On the Boundary Layer Flow over a Moving Surface in a Fluid with Temperature-Dependent Viscosity ()
1. Introduction
Studies on heat and mass transfer in boundary layers over continuously moving or stretching surfaces have been increasing due to their wide variety of applications in manufacturing processes such as glass-fibre production, metal extrusion, materials-handling conveyors and paper production.
One of the earliest studies on boundary-layer flow past moving surfaces was initiated by Sakiadis [1], who investigated momentum transfer for a flow over a continuously moving plate in quiescent fluid. The results of Sakiadis were later verified experimentally by Tsou et al. [2]. Over recent years studies of boundary layer past moving or stretching surfaces in otherwise quiescent fluids included the work of Ali [3] who investigated similarity solutions for a thermal boundary layer over a power-law stretching surface with suction or injection; Elbashbeshy [4] who studied heat transfer over a stretching surface with suction or injection; Magyari and Keller [5] who studied similarity solutions for boundary layer flow over an exponentially stretching surface and Mureithi [6] who examined linear stability properties of a boundary layer flow over a moving surface in a streaming flow.
Studies on free-stream effects on boundary-layer flows over moving or stretching surfaces included the work of Abdelhafez [7] and Chappidi and Gunnerson [8] who independently considered flows over moving surfaces in which both the surface and the free stream moved in the same direction. In their studies, they formulated two sets of boundary value problems for the cases 
and
. Afzal [9] formulated a single set of equations using as reference velocity a composite velocity given by
. Later Lin and Huang [10] used Afzal’s formulation to study momentum and heat transfer for a flow over a surface moving parallel or reversely to the free stream with temperature dependent viscosity. A study by Afzal [11] investigated momentum transfer on a power law stretching surface with free-stream pressure gradient.
The current study investigates a boundary layer flow over a moving surface in a streaming flow with a temperature dependent dynamic viscosity,
. The Ling and Dybbs [12] model for 
is used in this study.
In Section 2, we formulate the problem. In Section 3, boundary layer equations are reduced to the self-similar form. In Section 4, numerical solutions for the self-similar boundary layer equations are presented and discussed and conclusions are drawn in Section 5.
2. Problem Formulation
An incompressible flow past an infinite surface continuously moving with velocity 
in a streaming flow with velocity 
and with temperature dependent viscosity
, is investigated. The fluid is of density
, thermal conductivity 
and specific heat capacity 
(at constant pressure). The boundary layer equations are
(1)
The boundary conditions for this flow are
(2)
for a flow over an impermeable surface
.
The following temperature dependent viscosity model due to Ling and Dybbs [12] is used here:
where 
is a constant, 
is the constant reference viscosity in the absence of heating. The case 
corresponds to the constant viscosity situation.
3. Self-Similar Boundary Layer Equations
The basic flow is rendered in non-dimensional form through setting
where 
is the reference velocity (Afzal et al. [9]), 
is the boundary-layer similarity variable and 
and 
are the scaled free-stream velocity and temperature, respectively. The parameter 
is the local Reynolds number defined as
.
In non-dimensional form, the Lings-Dybbs model becomes
where 
is the dimensionless dynamic viscosity and 
is the variable viscosity parameter. The case 
is equivalent to the case 
corresponding to constant viscosity.
From the equation of continuity we have
Figure 1. Schematic diagram for the problem.
where the parameter 
reduces to the pressure gradient parameter
We assume power-law variations in the free-stream velocity and wall velocity of the form 
so that.
and
. The dimensionless similarity boundarylayer equations take the form
(3)
(4)
with boundary conditions
(5)
where
The parameter 
is the Prandtl number and 
is the Eckert number. The flow is self-similar if one of the following is satisfied:
1. n = 0 for any Ec.
2. Ec = 0 for any n (negligible viscous dissipation).
remarks We have assumed that both the wall and the free stream move in the same direction so that
. The case when 
is corresponds to a wall moving in an otherwise quiescent fluid
, 
corresponds to flow over a stationary wall 
and 
is equivalent to 
so that the wall and the freestream move with the same speed. When
, the wall moves faster that the free-stream while the case when 
corresponds to the free-stream moving faster than the wall.
The surface shear stress and surface heat transfer are represented using the local skin friction factor, 
, and the local Nusselt number, 
, respectively defined as
4. Numerical Solution and Discussion of Results
The coupled self-similar boundary layer Equations (3) and (4) together with the boundary conditions (5) are solved numerically using a shooting method coupled with the fourth-order Runge-Kutta scheme.
The results presented here are for the cases when
. Self-similar solutions were obtained for two cases. Case one is the flow viscous dissipation and
. Case two corresponds to the case with 
but without viscous dissipation effects.
Figures 2 and 3 show that the effect of varying the fluid viscosity variation parameter 
on the temperaturedependent dynamic viscosity, 
and the streamwise velocity
, within the boundary layer. At any location within the boundary layer 
decreases with increase in the viscosity parameter,
. The boundary layer thickness is found to decrease with increase in
. The parameter 
is a measure of fluid viscosity variation.
The effect of varying viscous dissipation parameter, 
, and the velocity ratio 
on the temperature distribution in the boundary layer is shown in Figures 4 and 5. Figure 4 shows that for
, increasing 
results in temperature over-shoot near the wall, with peaks in creasing with increase in
. Figure 5 illustrates the effect of varying the velocity ratio on the temperature distribution. For the case when
, the results show that the temperature peaks are realized for 
and the peaks amplitudes increase with decrease in
.
Figure 2. Effects of viscosity variation parameter on the dynamic viscosity for 
Figure 3. Effects of viscosity variation parameter on velocity profiles for 
.
Figure 4. Effects of varying 
on temperature distribution for
.
Figure 5. Effects of viscosity variation on the temperature distribution profiles for 
.
The skin-friction is presented as a function of 
in Figures 6 and 7. These figures show the effect of varying 
on the local skin friction coefficient for the case when 
and
. For the case when 
, the local skin friction coefficient is positive and hence the fluid exerts a dragging force on the wall. For this case, increasing 
results in increase in the skin friction coefficient. For the case when
, the local skin friction coefficient is negative, which is an indication that the wall drags the fluid. Also, increasing 
results in a decrease in the skin friction.
In Figure 8, the effect of varying 
on local Nusselt number is shown as a function of 
for the cases when
. The results show that the heat transfer coefficient decreases with increase in 
and increases with increase in
.
The results shown in Figure 9 are interesting. It is shown that for
, the local Nusselt number is
Figure 6. The skin friction coefficient against 
for the case when
.
Figure 7. The skin friction coefficient against 
for the case when
.
Figure 8. The heat transfer coefficient against 
for the case when
.
Figure 9. The heat transfer coefficient against 
for the case when
.
negative, changing sign to positive for
. This explains the results observed in Figure 7 where temperature over-shoot were observed adjacent to the wall. These results show that for low enough values for
, the heat transfer from wall to the fluid is greatly enhanced resulting in temperature over-shoots adjacent to the wall.
The effect of varying 
on the velocity and temperature distribution within the boundary layer is presented in Figures 10 and 11 for the case when wall is moving faster than the free-stream. The results show that velocity boundary layer thickness decreases with increase 
and that the temperature peaks decrease with increase in
. This shows that the increasing 
results in a decrease in heat transfer from the wall to the fluid.
5. Conclusions
A self-similar boundary layer flow has been presented
for a flow over a continuously moving heated surface in a fluid with temperature dependent viscosity. The selfsimilar equations were solved numerically and the results are presented in graphs.
In this study the effects of varying the viscosity variation 
and the velocity ratio 
are investigated for the case when the surface moves in the same direction as the free-stream.
For low enough values for the velocity ratio ξ, the local heat transfer is found to be negative, indicating the heat transfer from the wall to the fluid is greatly enhanced near the wall as the Eckert number increases. This is seen in the temperature distribution profiles where temperature peaks are observed adjacent to the wall.
For the case when the wall moves faster than the fluid, the skin friction coefficient is negative, indicating that wall drags the fluid. The reverse occurs for the case when 
where the skin friction is positive and hence the free-stream exerts a dragging force on the boundary layer.