 9 for K = 1.6. Plots for other values of K may be made in the same manner. Unfortunately, there appears to be no way to reduce the equations with respect to K to reduce temperature profiles for all values of K to a single plot. The corresponding velocity profiles are W
wiRi -=P{
-
tanh (a1 In K ) tanh (a1 In p ) tanh (a1In K) [ l =tA tanh (a1 In p ) ]
When the initial slope is negative in the (1, t ) coordinate system, the positive sign should be used; a positive initial slope requires the negative sign. Velocity profiles for K = 1.6 and for rotation of the inner cylinder (up = 0) are illustrated in Figure 10. The maximum shear stress which may be applied is
I 1
Brinkman number fixes the value of applied shear stress uniquely.
I O
Circular Flow-One Adiabatic Wall
1 2
Equations. In practice, it is easy to thermostat the outer cylinder of a Couette viscometer and difficult to thermostat the inner one. The inner cylinder surface will then, at least approximately, be adiabatic. Therefore, the viscous heating problem will be solved here for the inner cylinder wall adiabatic. The equations are identical to those for isothermal walls except for the inner wall boundary condition on the temperature. Instead of Equation 34, therefore, Equation 33 must be solved subject to
09 08 07
e 06
05 0 4
0 3 0 2
dB1 dP
01
'10
1 1
1 2
13
14
15
Id
P
Figure 8. Temperature profiles for circular flow, isothermal walls, low temperature ranges. K = 1.6
= 0;
= 0
e(K)
(52)
0-1
The boundary conditions on the transformed Equation 39 then read
Solution. With the first of the conditions of Equation 53, the first integral of Equation 33 gives
where 90 is the temperature at the inner cylinder surface and is yet to be determined. The slopes ds/dt and dO/dp will always be negative. Therefore
-
(2 A 2 ) W = F
Figure 9. Temperature profiles for circular flow, isothermal walls, high temperature ranges. K = 1.6
s
ds
When transformed back into of Equation 55 yields q(a22
9 = In
(55)
-
r o ~ 8 0 = l n ( q o - ~ 2 (4 / ~ ~ ) es)1'2
cosh
(e,
- l)P2
+ sinh (az1n
(a2 In P>
where
p ) coordinates, integration
P)
1
(F)
112
a2 =
with q the solution of
q(az*
- 1 ) ~=*
[a2
cosh
\
(a2 In K )
+ sinh (a2In
i
I
I
I
\
K)J*
l
l
m
a IO
Figure 10. Velocity profiles for circular flow. Isothermal walls, inner cylinder rotating. K = 1.6
Values of PBr2, determined from Equation 35, are plotted against A 2 in Figure 11, Again, as in the case for plane flow, for each value of the applied shear stress below this maximum value, there are two possible temperature and velocity profiles. There is no ambiguity because of the multiple solutions since prescribing the
I 0-1;
1. .
, , 1
5
' '
' '
,
' '
'
'
'
,
15
10
' ' '
I
'
20
' '
1
' 25
A2
Figure 1 1 . Plot of PBr2 as a function of Xq-isothermal walls. K = 1.6 VOL. 7
NO. 2 M A Y 1 9 6 8
As before, q is double valued except at X Z = 5.18151, no solutions exist when X Z > 5.18151. Figure 12 is a plot o f p against X 2 for K = 1.6. Figures 13 and 14 show the temperature profiles. Velocity profiles given by
\
08
+
&
IO
07
tanh (a2 In p ) [a2 4-tanh (azIn K ) ] ) {tanh (a2In K) [a,f tanh (a,In p ) ]
WP
I
09
tanh (a2 In K) - tanh (ag In p ) tanh (a2 In K ) [a2 tanh (a2 In p ) ]
W
I
I D
(59)
< 06 I
are shown in Figure 15 for w2 = 0-Le., for rotation of the inner cylinder only. A plot of pBr2, determined from Equation 35, is shown in Figure 16. The maximum shear stress which may be applied is
m
0 5 0 4
I
A
,
3
4
, ,
03 0 2 01
(
T
~
(60)
=) 2.2763 ~ ~ (kTopo//3Ri2)'/2
0 I O
I I
12
The most interesting result of this investigation is that the temperature field arising in either plane or circular flow of a Newtonian liquid with exponential dependence of viscosity on temperature between parallel surfaces is double valued in
I0 '
IO'
q IOZ
IO
P '0
I
2
3
4
Figure 13. Temperature profiles for circular flowinner wall adiabatic, low temperature ranges. K = 1.6
5
Figure 12. Plot of q as a function of Xz. K = 1.6 ,
,
,
,
,
I 6 0
5 0
R 4 0
3 0
2 0 I 0
0
IO
14
15
I 6
Figure 15. Velocity profiles for circular flowinner wall adiabatic, inner cylinder rotating. K = 1.6
10-0
I 2
I3
14
15
16
Figure 14. Temperature profiles for circular flow-inner wall adiabatic, high temperature ranges. K = 1.6 l&EC FUNDAMENTALS
I
2
5
Figure 16. Plot of PBrp as a function of 12-inner wall adiabatic. K = 1.6
the applied shear stress. I t is unique in the velocity of the surface or in the Brinkman number. This arises physically in the following manner. As the velocity increases from zero, the applied shear stress tends to increase as a result of the increased shear rate, but tends to decrease because the viscosity decreases as the temperature increases. Initially, the effect of the shear rate increase is greater than the effect of the viscosity decrease. But the shear rate increases slowly compared with the exponential rate of decrease of the viscosity as the velocity continues to increase. Eventually, a velocity is reached at which the higher rate of shear is exactly offset by the lower viscosity. Then the applied shear stress is a maximum. Further increase of velocity now causes such great decrease in viscosity that the applied shear stress decreases. Evidently, this has not yet been observed in real experimental situations. No doubt this is because no one has extended the experiments to large enough Brinkman numbers. Nomenclature
dimensionless A = a constant, (p - l)r/2/p1/2, Brl = Brinkman number for plane flow, poVo2/kTo, dimensionless Br2 = Brinkman number for circular flow, po(w2 - wI)2R1Z/kTo, dimensionless h = distance between walls in plane flow, cm. k = thermal conductivity of liquid, g. cm.jsec.3 O K. m = integration constant, plane flow, isothermal walls, dimensionless n = integration constant, plane flow, one adiabatic wall, dimensionless P = integration constant, circular flow, isothermal walls, dimensionless 9 = integration constant, circular flow, inner cylinder wall adiabatic, dimensionless r = radial coordinate in circular flow, cm. Rr = radius of inner cylinder, circular flow, cm. Rz = radius of outer cylinder, circular flow, cm. s = transformed variable, 0 2 t , dimensionless t = transformed variable, In p , dimensionless T = temperature, O K. u = local velocity, plane flow, cm./sec. velocity of moving wall, plane flow, cm./sec. w = local velocity, circular flow, cm./sec. z = posit:on coordinate, plane flow, cm. a1 = a constant, (X2p/2)lI2, dimensionless
-
I1
P
238
13
P
Conclusions
v =
a ? = a constant, ( X 2 q / 2 ) ' j 2 , dimensionless /3 = exponent in viscositrtemperature relationship, dimen{
=
0
= = =
K
A1
A? =
T
= = =
w1
=
w?
= =
p p
sionless reduced position (coordinate. z / h , dimensionless reduced temperature. p(T - To)/To,dimensionless cylinder radius ratio, RZ/RL, dimensionless constant in plane flow differential equations, dimensionless constant in circular flow differential equations, dimensionless viscosity of liquid, g./cm. sec. reduced radial ccordinate, r,'Rl, dimensionless shear stress, dynes/sq. cm. angular velocity of inner cylinder, circular flow, cm.-' angular velocity of outer cylinder, circular flow, cm.-' (subscript) denotes value of a quantity at a surface
literature Cited
Aslanov, S. K., Inzh. Fiz. Zh. Akad. Nauk Belorussk. SSR 6 , 8 (1963). Bratu, G., Bull. SOC.Math. France 42, 113 (1914). Hausenblas, H., Ingr.-Arch. 18, 151 (1950). Kaganov, S. A., Inzh. Fiz. Zh. Akad. A'auk Belorussk. SSR 8 , 307 (1 965 - - 'i. /. \ - -
Kearsley, E. A., Trans. SOC. Rheol. 6 , 253 (1962). Nahme, R., Ingr.-Arch. 11, 191 (1940). Philippoff, W., Z. Physik 43, 373 (1942). Turian, R. M., Bird, R . B., Chem. Eng.Sci. 18, 689 (1963). RECEIVED for review August 2, 1967 ACCEPTED December 18, 1967 It'ork supported by grants from the National Science Foundation, $5 GK 1714 and GK 839.
EVALUATION OF POWER-MODEL LUBRICANTS
IN A N INFINITE JOURNAL BEARING ROBERT EHRLICH' AND JOHN C. SLATTERY Department of Chemical Engineering, Northwestern University, Evanston, Ill.
A journal bearling consists of two right circular cylinders, one enclosed by the other, whose axes are parallel. The forces exeirted upon the cylinders are such that the axes do not coincide. A flow is set up in the annular space as the inner cylinder is rotated with respect to the outer cylinder. An "infinite" bearing means that end effects may b e neglected. Two previously available inequalities are used to obtain upper and lower bounds on the z-component of the torque which a power-model fluid exerts on the inner cylinder. The results suggest that for a wide variety of reasonable designs, a nowNewtonian fluid is a better lubricant for this geometry than a Newtonian fluid. The results are consistent with the limited experimental data available.
equation of continuity and the stress equation of motion are not sufficient to describe the motion of a particular material under a given set of boundary conditions. As a minimum of additional information, we require a description of stress in the material as a function of deformation; this is usually referred to as a constitutive equation for stress. The subject of constitutive equations has been briefly reviewed (Slattery, 1965; Wasserrnan and Slattery, 1964). It is sufficient to say that the simplest of all the constitutive equations proposed for incompressible fluids is the incompressible ReinerRivlin model (Reiner, 1945; Rivlin, 1947, 1948; Serrin, 1959): THE
c = T +,bI
= cud
+ @d
d
(11
where (Y
= (~(11, 111)
and
B
= S(I1, 111)
(2)
I11
(3)
Here we define
I1
tr(d
*
d)
.=
d,,d3",
det(dl,)
By d we mean the rate of cleformation tensor,
d
=
+
'/'~[vv (vv)']
(4)
1 Present address, Gulf Research and Development Co., P. 0. Drawer 2038, Pittsburgh, Pa. 15230.
A popular subclass of Equation 1 is formed by the incompressible, generalized Newtonian fluids, T =
(5)
2pd
where p = /.L(y2),
y = di1
(6)
Two special cases of Equation 5 are the incompressible Newtonian fluid, 1.1
= constant
(7)
and the power-model fluid, p = m(2
(8)
yZ)P--1)/2
While the power-model fluid cannot fully represent the behavior of any real fluid now known (an immediate objection is that it does not predict nonzero and finite limiting viscosities at very l o i v and at very high rates of deformation), over a limited range of stress it often gives a satisfactory approximation for the stress us. rate of deformation curve from a viscometric study. By a journal bearing we mean two right circular cylinders, one enclosed by the other, whose axes are parallel. By an infinite journal bearing, we mean that the cylinders are infintely long or, at least, that end effects are negligibly small. A flow is set up in the fluid contained in the annular space VOL 7
NO. 2
MAY
1968
239
