Unsteady Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity ()
1. Introduction
Due to number of applications in industrial manufacturing process, the problem of boundary layer flow past a stretching plate has attracted considerable attention of researchers during the past few decades. Examples of such technological process are hot rolling, wire drawing, glass-fiber and paper production. In the process of drawing artificial fibers the polymer solution emerges from orifice with a speed which increases from almost zero at the orifice up to a plateau value at which remains constant. The moving fiber, which is of great technical importance, is governed by the rate at which the fiber is cooled and this, in turn affects the final properties of the yarn. A number of works are presently available that follow the pioneering classical work of Sakiadis [1], F. K. Tsou, E. M. Sparrow, R. J. Goldstein [2] and Crane [3]. Table 1 lists some relevant works that pertain to cooling liquids, i.e., heat transfer for stretching surface.
There are liquid metals whose thermal conductivity varies with temperature in an approximately linear manner in the range from 0˚F to 400˚F. In 1996, T. C. Chiam [21] considered heat transfer problem with variable thermal conductivity in stagnation-point flow towards stretching sheet. Naseem Ahmad and Kavita Marwah [22] also studied boundary layer flow of Walters Liquid B Model with heat transfer for linear stretching plate with variable conductivity numerically. In 2010, Ahmad and Mishra investigated unsteady boundary layer flow and heat transfer over a stretching sheet [23].
In almost all the problems of stretching sheet with heat transfer where closed form solution is obtained, the thermal conductivity of liquid has been taken constant. The present paper is the extension of the work done by N. Ahmad and M. Mishra [23] assuming that the thermal conductivity varies in linear manner with temperature. The temperature field has two parts: one mean temperature, other is due to variable thermal conductivity. Both the parts mean temperature and temperature due to variable thermal conductivity have been analyzed thoroughly for some new recommendations.
2. Mathematical Formation and Solution
The problem considered here is the unsteady boundary layer flow due to a stretching flat plate in a quiescent viscous incompressible fluid. The flow is two dimensional where x-axis is along the plane of moving plate and y-axis is normal to it, respectively. We assume that the surface is moving continuously with the velocity
and t < 1/a in the positive x-direction. Under these assumptions, the boundary layer flow along moving plate is governed by the equations
(2.1)
(2.2)
Table 1. Some relevant works that pertain to cooling liquids.
where u, the horizontal velocity component; v, the vertical velocity component;
, the kinematic viscosity The relevant boundary conditions are:
y = 0, u = us, v = 0 y → ¥, u = 0 Introducing the dimensionless variables
the Equations (2.1) and (2.2) reduce to
(2.3)
(2.4)
with boundary conditions
(2.5)
where bar has been dropped for convenience.
Setting the similarity solution of the form
and using continuity equation, we have
(2.6)
Putting u and v in the Equation (2.4), we have
(2.7)
and the relevant boundary conditions become
(2.8)
(2.9)
Boundary conditions suggest that the velocity function f may be of the form 
where r is unknown to be determined. Thus
and the Equation (2.7) gives
. Therefore, we have the velocity components as follows:
and
(2.10)
3. Skin Friction
The wall shear stress at the stretching plate is given by
Thus, the skin friction is
(3.1)
4. Heat Transfer Problem
In absence of viscous dissipation and heat generation, the energy equation for two dimensional heat flow is given b
(4.1)
subject to boundary conditions
(4.2)
where TP is plate temperature, T¥ is temperature of surrounding fluid, CP is specific heat at constant pressure and k is thermal conductivity.
4.1. Case A: Prescribed Power Law Surface Temperature (PST)
Let the surface temperature be of the form
while the temperature outside the dynamic region be
. Now, we define the dimensionless temperature by 
For liquid metals, it has been found that the thermal conductivity varies with temperature in an approximately linear manner in the range from 0˚F to 400˚F. Thereforewe assume k as 
where
. Nowsubstituting u and v in the Equation (4.1) and changing the independent variable y to
, we have
(4.3)
with boundary conditions
(4.4)
The Equation (4.3) can be rewritten as
(4.5)
From Equation (4.5), we note that the heat transfer takes place in two parts according to 
and
. If
, then we have the main heat transfer due to constant thermal conductivity i.e.
(4.6)
(4.7)
and if
, then we get the first correction equation to main heat transfer as
(4.8)
, as
(4.9)
The solution of the Equation (4.6) is
where
is incomplete gamma function Equation (4.8) is a non linear differential equation of order two. Let the solution of this equation be of the form:
Putting this solution in Equation (4.8) we have
(4.10)
The roots of this equation are 0 and 1/2. Therefore
(4.11)
The solution (4.11) of the Equation (4.8) does not satisfy the condition 
as
. This condition is met only for
. Therefore, the heat transfer in case 
takes place within the dynamic region
.
4.2. Nusselt Number
The coefficient of convective heat transfer is given by
(*)
Therefore the Nusselt number is (see Table 2):
(4.12)
4.3. Case B: Prescribed Power Law Surface Heat Flux (PHF Case)
The power law heat flux on the surface of stretching plate is considered to be a quadratic power of x in the form
at
(4.13)
, as
(4.14)
where D is a constant, k is the thermal conductivity. Now we define dimensionless temperature 
by
(4.15)
where
when 
where 
Writing the Equation (4.2) in terms of
, we get the following differential equation
(4.16)
together with boundary conditions:
(4.17)
Equating the terms independent of and the terms involving from Equation (4.16), we get the following two boundary value problems:
(4.18)
and
(4.19)
where the nomenclature 
is main heat transfer when thermal conductivity is constant and 
is correction to the heat flow due to variation in thermal conductivity in PHF case. The solution of the Equation (4.18) together with the boundary conditions (4.18) is
(4.20)
The general solution of the Equation (4.19) is
(4.21)
where we again observe that the dynamic region for this temperature field is
. Hence, the boundary conditions (4.17) have been modified as
, and 
as
(4.22)
Using boundary conditions (4.22), the solution (4.19) becomes
4.4. Nusselt Number (See Table 3)
Recalling (*), we have
5. Discussion and Results
The problem of unsteady boundary layer flow of viscous
incompressible fluid overstretching plate has been analyzed. The velocity field has been obtained by similarity transformation method. Later, the heat flow problem has been studied by considering PST and PHF cases. We summarize the results as in Figure 1 to Figure 6.
Figure 1 shows that horizontal component of velocity u. It increases as time progresses within the dynamical region [0, 1]. We also see that u is maximum in the immediate neighborhood of stretching plate and it starts decreasing as 
. In fully developed flow, as time goes on progressing, the velocity progresses too.
Figure 2 is a graph of v versus y for different instant of time. The vertical component v is almost constant within [0, 1] and later it starts increasing. v progresses as we march away from the slit and it also increases as time progresses.
Figure 3 is to study the variation of mean temperature field 
with respect to Prandtl number Pr. We see that as Pr increases, 
decreases within dynamical region
[0,1] in PST case. As 
decreases, i.e. kinematic viscosity 
increases in turn viscosity of fluid increases. In case of more viscosity, generally flow of heat becomes slow. It is supported by our study.
Figure 4 is expressing the trend of temperature field 
due to variable thermal conductivity. We see that the contribution of this temperature is more near the moving plate than as we go away from the plate in PST case. 
Figure 1. Velocity field at different instant of time.
Figure 2. Velocity component v at different instant of time.
Figure 3. Mean temperature field for different values of Prandtl number Pr in PST case.
Figure 4. Temperature distribution due to variation in thermal conductivity.
Figure 5. Skin friction with respect to time t.
Figure 6. Temperature field for different values of Prandtl number in PHF case.
is independent of Prandtl number Pr.
Figure 5 is the graph of ReCf versus time. We see that as time progresses the skin friction increases .We mean that as time progresses the velocity increases, in turn skin friction increases.
Figure 6 is temperature field in PHF case. We see that it approaches to zero asymptotically.
We see the gm for Pr = 5.10 of liquid oxygen at 56˚K, Pr = 2.15 of para-hydrogen at 14˚K and Pr = 1.54 of liquid ammonia at 10˚C. It has been observed that gm increases absolutely as Pr increases, but for Pr = 9.42 of water at 10˚C, gm behaves in a different way due to its density.
Nusselt number in PST case is given by the Equation (4.12) where we observe referring Table 2 that it depends on Prandtl number and time t as well. From Table 1 we find that as Prandtl number increases the Nusselt number also increases but Nusselt number decreases as time increases.
Referring Table 3 for PHF case, we see the same pattern as in PST case from Table 2.
We have not got Prandtl number from the temperature field due to variation in thermal conductivity.