On the Boundary Layer Flow over a Moving Surface in a Fluid with Temperature-Dependent Viscosity

This paper examines a boundary layer flow over a continuously moving heated flat surface with velocity   U x  in a streaming flow with velocity and with temperature dependent viscosity,   U x    T  . The momentum and the energy equations are coupled through the viscous dissipation term. The coupled boundary layer equations are transformed into a self-similar form using an appropriate similarity variable. An efficient numerical technique is used to solve the self-similar boundary layer equations. It is shown that at low enough values for the velocity ratio  , an increase in viscous dissipation enhances greatly the local heat transfer leading to temperature overshoots adjacent to the wall. The viscosity variation parameter is shown to have significant effects on the temperature dependent viscosity and the velocity and temperature distribution within the boundary layer.


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 U U    and U U    . Afzal [9] formulated a single set of equations using as reference velocity a composite velocity given by U U U     .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.

Problem Formulation
An incompressible flow past an infinite surface continuously moving with velocity U  in a streaming flow with velocity and with temperature dependent viscosity , is investigated.The fluid is of density thermal conductivity and specific heat capacity The boundary conditions for this flow are 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 0   corresponds to the constant viscosity situation.

Self-Similar Boundary Layer Equations
The basic flow is rendered in non-dimensional form through setting In non-dimensional form, the Lings-Dybbs model be- where    is the dimensionless dynamic viscosity and is the variable viscosity parameter.The case is equivalent to the case From the equation of continuity we have where the parameter  reduces to the pressure gradient We assume power-law variations in the free-stream velocity and wall velocity of the form so that.

0
, The dimensionless similarity boundarylayer equations take the form with boundary conditions 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.

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 0    .The case when 0   is corresponds to a wall moving in an otherwise quiescent fluid     0 corresponds to flow over a stationary wall and so that the wall and the freestream move with the same speed.When 0 12    , the wall moves faster that the free-stream while the case when 1

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 The effect of varying viscous dissipation parameter, , and the velocity ratio Ec  on the temperature distri- bution in the boundary layer is shown in Figures 4 and 5. Figure 4 shows that for 0.1

Ec
, 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   negative, changing sign to positive for 0.3   .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.

Conclusions
A self-similar boundary layer flow has been presented 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 0.5 1    where the skin friction is positive and hence the free-stream exerts a dragging force on the boundary layer.

f
is the boundary-layer similarity variable and   and     are the scaled free-stream velocity and temperature, respectively.The parameter x Re is the local Reynolds number defined as

Figure 1 .
Figure 1.Schematic diagram for the problem.

2 1 
 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, f C , and the local Nusselt number, x Nu , respectively defined as 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   f   , 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 peaks amplitudes increase with decrease in  .