The steady laminar boundary layer flow over a continuously stretching plate
was first studied by Crane (1970), who obtained an exact
solution to the Navier-Stokes equations. This problem was then extended by Afzal
and Varshney (1980) to a general power law of stretching velocity uw~xm,
where, x is the distance from the issuing slit and m is a constant. The value
m = 0 corresponds to a moving surface with constant velocity studied by Sakiadis
(1961), while m = 1 corresponds to the problem posed by Crane
(1970). The development of the boundary layer due to a stretching permeable
sheet was studied by Gupta and Gupta (1977), who reported
an exact solution for the flow field and a solution in incomplete gamma functions
for the thermal field. Ali (1995) studied the general
case when the sheet is stretched with stretching velocity of the form xm.
The boundary layer flow past a stretching plane surface in the presence of
a uniform magnetic field, which has practical relevance in polymer processing,
was studied by Pavlov (1974). Andersson
(1995) then demonstrated that the similarity solution derived by Pavlov
(1974) is not only a solution to the boundary layer equations, but also
represents an exact solution to the complete Navier-Stokes equations. Liu
(2005) extended Andersson's result by finding the temperature distribution
for non-isothermal stretching sheet, both in the prescribed surface temperature
and prescribed surface heat flux conditions, in which the surface thermal conditions
are linearly proportional to the distance from the origin.
The heat transfer aspect for the problem posed by Crane
(1970) was studied by Grubka and Bobba (1985), who
reported the solution for the energy equation in terms of Kummer's functions.
Several closed-form analytical solutions for specific conditions were also reported.
Chen and Char (1988) investigated the effects of suction
and injection on the heat transfer characteristics of a continuous, linearly
stretching sheet for both the power law surface temperature and the power law
surface heat flux variations. Char (1994) then studied
the case when the sheet immersed in a quiescent electrically conducting fluid
in the presence of a transverse magnetic field. The effect of thermal radiation
on the heat transfer over a nonlinearly stretching sheet immersed in an otherwise
quiescent fluid has been studied by Bataller (2008).
The unsteady boundary layer flow over a stretching sheet has been studied by
Devi et al. (1991), Elbashbeshy
and Bazid (2004) and quite recently by Tsai et al.
(2008). The objective of the present study is to find the similarity solution
for MHD flow over a linearly stretching sheet with prescribed surface temperature.
The governing partial differential equations with three independent variables
are transformed to ordinary differential equations using similarity transformation,
before being solved numerically by the Keller-box method. The results obtained
are then compared with those of Grubka and Bobba (1985)
and Liu (2005) as well as the series solution for the
steady-state flow case to support their validity.
Consider an unsteady, two-dimensional laminar boundary layer flow over a continuously
stretching plate immersed in an incompressible electrically conducting fluid.
At time t = 0, the plate is impulsively stretched with the velocity Uw
(x, t) along the x-axis, keeping the origin fixed in the fluid of ambient temperature
T∞. The stationary Cartesian coordinate system has its origin
located at the leading edge of the plate with the positive x-axis extending
along the plate, while the y-axis is measured normal to the surface of the plate.
A transverse magnetic field of strength B is assumed to be applied in the positive
y-direction, normal to the surface. The induced magnetic field is assumed to
be small compared to the applied magnetic field and is neglected. The simplified
two-dimensional equations governing the flow are:
where, u and v are the velocity components along the x and y axes, respectively, T is the fluid temperature in the boundary layer, t is time and v, ρ and α are the kinematic viscosity, fluid density and thermal diffusivity, respectively. We shall assume that the boundary conditions of Eq. 1-3 are:
We assume that the stretching velocity Uw (x,t) and the surface temperature Tw (x, t) are of the form:
where a, b and c are constants with a>0, b≥0 and c≥0 (with ct<1)
and both a and b have dimension time-1. It should be noticed that
at t = 0 (initial motion), Eq. 1-3 describes
the steady flow over a stretching surface. This particular form of Uw
(x, t) and Tw (x, t) has been chosen in order to be able to devise
a new similarity transformation, which transforms the governing partial differential
Eq. 1-3 into a set of ordinary differential
equations, thereby facilitating the exploration of the effects of the controlling
parameters (Andersson et al., 2000). Further,
to obtain similarity solutions of Eq. 1-4,
we assume that the unsteady magnetic field B is of the form ,
where B0 is a constant.
The continuity Eq. 1 is satisfied by introducing a stream function ψ such that:
The momentum and energy equations can be transformed into the corresponding ordinary differential equations by the following transformation:
The transformed ordinary differential equations are:
where, primes denote differentiation with respect to η, M = σB20/(ρa) is the magnetic parameter, A = c/a is a parameter that measures the unsteadiness and Pr is the Prandtl number. The boundary conditions (4) now become:
f(0) = 0, f'(0) = 1, θ (0) = 1
We note that when A = 0 (steady-state flow), the problem under consideration
reduces to those considered by Liu (2005) for prescribed
surface temperature case, while when A = 0 and M = 0 (without magnetic field)
the present problem reduces to those of Grubka and Bobba
(1985), for which an exact analytical solution was reported.
The physical quantities of interest are the skin friction coefficient Cf and the local Nusselt number Nux, which are defined as:
where, the surface shear stress τw and the surface heat flux qw are given by:
with μ and k being the dynamic viscosity and thermal conductivity, respectively. Using the non-dimensional variables Eq. 7, we obtain:
where, Rex = Uwx/v is the local Reynolds number.
When A = 0, the problem under consideration reduces to the steady-state flow, where the exact solution for the flow field is given by:
and the solution for the thermal field in terms of Kummer's functions is:
where, γ = Pr/(1+M) and F (a, b, z) denotes the confluent hypergeometric
function (Abramowitz and Stegun, 1965), with:
By using Eq. 14 and 15, the skin friction
coefficient f" (0) and the local Nusselt number -θ' (0) can be shown to
be given by:
Moreover, when Pr = 1 and M = 0, the solution θ (η) given by Eq. 15 can be expressed as θ(η) = f' (η) = e-η, which implies:
RESULTS AND DISCUSSION
The nonlinear ordinary differential Eq. 8 and 9
subject to the boundary conditions Eq. 10 were solved
numerically by means of an implicit finite-difference scheme known as the Keller-box
method as described in the books by Keller (1971) and
Cebeci and Bradshaw (1988), for several values of A, M
and Pr. This method has been successfully used by the present author to solve
various boundary-value problems (Ishak et al., 2010a-d).
The step size Δη in η and the position of the edge of the boundary layer η∞ have to be adjusted for different values of parameters to maintain the necessary accuracy. In this study, the values of Δη between 0.001 and 0.1 were used, depends on the values of the parameters used, in order that the numerical values obtained are independent of Δη chosen, at least to four decimal places. However, a uniform grid Δη = 0.01 was found to be satisfactory for a convergence critirion of 10-5 which gives accuracy to four decimal places, in nearly all cases. On the other hand, the boundary layer thickness η∞ between 6 and 50 was chosen where the infinity boundary conditions are achieved. To assess the accuracy of the present method, comparison with previously reported data available in the literature, as well as the series solution for the steady-state case, is made for several values of A, M and Pr, as given in Table 1, which shows an excellent agreement.
Figure 1 and 2 present the velocity profiles
for different values of A and M, respectively, when the other parameter is fixed.
From both figures, it is seen that the velocity gradient at the surface increases
(in magnitude) with both A and M. Thus, the magnitude of the skin friction coefficient
|f"(0)| increases as A or M increases. This observation is in agreement
with the series solution for the steady-state case given by Eq.
16. Figure 1 and 2 as well as Eq.
16 show that the velocity gradient at the surface f"(0) is negative for
all values of parameters considered. Physically, negative values of f" (0) means
the solid surface exerts a drag force on the fluid. This is not surprising since
the development of the velocity boundary layer is caused solely on the stretching
plate. Further, the velocity is found to decrease as the distance from the surface
increases and reaches the boundary condition at infinity asymptotically.
|| Values of -θ'(0) for various values of A, M and Pr
||Velocity profiles f' (η) for different values of A when
M = 1
||Velocity profiles f' (η) for different values of M when
A = 1
We note that, the Prandtl number Pr gives no influence to the development
of the velocity boundary layer, which is clear from Eq. 8.
In many practical applications, the characteristics involved, such as the heat
transfer rate at the surface are vital since they influence the quality of the
final products. The temperature profiles presented in Fig. 3
show that the temperature gradient at the surface increases (in magnitude) as
A increases, which implies an increase of the heat transfer rate at the surface
-θ' (0). The same phenomenon is observed for the variation of θ(η)
with Pr as can be seen from Fig. 4, i.e., the heat transfer
rate at the surface increases with increasing Pr. This can be explained as a
higher Prandtl number fluid has a relatively low thermal conductivity, which
reduces conduction and thereby the thermal boundary layer thickness and as a
consequence increases the heat transfer rate at the surface (Chen
and Char, 1988). Conversely, the absolute value of the temperature gradient
at the surface decreases with an increase in M, as shown in Fig.
5. Thus, the heat transfer rate at the surface decreases with increasing
M. Though the magnetic parameter does not directly enter into the energy equation,
it actually affects the velocity distribution and therefore increases the temperature
profile indirectly (Liu, 2005).
||Temperature profiles θ(η) for different values of
A when Pr = 7 and M = 1
||Temperature profiles θ(η) for different values of
Pr when M = 1 and A = 1
||Temperature profiles θ(η) for different values of
M when Pr = 7 and A = 1
It can be observed from Fig. 3-5 that
θ(η) decreases to zero so as to meet the far field boundary condition
θ(∞) = 0 asymptotically for all values of parameters considered.
Finally, we note that in the absent of the magnetic field, the velocity profile is identical to the temperature profile, i.e., f' (η) = θ(η), for Prandtl number unity.
The problem of unsteady MHD boundary layer flow and heat transfer due to a
stretching plate immersed in an electrically conducting fluid was investigated
numerically. The effects of the governing parameters, namely the unsteadiness
parameter A, magnetic parameter M and Prandtl number Pr on the fluid flow and
heat transfer characteristics were thoroughly examined. The numerical results
obtained are comparable very well with previously reported cases, as well as
the series solution for the steady-state case. It was found that the heat transfer
rate at the surface -θ'(0) increases with A and Pr, but decreases with
This study was supported by a research grant (Project Code: 06-01-02-SF0610) from the Ministry of Science, Technology and Innovation (MOSTI), Malaysia.