INTRODUCTION
Lubrication flows are flows between nearly parallel walls of a small inclination
with respect to each other, as well as thin film flows under nearly planar interfaces.
Important operational flows such as journalbearing and pistonring lubrication
of engines and processing flows such as application of thin films wire or roll
coating and multilayer extrusion can be analyzed as lubrication flows. Lubrication
of journalbearing and pistonring is performed to reduce the friction of two
bodies in near contact and is usually accomplished by viscous fluids moving
through the narrow but variable distance between two bodies. This thin viscous
film should create to somehow extremely large pressure difference in order to
prevent the contact of two bodies. Hence the properties of lubricants are usually
modified by additives to provide various requirements of machinery systems.
The effects of elastic fluids on velocity and pressure field studied for viscoelastic
fluids obeying Upper Convected Maxwell model (UCM) (Tichy, 1996) and second
order model (Sawyer and Tichy, 1998). Advantages of PhanThienTanner (PTT)
model (PhanThien and Tanner, 1977; PhanThien, 1978) encouraged many authors
to use this model to investigate the problems. Alves et al. (2001) presented
an analytical solution of fluids following single mode PTT model for steady
flows through pipe and parallel plates. Hashemabadi et al. (2003) also
provided an analytical solution for dynamic pressurization of viscoelastic fluids
obeying SPTT between two parallel plates for simulation in single screw extruder.
Mirzazadeh et al. (2005) presented an analytical solution for purely
tangential flow of PhanThienTanner (PTT) viscoelastic fluid model in a concentric
annulus with relative rotation of the inner and outer cylinders. Oliviera and
Pinho (1999) have published a series of analytical solutions for PTT model for
flows through ducts.
The SPTT nonNewtonian model for lubrication flows has not yet been considered. Therefore, the objective of the present investigation is to solve analytically the lubrication flows of a PTT viscoelastic fluid flowing between two plates with small inclination.
PROBLEM DESCRIPTION AND MATHEMATICAL FORMULATION
Figure 1 illustrates two plates, the stationary upper one has a small slope respect to the other and the lower one is horizontal and moving with a constant velocity U. The flow is assumed to be laminar, low Reynolds number, steady state, isothermal and incompressible. The plates are subjected to no slip condition and the gravitational forces for the flow domain are negligible. The necessary conditions of lubrication flow in X direction are:

Fig. 1: 
Schematic
diagram of flow domain 
The continuity and momentum equation in lubrication approximation can be developed
respectively by an order of magnitude of full two dimensional NavierStokes
equations as follows:
The PTT constitutive equation is given by Bird et al. (1987):
Where Z usually is suggested in exponential form and reported as follows:
As mentioned previously, Reynolds number is assumed to be low, so small molecular deformation occurs and therefore considering the linearized form of Eq. 4 as follows is acceptable (Tanner, 2000):
where ε is related to the elongational behavior, η is the viscosity coefficient of the model, λ is the relaxation time and trτ is the trace of stress tensor τ.ξ is a constant parameter of PTT model. It is related to the slip velocity between the continuum medium and molecular network. For weak flows where the rate of deformation of fluid elements is not noticeable, the constant ξ is equal to zero (Alves et al., 2001) and the model is called simplified PhanThien Tanner (SPTT). τ_{(1)} is the convected time derivative of stress tensor and is defined by:
Where velocity gradient tensor ∇μ, for the problem depicted above, is simplified as:
By substituting Eq. 6 into Eq. 3, finally, it reduces to
It was found from Eq. 9 that τ_{yy} = 0, hence the trace of stress tensor will be equal to τ_{xx}. The shear stress (τ_{yx}) can be obtained by integrating Eq. 2:
Where τ_{0} indicates the shear stress at the moving wall. By dividing Eq. 8 by Eq. 10 and using Eq. 11, we get
By substituting Eq. 11 into Eq. 10, the following equation is found for the velocity gradient:
By using the linear form of stress coefficient and exerting the dimensionless terms into Eq. 13, the dimensionless velocity gradient yields:
Where the dimensionless terms are defined as
Where the dimensionless group De is the Deborah number that is a measure for
elasticity of the fluid. The term εDe^{2 }is referred to as the
viscoelastic dimensionless group. G is a dimensionless group for the pressure
drop and finally, τ*_{0} is the dimensionless shear stress at the
moving wall. Nondimensionalized boundary conditions are written as
Where h* (x*) indicates the dimensionless height profile at any point in xdirection for upper plate.
RESULTS AND DISCUSSION
By integrating Eq. 14 and using boundary condition (15), the dimensionless velocity profile is obtained:
When εDe^{2} approaches to zero, Eq. 17 converts to Newtonian velocity profile for flow between two plates depicted in Fig. 1 (White, 1991). To obtain the unknown τ*_{0} in Eq. 17, we should substitute boundary condition (16) into Eq. 17, finally, a cubic equation in the term of dimensionless shear stress at the moving wall (τ*_{0}) yields. Solution of this equation leads to one real root which is acceptable and two complex roots which are not acceptable:
Where
From Eq. 12 can be concluded it means the dimensionless
shear stress on lower plate is depend to elastic effects of viscoelastic fluid.
In Eq. 18, G, the dimensionless pressure gradient is unknown.
The correct pressure gradient must satisfy the continuity equation everywhere
between two plates. By integrating of Eq. 1, we get:
Where in this particular case we are assuming that the ycomponent of velocities v*(0) and v*(h*) are zero at both walls. By Leibniz formula, Eq. 19 can be rearranged as follows:
Substitution of dimensionless xvelocity, Eq. 17 and integration of Eq. 20, obtains a nonlinear second order differential equation for the dimensionless pressure:
Where
If we assume the pressure out of the channels is equal to gage pressure, =
0, Therefore the appropriate boundary conditions are:
If the dimensionless distance between two plates depends upon position in the linear form, then we have:
Where
For Newtonian fluids, while εDe^{2} approaches to zero, Eq.
21 can be simplified as follows:
With using of boundary conditions, Eq. 22, the dimensionless pressure distribution for Newtonian fluids between the plates can be derived as (White, 1991):
Figure 2a illustrates the dimensionless pressure p*, along the channel for various values of inclination β for the case of Newtonian fluid. For small inclination β, the pressure distribution is nearly symmetric and the maximum dimensionless pressure occurs at x* ≈ 0.5. As the slope of upper plate promotes, p*_{max} increases and moves toward the exit plane.
Equation 21 is a highly nonlinear ordinary differential
equation and cannot be solved analytically, however it is instructive to study
limiting case while dimensionless pressure gradient G, approaches to zero. We
can presume the dimensionless pressure gradient is negligible while the values
of β is less or the dimensionless viscoelastic group εDe^{2}
is high, therefore the differential Eq. 21 is simplified
as follows:
And also Eq. 18 is simplified as follows
Where
Two times of integrating of Eq. 26 and using of boundary
conditions Eq. 22, leads to an expression for dimensionless
pressure:

Fig. 2: 
(a)
Dimensionless pressure distribution along the channel for Newtonian fluid
(εDe^{2} = 0), (b) Differences of numerical and analytical
results on dimensionless pressure 
For small values of β and εDe^{2}, the results show good
agreement between numerically solution of Eq. 21 and approximate
solutions of Eq. 28, (Fig. 2b).
The dimensionless pressure distribution derived from numerical solution of
Eq. 21 has been shown in Fig. 3 for various
values of β and εDe^{2} as effective parameters. From the
dimensionless pressure profiles (Fig. 3) we anticipate two
flow regions, a region where the pressure gradient is positive (0 < x* <
x*_{max}) and a region where the pressure gradient is negative (x*_{max}
< x* < 1). For very small dimensionless viscoelastic group εDe^{2},
the pressure distribution is the same as Newtonian fluids (White, 1991). As
a limiting case, while β approaches to zero, the resulting solution converts
to the solution reported for two parallel plates according to SPTT fluid flow
(Hashemabadi et al., 2003).

Fig. 3: 
The
effect of β and εDe^{2} on pressure distribution along
the channel 
Totally, as the degree of contraction (β)
increases, p*_{max} increases and moves toward the end of channel. It must be noted that the increasing of dimensionless pressure from β
= 0.1 to β = 0.3 is larger than the increasing of pressure while the
inclination of upper plate changes from β = 0.01 to β = 0.1.
For constant angle of upper plate, the results show while εDe^{2 }increases,
the maximum pressure decreases. In other words, existence of elasticity effects
and normal stresses could change the pressure distribution between two plates
how to be more uniform respect to Newtonian fluids. The dimensionless apparent
viscosity is defined as the ratio of dimensionless shear stress to dimensionless
shear rate:
Figure 4 shows the dimensionless viscosity profile at two cross sections x* = 0.25 and 0.75. As it’s shown, the average viscosity at the entrance of the channel is higher than the average viscosity at the end region.
The typical velocity profiles have been illustrated at three cross sections of channel in Fig. 5. It can be concluded the velocity magnitude decreases relative to the Couette flow at entrance region, increases at exit region and near to location of maximum pressure, the flow is almost Couette and the velocity profiles are linear at cross section. The combined effects of Couette and Poissule flow in exit region causes lower apparent viscosity (Fig. 4) and this create higher velocity gradient at two walls, meanwhile there is no changes in velocity gradient at lower plate in entrance region. On the other hand, the dimensionless pressure gradient decreases with increasing of dimensionless viscoelastic group, so the velocity profile approaches to the velocity profile of Couette flow for this situation (Fig. 5).
The skin friction coefficient is defined as (Shah and London, 1978):
By using the dimensionless groups, the product of skin friction factor and Reynolds number becomes as follows:
Where , the dimensionless average wall shear stress can be obtained by
The dimensionless average velocity can be obtained by the following equation:
From substitution of Eq. 17, the non dimensional average velocity can be obtained:

Fig. 4: 
Variation
of apparent viscosity across the channel for β = 0.5 at x* = 0.25
and x* = 0.75 
For Newtonian fluids, the dimensionless average velocity equals to that for
parallel plate the solution converts to the previous solution (Hashemabadi et
al., 2003). Figure 6 shows variation of production of
skin friction factor C_{f, x} * and Reynolds number for different values
of β and εDe^{2}.

Fig. 5: 
Velocity
profile at three cross sections in the duct β = 0.5, εDe^{2}
=0.01 (solid line), εDe^{2} = 0.5 (dashed line) 

Fig. 6: 
Effects
of viscoelastic group εDe^{2}, on C_{f.x}* Re for
various values of β 
The results illustrate while the inclination is small, the magnitude of C_{f,
x} * Re becomes nearly constant and when the inclination increases, the
C_{f, x}* Re reduces. Also the results show when the viscoelastic group
increases, due to shear thinning effects of fluid the value of C_{f, x}*
Re decreases. For Newtonian fluids
when :
When :
When the plates are parallel and G approaches to negative infinity, Eq.
35 approaches to 24 which is equivalent to the purely Poiseuille flow. Also
when the plates are parallel and G approaches to zero, then Eq.
36 approaches to 8 which is equivalent to the purely Couette flow.
CONCLUSIONS
The analytical solution was suggested for lubrication flow of nonlinear viscoelastic
fluid between two plates that make an angle with respect to each other. It was
assumed that fluid obeys the simplified PhanThien Tanner (SPTT) model. Effects
of elasticity and elongational behavior of fluid on pressure distribution through
the channel and velocity profile were investigated. The results show with increasing
of the viscoelastic dimensionless group (εDe^{2}), the velocity
gradient at moving wall decreases and as a result, the friction factor of the
fluid at this wall decreases and the amount of maximum pressure relative to
Newtonian fluids reduces and the pressure distribution through the channel becomes
more uniform for a given upper plate inclination β. When β decreases,
the pressure distribution becomes symmetric and for two parallel plate dimensionless
pressure gradient approaches to zero. Effect of inclination of upper plate (β)
on pressure distribution for higher viscoelastic group is more considerable.
Influence of viscoelasticity on skin friction coefficient is noticeable and
results show when the contraction is slight, magnitude of C_{f.x}* Re
becomes nearly constant and when the contraction is steeper, the changes of
C_{f.x}* Re take the falling form. Also the results show when elasticity
of the fluid increases, the values of C_{f.x}* Re decrease.
Nomenclature
a, a_{0}, a_{1 }and a_{2} 
Parameters of Eq. 21 
D_{h} 
Hydraulic diameter 
D 
Deborah number (λU/h_{0}) 
G 
Dimensionless pressure gradient 
h 
Gap between two plates through the channel 
P 
Pressure 
Re 
Reynolds number (ρuD_{h}/η) 
U 
Velocity of moving plate 
u 
Velocity profile 
x 
Axial coordinate 
y 
Lateral coordinate 
Z 
Stress coefficient function 
Greek symbols
α 
Parameter of Eq. 18 
β 
Parameter of Eq. 23 
ε 
Elongational parameter of PTT model 
η 
Viscosity coefficient of PTT model 

Shear rate tensor 
λ 
Relaxation time in PTT model 
ξ 
Slip parameter 
τ 
Stress tensor 
Subscripts
e 
Values at entrance of channel 
L 
Values at end of channel 
0 
Values at the moving wall 
w 
At the wall 
Superscripts
* 
Refers to dimensionless quantities 
 
Refers to average quantities 
† 
Transpose of tensor 