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 a t a surface
literature Cited
Aslanov, S. K., Inzh. Fiz. Z h . 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 a t 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 a n infinite journal bearing, we mean that the cylinders are infintely long or, a t 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
between the two cylinders as the inner cylinder is rotated with respect to the outer cylinder. With no load on the inner cylinder, the axes of the cylinders coincide. As the bearing is loaded, the cylinders take on an eccentricity a t which the hydrodynamic forces exerted by the fluid on the inner cylinder balance the load. The eccentricity of the bearing is the distance between the axes of the cylinders; the eccentricity ratio i E the ratio of the eccentricity to the nominal clearance of the bearing. Nominal clearance is defined as the difference between the radii of the cylinders. Previous workers have obtained solutions for the flow of an incompressible Newtonian fluid in both finite and infinite journal bearings. For the infinite case. Sommerfeld (1 904), following the suggestion of Reynolds (1886) in neglecting the inertial terms in the equation of motion, assumed that clearance and eccentricity are much smaller than the average of the bear'ng radii. [Milne (1959) has shown that it is reasonable to neglect inertial effects under normal operating conditions.] Subject to these samc restrictions, flow in a finite bearing has been analyzed as well : Raimondi and Boyd (1 958) obtained a numerical solution, Hays (1959) used a variational method, and Tao (1959) presented an exact solution in terms of Heun functions. Using only the assumption that inertial terms in the equation of motion may be neglected, Joukoivsky and Tschaplygin (1904), Wannier (1950), Farris (1962), and Farris and Slattery (1963) obtained exact solutions for flow in an infinite journal bearing. Horoivitz and Steidler (1960. 1961) used an experimentally obtained apparent viscosity-rate of shear curve to estimate that less friction is obtained for a given load with a non-Newtonian lubricant than with a Newtonian fluid of the same zero-shear viscosity. Tanner (1963) analyzed flow of a power-model fluid in a short journal bearing. H e found that, with no eccentricity ratio limitation, a power-model fluid gives the least friction for a qiven load; but. if eccentricity ratio has an upper limit less than unity. a Newtonian fluid may give less friction. Our object here is to say for the infinite journal bearing under what conditions, if any, a lubricant obeying the power model would be a better lubricant than a Newtonian fluid. I n principle, this knowledye should influence the formulation of industrial lubricants (at least for situations approximating long journal bearings). I n what folloivs, we attempt to obtain the friction-load characteristics for slow flow of a power-model fluid in an infinite journal bearing. For any given computation, a ratio of cylinder radii and an eccentricity ratio is chosen; this fixes the bearing geometry and the parameters of the bicylindrical coordinate system (see Appendix A). The bounding principles suggested by Hill (1956) and by Hill and Power (1956) are used to obtain upper and lower bounds to the torque on the inner (rotating) cylinder. The trial functions used with the bounding principles are further used to approximate the corresponding load on the bearing. Extremum Principles
The extremum principles used here have been developed previously by Hill (1956) and by Hill and Power (1956) and later, from a different point of view, by Johnson (1960, 1961) [the velocity extremum principle was presented by Pawlowski (1954) and by Bird (1960)]. Since the presentations by Hill (1956) and by Hill and Power (1956) were compact, we begin by giving those portions of the proofs of these extremum principles lvhich we believe are helpful in understanding our computations for the infinite journal bearing. 240
l&EC FUNDAMENTALS
I n developing these extremum principles, the following assumptions are made. The fluid is incompressible,
v * v = o
(9)
Inertial effects are neglected in the equation of motion, which reduces to V
*
(T
- prpI)
=
0
(10)
We assume here that the external force per unit mass is representable in terms of a potential, f = -vp. Assumptions concerning fluid behavior are stated as needed below. Velocity Extremum Principle. Here we assume that the components of the extra stress tensor, o, may be represented in terms of a scalar-valued function E of the rate of deformation tensor, T,,
=
bE -
Mi'
(11)
For a generalized Newtonian model, Y2
E
=
~ ( rdr2 ?
E(?) =
(1 2)
We also require E to be a convex function in the sense that (13) A necessary and sufficient condition that E be a convex function is that the quadratic form associated with the Hessian matrix [bZE/bdi5bP" J be nonnegative; for a generalized Newtonian fluid, it is sufficient that
1 dE 2 p = - - > o Y dr
-
(1 4)
and (15)
After integrating Equation 13 over the volume of fluid in a given system and rearranging, we arrive a t
If we understand that v* is an approximate velocity distribution which satisfies the equation of continuity and the boundary conditions on velocity, and which is continuous throughout a multiply connected domain (y* is defined in terms of v*), this becomes
J, E(y*)dY - l-sn (v* - v) (T - PPI) n dS 2 *
where S represents the entire bounding surface of the system and S, is that portion of S upon which velocity is specified. We find below that the integral on the right has physical significance in the problem we consider here.
Stress Extremum Principle. T h e stress extremum principle is in some sense the complement of the velocity extremum principle above. We begin by assuming that the rate of deformation tensor d ma)7 be represented in terms of a scalarvalued function E, of the extra stress tensor,
I n the first line here, we made use of the symmetry of t (Equations 4 and 9) ; at the second line, we integrated by parts and applied the generalized divergence theorem; finally, we recognized the equation of motion, Equation 10. This allows us to express the integral of Equation 28 over the volume of a system as
For the generalized Newtonian fluid, we may also write
E, =
&(u)
l*
=
and Inequality 24 becomes
By analogy with the development of the velocity extremum principle above, we require E , to be a convex function,
A necessary and sufficient condition for Equation 21 to hold is that the quadratic form associated with the Hessian matrix [bZEc/dsi~dsmfl] be nonnegative or 1 1 dE, -. u du 2 0 2~
(31; Inequalities 18 and 31 give upper and lower bounds for the same quantity. Physical Significance of Extremum Principles
When E(?) is a homogeneous function of degree q, by Euler's theorem (Kaplan, 1952)
qE =
and
(23)
dE - d l j = z,,dt5 = t7(e dd'j
d)
*
(32)
But the quantity on the right is the rate of dissipation of energy per unit volume by viscous forces. For a power-model fluid,
After integrating Equation 21 over the volume of fluid in a given system and rearranging, Hill (1956) obtains
Jv
E,(u*)dV
-
J' v
i(T* - T)
. n dS 2
S"
E,(u)dV
(24)
Here we understand T* to represent an approximate stress distribution which is continuous throughout a multiply connected domain and whimch satisfies the equation of motion (u* is defined in terms of 7*). Let us see under what conditions we can relate the integrals on the right sides of Inequalities 18 and 24. From Equations 5 and 20, dE, u - 2p = y (25) du Similarly, from Equations 5 and 12,
which is a homogeneous function of degree n that, for a power-model fluid,
+ 1.
This means
An application of the integral mechanical energy balance (Bird, 1957) to the incompressible fluid contained between two cylinders, one of which is rotating, shows that
r
r
O n the surface of the moving cylinder,
(36)
v = R X p '
where p' may conveniently be taken as the positive vector measured from some point on the axis of the rotating cylinder and R is the (constant) angular velocity of the cylinder. This means that
These equations allow us to write
l$ + du
dy
ydu
:=
+
udy =
L'dbr) or E,
+ E = uy = 2py2 = t r ( t
*
d)
(27)
tr(t
tr(['T - p q I ]
d)dV =
*
- ppI)
*
n)dS = R
p'X{T n ) d S = ss.
(28)
If we integrate the term on the right of Equation 28 over the volume of the system, we have
sv
p'X{(T
Vv)dV =
where is the vector torque exerted by the fluid on the rotating cylinder whose surface is S,. For the power-model fluid, Equations 34 and 37 yield
S V
v
(T
- p q I ) . n d$ - i
7 v
*
[V
r
J
v S
(T
(T
- pqI)]dV =
- ppI)
n dS
(29)
-R
E
=
(n
+ 1) J V E ( y ) d V
(38)
I n what follows, we compute upper and lower bounds to the left side of Equation 38 using Inequalities 18 and 31. VOL. 7
NO.
2
MAY 1968
241
Figure 1. Bicylindrical coordinate system
/ Upper Bound Analysis
I n the bicylindrical coordinate system described in Appendix
A and illustrated in Figure 1 (Moon and Spencer, 1961)) the rotating inner cylinder of the bearing is represented by u = u1 and the outer cylinder by u = us. T h e boundary conditions on velocity are
u =
U I : U, =
0, U, = QRi
Parameter s is defined in Appendix A.
(39)
(2 n * ) ( n + l ) / z d u du
and
vu = 0 , u o = 0
(40) Here il is the magnitude of the angular velocity of the inner cylinder, whose radius is R 1 . We require the velocity trial function, v*, to be a periodic function of u with period 27r (in order that it be continuous throughout the doubly connected domain) ; it must satisfy Equations 9, 39, and 40 as well. Such a trial function may be formulated by adding the exact solution for a Newtonian fluid in Appendix B to a function of u, u, and several parameters, at (i = 1 , 2, , , .). This latter function would represent the deviation of the results for a power-model fluid from those for a Newtonian fluid. Defining a dimensionless stream function, $, as u = us:
From Equation 33,
Quantity
(45)
n*is a dimensionless trial second invariant defined as 3*=
(-)
sinh uI
11*
For this situation, the surface integral in Inequality 18 is identically zero, n
n
(47) Inequality 47 together with Equations 38 and 45 allows us to obtain an upper bound on the magnitude of the z-component of the torque which the fluid exerts on the inner cylinder (Equations 36 and 39 show that the z-component of SL must be -a. Consequently, Sz must be positive in sign.)
we take the dimensionless trial stream function to be of the form
o$*
=f(u)
+ M u ) + g(u,
(42) The quantities a, p , f ( u ) , and h(u) are defined in Appendix B. The function g(u, u, ai) is to represent the deviation of b$ for a power model fluid from that for a Newtonian fluid. If it is assumed that four parameters sufficiently describe this deviation, one arbitrary expression for g(u, u, at) of the required form is g(u, u, a t ) = (u
-
U1)YU
- U2)2[Ul
+
u, 4 )
a2(u
- UP)
adu
+
a3
cos u
- u2)2/21
l&EC FUNDAMENTALS
(49) then
+
(43)
In terms of the trial stream function, $*, the second invariant of the rate of deformation tensor, Equation 3, is 242
If @ is a dimensionless torque defined by
The four parameters ai (i = 1, 2, 3 , 4 ) which enter Inequality 50 through Equation 43 are specified by requiring Inequality
50 to yield a minimum upper bound. When this requirement is applied to the right side of Inequality 50, a set of four equations, which must be solved simultaneously, result for the ai. This set of equations was solved by a Newton-Raphson iteration, with Simpson’s rule used to evaluate the various definite integrals involved. The resulting minimum upper bound for is presented in Table I as a function of the power-model index, n.
Table 1.
Dimensionless Torque and Load for a Bearing with R2/R1 = 1.002
n 1 .ooa
zz
Lower Bound Analysis
A trial stress distribution suitable for use with Inequality 31 must satisfy Equation 10 and must be periodic in u with period 21r (in order that T* be continuous throughout the doubly connected domain). From the solution to the Newtonian problem (Appendix B), a stress distribution which satisfies these requirements is
,[e + 2 ( A cosh 2u + B sinh 2 u ) ] Tu,* = -2@[csinh u -+ 6 cosh u 2 cos ,(A sinh 2u + cosh 2 u ) ] (51) p - PO = 2 sin u [ 2 ( A cosh u + B sinh u ) + G sinh u - 2 cos u(A cosh 2u + B sinh 2 u ) ] YUu* =
-fUu*
0.80
= 2P sin
where dimensionless modified pressure and the dimensionless physical components of the extra stress tensor are defined as
and (53) The quantities A, B , 6 , and are parameters yet to be determined. W e define here for convenience a modified pressure P =p pp; Po refers to the modified pressure at some reference point. From Equations 5, 8, ;and 20, we have
0.70
c
0.95 0.90 0.80 0.70 0.60
0.98 0.95 0.90 0.80 0.70 0.60 0.98 0.95 0.90
0.80 0.70 0.60 0.60 0.98 0.95 0.90 0.80 0.70 0.60 0.50 0.98 0.95 0.90 0.80 0.70 0.60 a Dimensionless torque Equations B78 and B79.
Wlfrom ( X 106), iz ( X 70a) Lower LOWET UMer bound” bound Bound0 19.467 ... 9.889 13.455 ... 6,932 9.056 ... 4.765 7.007 ... 3.714 5.734 ... 2.998 8.55 8.63 4.50 6.22 6.25 3.29 4.37 4.39 2.34 3.47 1.84 3.48 2.89 2.89 1.49 5.15 5.40 2.81 3.74 3.86 2.03 2.86 2.92 1.55 2.11 2.14 1.14 1.71 1.73 0.910 1.45 1.46 0.736 2.06 2.29 1.16 1.62 1.74 0.907 1.31 1.37 0.727 1.01 1.04 0.551 0.844 0.860 0,442 0.730 0.738 0.356 0.808 0.987 0.466 0.694 0.790 0.396 0.335 0.595 0.646 0.484 0.508 0.262 0.415 0.210 0.429 0.366 0.373 0.167 0.309 0.432 0.180 0.292 0.361 0.168 0.267 0.305 0.150 0.230 0.121 0.248 0.204 0.214 0.0960 0,184 0.189 0.0754 and load for n = 1.00 from exact solution,
+
(54) This, together with Equa.tions 51, allows us to express the first term on the right of Inequality 31 as n+l
[i
(n +I) / zn
(E*)~]
du dv
(55)
v
- (T*- pp1) -
11
Bearing Load
T h e dimensionless orthogonal, Cartesian components of the force which the fluid exerts on the (inner) rotating cylinder are
T h e second term on the right of Inequality 31 gives
-
The four parameters A, B, 6, and 6 are determined by requiring the lower bound of Inequality 57 to be a maximum lower bound. When this condition is applied to the right side of Inequality 57, a set of four equations for these parameters result. These four equations were solved simultaneously by a Newton-Raphson iteration, with Simpson’s rule being used to evaluate the various definite integrals involved. T h e resulting maximum lower bound for 3z is presented in Table I as a function of the power-model index, n.
dS =
mL’n+1R12L 4n
ct(6 sinh u1 + 6 cosh ul)
(56)
Inequality 31 and Equations 38, 55, and 56 yield as a lower bound for the dimensionless - j z defined by Equation 49: and
nfl
4n a ( n
+ l)(6sinh u1 f
cosh u1)
(cash u
COS
u
-
PZ
(57) VOL 7
NO. 2
Tun}
du
(59)
MAY I968
243
U-UI
Figure 2. Ratio of dimensionless torque to dimensionless load (friction coefficient) as a function of dimensionless load for an infinite journal bearing with R2/R1 = 1.002
where
T h e trial stress distribution used in the lower bound calculation may be used to estimate these components as
w, = 0
(61)
and
IFu = 4~ a ( 6 + 2 sinh ul(cosh - sinh ul) X u1
[C?
+ 2(A cosh 2ul -I-
sinh 2 u l ) I J (62)
In arriving a t Equations 61 and 62 we neglect the effect of the external force (gravity) and take p = p . Results
Because of the quantity of computations required for each point, it was impractical to obtain a grid of calculated results which would represent all bearing geometries and all powermodel coefficients which are of practical interest. Since this work is concerned with the effects of variation in the powermodel exponent and since previous work indicates that eccentricity variation is of interest, the ratio of the bearing radii is taken to be fixed a t R2/R1 = 1.002. It is assumed that this ratio is representative of most bearings which are of interest and that conclusions drawn from data obtained for a bearing with this ratio will hold, a t least qualitatively, for other bearings. No computations were carried out for B < 0.6, because the Newton-Raphson iteration converged too slowly. For n < 0.5, the upper and lower bounds t o 5, were too far apart to be meaningful. We show in Figure 2 a plot of 3JTFu (often referred t o as the bearing friction coefficient) as a function of IT, for various values of the eccentricity ratio, e, and the power-model ex244
l&EC FUNDAMENTALS
ponent, n. Actually, this is an estimated bearing friction coefficient based upon the lower bound to the z-component of torque and the y-component of load computed from the lower bound stress trial function. I t appears that, for a given dimensionless load, a decrease in the friction coefficient is obtained by decreasing n at the expense of an increase in the eccentricity. I n any given situation, an upper limit is placed upon the eccentricity ratio. The lower bound trial pressure distribution in Equation 51 (modeled after the Newtonian pressure distribution, Equation Bl6) indicates that (P - PO)is negative for s < u < 2s. Depending upon the magnitude of PO,the pressure may be less than the vapor pressure of the lubricant in a portion of this region, which would cause vaporization or cavitation. Since this effect increases with increasing eccentricity ratio, an upper limit must be placed upon the eccentricity ratio to avoid this breakdown of the lubricant film. This suggests that the best comparison of Newtonian lubricants to non-Newtonian lubricants would appear to be on the basis of a fixed bearing geometry (including eccentricity ratio), angular velocity, and load. A reasonable design eccentricity ratio might be e = 0.9; for this case, it would appear that there is a n optimum power model exponent on the order of n = 0.55. O n the other hand, if one chose to design for e = 0.6, a Newtonian fluid would be the most desirable lubricant. O u r conclusions for e = 0.6 are in agreement with those of Tanner (1963), who presented an approximate analysis of a short journal bearing filled with a power-model fluid. But our conclusions above for e = 0.9 differ sharply from those of Tanner. I t is likely that this difference should be attributed to the difference in geometry, which means that no sweeping conclusions as to the relative value of Newtonian and nonNewtonian lubricants can be made on the basis of studies in one or two geometries. O n the other hand, we are not able to assign error bounds to our load computation and we used the lower bound on the torque in preparing Figure 2. I n the experiments of Dubois, Ocvirk, and Wehe (1960),
a small reduction in the friction coefficient was obtained by replacing a Newtonian lubricant by a non-Newtonian lubricant, while holding load, angular velocity, and m fixed (rn is viscosity for a Newtonian fluid). The effective value of n appeared to be 0.9 for their experiments. At an eccentricity ratio of about 0.9 Figure 2 predicts a 57, reduction in the friction coefficient. Tanner's (1963) results for the short journal bearing are in substantial agreement here with a prediction of a less than 10% reduction.
0-COMPONENT OF THE EQUATION OF MOTION
bo, P bou bv, - + - v , - + + ~ , - + + , bV~ sinh u __
The authors thank the National Lubricating Grease Institute (R.E.) and the National Science Foundation (J.C.S., NSF-G-20385) for their Einancial support during a portion of this work. Bicylindrical Coordinates (Farris, 1962)
The journal bearing is conveniently discussed in terms of the bicylindrical coordinate transformation x =
=
V,V,]
P dP --+ P s
bu
- s sin v
s sinh u
P = cosh u 7
~
P
COS
u
P -brzu -P. b- ~+, -, - + -bu
s
vuz
I P
bT0u
bu
s
bTuo
I broz bz
bv
s
-
s
dv
drzz bz
sinh u
sin v
- __
Tzu
5
+
Tzu
Pfz
(A101 For a n incompressible Newtonian fluid, the physical components of the extra stress tensor are:
(Al)
which is illustrated in Figure 1. The inverse transformation is
T , = ~
bV 2s.[P 2 - (sinh u)u, bu
rzz = 2p
bvz bz
+ p b2 V + (sinhu)v, + (sinv)vu]
We have for the components of the metric tensor:
(All)
bV
S2
guu = g,, =
+ sin sv
Z-COMPONENT OF THE EQUATION OF MOTION.
Acknowledgment
Appendix A.
dv,
bu
s
at
p gzr = 1,
g f j = 0 (i # j )
(-43)
Physically, a bearing configuration is specified by
e,
R1, and
Rz,where c is the eccentricity ratio, e =
dz
- dl
Rz
- Ri
Appendix B. Exact Solution for an Incompressible Newtonian Fluid (Farris, 1962)
In terms of the bicylindrical coordinate system, a bearing configuration is specified by s, 711, and U Z :
Working in terms of the bicylindrical coordinate system, we define a stream function by vu =
Two useful expressions are:
p-)b$ bV
u, =
- p - a$ bU
Pressure and the external force potential may be eliminated between the u- and v- equations of motion to obtain
I n this coordinate system, the equation of continuity and the three components of the equation of motion have the following forms. CONTINUITY.
This differential equation is to be solved with Equations 39 and 40 as boundary conditions; as additional constraints, we require vu, v,, and p to be periodic functions of u with period 27. This boundary value problem has as a solution
+Mu) 033) = E sinh u + F cosh u + Gu sinh u + Hu cosh u P$
=f(u)
where f(u) P
-- (vu sinh u S
+ u , sin u )
=
0 (A7)
U-COMPONENT OF THE EQUATION OF MOTION.
av, p sinh u -++~ " "dv,- + -p" , -dv' + ~ zbv,- ~ + __ uc.2 bU
s
bu
32
S
+ B cosh 3 u ] h(u) = A sinh 2u + B cosh 2u + Cu A = 2 RiQciG cash + B = -2 RlQciG sinh (ui + '/z[A sinh 3u
( ~ i
UZ)
UZ)
C = -4 RiQci6 cash (ui 6[3 cosh ul cosh u z 4 sinh u z sinh (u1
-
sinh
-~ ul
035) (B6) 037) (B8)
2 )
sinh
(B4)
UZ]
+
- U Z ) [ U Z cosh u i sinh uz U I sinh u1
VOL. 7
cosh
UZ]
NO. 2 M A Y 1 9 6 8
245
F =
R1Qa[4(Ul
6 sinh
G =4
RlQa
sinh
u2
sinh2 (u,
(UI
+
UZ)]
- up)
(u1
(B10) (B11)
H=O a = [4 sinh
a
- u2) sinh2 u2 sinh u1 sinh (u1 - u2) -
(B12)
- UZ){ (sinh2 u1 + sinh2U , ) ( U I - UZ) 2 sinh U I sinh U S sinh (u1 - u 2 ) } ] - l
(B13)
5
u, v
- u2) sinh u1 - sinh u2 sinh (UI - u2)
(u1
(B14)
u2
V vu,
6 =
uv
V
w,
wz,
I n terms of the constants above, pressure and the components of the extra stress tensor are found to be
+ B sinh u ) + G sinh u 2 cos v ( A cosh 2u + B sinh 2u)l (B15) = rUu= ? !’ Bh’ sin u = 2P - P sin v[C + S 2(A cosh 2u + B sinh 2 u ) ] (Bl6)
P - Po = 2P - sin v [ 2 ( A cosh u S
-T,,
S
and -Tun
=
@
[f” - f
S
2PP - [C sinh u S
+ G cosh u - 2 cos v ( A sinh 2u -I- B cosh 2u)] (B17)
Since h = 0 for u = u2 and for u = u1, the components Tu, and T,, vanish on the bearing surfaces. The orthogonal, Cartesian components of the bearing load (force which the fluid exerts on the rotating inner cylinder beyond any effect of the external force vector) are found to be
w, = 0, w, =
Sz =
-4*
__ (C sinh U I
RiQ
RlR
+ G cosh
u1)
= four dimensionless coefficients in Equa-
tion 51; they reduce to (Rlna)-l times the values given in Appendix B for a Newtonian fluid ai (i = 1, 2, 3, 4) = four coefficients in Equation 43 = rate of deformation tensor, Equation 4 d = covariant (contravariant) components of dij (dii) the rate of deformation tensor = potential defined by Equation 11 E = E(Y) = potential defined by Equation 19 E, = E ( u ) I = identity tensor L = length of cylinders = coefficients of power model, Equation 8 m, n = outwardly directed unit normal vector n P = pressure = position vector measured from some PI point on axis of rotating cylinder Po = reference pressure = modified pressure, P = p pp P Ri(R2) = radius of inner (outer) cylinder as indicated in Figure 1 = defined by Equation A51 S = total bounding surface of system S = portion of bounding surface upon which S, velocity is specified T = stress tensor
+
246
I&EC FUNDAMENTALS
= defined by Equation B13
P
defined by Equation A1 defined by Equation 6 viscosity function, Equation 5 density = defined by Equation 16 = extra stress tensor, Equation 1 = covariant (contravariant) components of extra stress tensor = eccentricity ratio = physical components of extra stress tensor in bicylindrical coordinates = extra force potential; external force per unit mass is -vp = stream function, defined by Equations Bl = angular velocity vector = magnitude of angular velocity of inner rotating cylinder; z-component of angular velocity vector in this case = defined by Equations 3 = determinant of matrix of components of d = trial function = dimensionless variable
Y
P = P
dr2)
U
7 T ij
(Ti j) +uu,
7”Y
9
$
n R 11, I11 det ( d i j )
*
-
= = = =
literature Cited
~
Nomenclature
A, B , e, d
a
41 G
T h e r-component of the torque which the fluid exerts on the inner cylinder is given by -
a
GREEK LETTERS
E T,“,
+ 2h‘ sinh u + Ph”] =
fluid on inner rotating cylinder r-component of Q dimensionless r-component of defined by Equation 49 = bicylindrical coordinates defined by Equations A1 = values of bicylindrical coordinate u on inner and outer cylinders, respectively = velocity vector = physical components of velocity vector in bicylindrical coordinate system = volume of system = orthogonal, Cartesian components of load (force) which fluid exerts on inner cylinder
= =
’J*
u1,
and
= moment of force (torque) exerted by
Bird, R. B., Chem. Eng. Sci. 6, 123 (1957). Bird, R. B., Phys. Fluids 3, 539 (1960). Dubois, G. B.,Ocvirk, F. W., Wehe, R. L., Natl. Aeronaut. Space Admin., Tech. Note D-427 (1960). Farris, G. J., “Flow in a Journal Bearing,” M.S. thesis, Northwestern University, Evanston, Ill., 1962. Farris. G. J.. Slatterv. J. C.. N.L.G.Z. Sbokesman 27. 263 (1963). Hays,D. F.,’J. Basiczng. (?ram. ASMh, Ser. D ) 81D, 13‘(195$). Hill, R., J . Mech. Phys. Solids 5 , 66 (1956). Hill, R., Power, G., Quart. J . Mech. A@$. Math. 9, 313 (1956). Horowitz,H. H., Steidler, F. E., A.S.L.E. Trans. 3, 124 (1960). Horowitz, H. H.. Steidler, F. E.. A.S.L.E. Trans. 4, 275 (1961). Johnson, M. W.,’Jr., Phys.. Fluids 3, 871 (1960). Johnson, M. W., Jr., Trans. Soc. Rheol. 5 , 9 (1961). Joukowsky, N., Tschaplygin, S., Jahr. Fortschr. Math. 35, 766 (1904). Kaplan,’ Wilfred, “Advanced Calculus,” p. 96, Addison-Wesley, Cambridge, Mass., 1952. Milne, A. A., J . Basic Eng. (Trans. ASME, Ser. D ) 81D, 239 flc)clc)\ \-,“-,.
Moon, Parry, Spencer, D. E., “Field Theory Handbook,” SpringerVerlag, Berlin, 1961. Pawlowski, J., Kolloid Z . 138, 6 (1954). Raimondi, A. A , , Boyd, J., A.S.L.E. Trans. 1, 159 (1958). Reiner, Markus, Am. J . Math. 67, 350 (1945). Reynolds, O., Trans. Roy. SOC. London, Ser. A, 177, 157 (1886). Rivlin, R. S., h’ature 160, 611 (1947). Rivlin, R. S., Proc. Roy. SOC. London, Ser. A 193,260 (1948). Serrin, James, “Handbuch der Physik,” V I I I / l , Fliigge, ed., p. 231, Springer-Verlag, Berlin, 1959. Slattery, J. C., A.Z.Ch.E. J . 11, 831 (1965). Sommerfeld, A., Z. Math. Physik 50, 97 (1904). Tanner, R. I., Australian J . Appl. Sci. 14, 129 (1963). Tao, L. N., Quart. A$$. Math. 17,129 (1959). U’annier, G. H., Quart. Appl. Math. 8, 1 (1950). Wasserman, M. L., Slattery, J. C., A.2.Ch.E. J . 10, 383 (1964). RECEIVED for review September 11, 1967 ACCEPTEDJanuary 11, 1968
