We analyze the lubrication flow of a viscoelastic fluid to account for the time dependent nature of the lubricant. The material obeys the constitutive equation for Phan-Thein-Tanner fluid (PTT). An explicit expression of the velocity field is obtained. This expression shows the effect of the Deborah number $(De=λU/L,$ λ is the relaxation time). Using this velocity field, we derive the generalized Reynolds equation for PTT fluids. This equation reduces to the Newtonian case as $De→0.$ Finally, the effect of the Deborah number on the pressure field is explored numerically in detail and the results are documented graphically.

1.
Tichy
,
J. A.
,
1996
, “
Non-Newtonian Lubrication With the Convective Maxwell Model
,”
ASME J. Tribol.
,
118
(
2
), pp.
344
349
.
2.
Sawyer
,
W. G.
, and
Tichy
,
J. A.
,
1998
, “
Non-Newtonian Lubrication With the Second-Order Fluid
,”
ASME J. Tribol.
,
120
, pp.
622
628
.
3.
Huang
,
P.
,
Li
,
Zhi-Heng
,
Meng
,
Yong-Gang
, and
Wen
,
Shi-Zhu
,
2002
, “
Study on Thin Film Lubrication With Second-Order Fluid
,”
ASME J. Tribol.
,
124
, pp.
547
552
.
4.
Phan-Thein
,
N.
, and
Tanner
,
R. I.
,
1977
, “
A New Constitutive Equation Derived From Network Theory
,”
J. Non-Newtonian Fluid Mech.
,
2
, pp.
353
365
.
5.
Bird
,
R. B.
,
Dotson
,
P. J.
, and
Johnson
,
N. L.
,
1980
, “
Polymer Solution Rheology Based on a Finitely Extensible Bead-Spring Chain Model
,”
J. Non-Newtonian Fluid Mech.
,
7
, pp.
213
235
.
6.
Giesekus
,
H.
,
1982
, “
A Simple Constitutive Equation for Polymer Based on the Concept of the Deformation Dependent Tensorial Mobility
,”
J. Non-Newtonian Fluid Mech.
,
11
, pp.
69
109
.
7.
Quinzani
,
L.
,
Armstrong
,
R. C.
, and
Brown
,
R. A.
,
1995
, “
Use of Coupled Birefringence and LDV Studies of Flow Through a Planar Contraction to Test Constitutive Equations for Concentrated Polymer Solutions
,”
J. Rheol.
,
39
, pp.
1201
1228
.
8.
Baaijens
,
F. P. T.
,
1993
, “
Numerical Analysis of Start-Up Planar and Axisymmetric Contraction Flows Using Multi-Mode Differential Constitutive Models
,”
J. Non-Newtonian Fluid Mech.
,
48
, pp.
147
180
.
9.
Azaiez
,
J.
,
Guenette
,
R.
, and
,
A.
,
1996
, “
Numerical Simulation of Viscoelastic Flows Through a Planar Contraction
,”
J. Non-Newtonian Fluid Mech.
,
62
, pp.
253
277
.
10.
Bolach
,
H.
,
Townsend
,
P.
, and
Webster
,
M. F.
,
1996
, “
On Vortex Development in Viscoelastic Expansion and Contraction Flows
,”
J. Non-Newtonian Fluid Mech.
,
65
, pp.
133
149
.
11.
White
,
S. A.
, and
Baird
,
D. G.
,
1988
, “
Numerical Simulation Studies of the Planar Entry Flow of Polymer Melts
,”
J. Non-Newtonian Fluid Mech.
,
30
, pp.
47
71
.
12.
Phan-Thein
,
N.
,
1978
, “
A Nonlinear Network Viscoelastic Model
,”
J. Rheol.
,
22
, pp.
259
283
.
13.
O’Brien, S. B. G., and Schwartz, L. W., 2002, “Theory and Modeling of Thin Film Flows,” Encyclopedia of Surface and Colloid Science, pp. 5283–5297.