UNSTEADY FREE CONVECTIVE FLOW
PAST A MOVING VERTICAL POROUS
PLATE WITH NEWTONIAN HEATING
SANKAR KUMAR GUCHHAIT
Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, West Bengal, India
e-mail: asitiitk@gmail.com
SANATAN DAS
Department of Mathematics, University of Gour Banga, Malda 732 103, West Bengal, India
e-mail: tutusanasd@yahoo.co.in
RABINDRA NATH JANA
Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, West Bengal, India
e-mail: jana261171@yahoo.co.in
Abstract:
The unsteady free convective flow past a vertical porous plate with Newtonian heating has been studied. The governing equations have been solved numerically by Crank-Nicolson implicit finite-difference scheme. The variations of velocity and fluid temperature are presented graphically. It is found that the fluid velocity decreases with an increase in Prandtl number. Both the fluid velocity and the fluid temperature increase with an increase in suction parameter. An increase in Grashof number leads to rise in the fluid velocity. Further, it is observed that the shear stress and the rate of heat transfer at the plate increase with an increase in either Prandtl number or suction parameter or time.
Keywords: Free convection flow; Prandtl number; Porous plate; Suction, Grashof number; Newtonian heating.
Introduction
boundary-layer flow past an impulsively started vertical surface with Newtonian heating has been studied by (Chaudhary and Jain, 2007). (Mebine and Adigio, 2009) have analyzed the unsteady free convection flow with thermal radiation past a vertical porous plate with Newtonian heating. (Narahari and Ishak, 2011) have investigated the radiation effects on free convection flow past a moving vertical plate with Newtonian heating. Recently, (Das et al., 2012) have presented the radiation effects on unsteady free convection flow past a vertical plate with Newtonian heating.
In this paper, we study the unsteady free convection flow past a vertical porous plate with Newtonian heating. At time t0, both the fluid and the plate are at rest with constant temperature T. At time t> 0, the plate is given an impulsive motion in the vertically upward direction against gravitational field with a uniform velocity U0 while fluid is sucked from the plate with velocity v=v0, where v0> 0 is the suction velocity
and v0< 0 is the blowing velocity at the plate. It is assumed that rate of heat transfer from the surface is
proportional to the local surface temperature T. It is found that both the velocity u as well as the temperature
of the fluid increase with an increase in either Prandtl number Pr or suction parameter S. It is also found that an increase in Grashof number Gr leads to rise the fluid velocity. Further, it is observed that the shear stress x and the rate of heat transfer at the plate ( = 0) increase with an increase in either Prandtl number Pr
or suction parameter S or time .
Formulation of the problem and its solutions
Consider a two-dimensional unsteady flow of an incompressible viscous fluid past an impulsively stared infinitely long vertical plate and subjected to a thermal radiation. The
x
-axis is taken along the vertical plate in an upward direction and y-axis is taken normal to the plate. At time t0, both the fluid and plate are at rest with constant temperature T. At time t> 0, the plate is given an impulsive motion in the vertically upward direction against gravitational field with a uniform velocity U0. It is assumed that rate of heat transfer from thesurface is proportional to the local surface temperature T. Since the plate is considered infinite in the x -direction, all the physical variables are the functions of y and t only. The equation of continuity is v= 0
y
which on integration gives v= constant =v0, where v0> 0 for the suction and v0 < 0 for the blowing at the
plate.
Figure 1: Geometry of the problem.
Under usual Boussinesq approximation, the momentum and energy equations are
2
0 = 2 ( ),
u u u
v g T T
t y y
(1)
2
0 = 2 ,
p
T u k T
v
t y c y
(2)
neglected for small velocities in the energy equation (2). The initial and boundary conditions are
0, for 0 and 0,
u T T y t
0
= , T q at = 0 at 0,
u U T y t
y k
(3)
0, as for > 0,
u T T y t
where q is the constant heat flux. We introduce dimensionless variables
2
0 0
0
( ) , , , .
y U tU u T T u
U T
(4)
On the use of (4), equations (1) and (2) can be written in a dimensionless form as
2
2 ,
u u u S Gr
(5)
2 2 1 , u S Pr
(6)
where 0 0
= v
S
U is the suction parameter, =
p
c Pr
k
, the Prandtl number, 3
0
= g T
Gr U
, the Grashof number
and the characteristic velocity U0 is defined by
q k
. The corresponding boundary conditions for u and are
0, 0 for 0 and 0,
u
1, d (1 ) at 0 for > 0,
u d
(7)
0, 0 as for > 0.
u
Numerical Solution
One of the most commonly used numerical methods is the finite difference technique, which has better stability characteristics, and is relatively simple, accurate and efficient. Another essential feature of this technique is that it is based on an iterative procedure and a tridiagonal matrix manipulation. This method provides satisfactory results but it may fail when applied to problems in which the differential equations are very sensitive to the choice of initial conditions. In all numerical solutions the continuous partial differential equation is replaced with a discrete approximation. In this context the word discrete means that the numerical solution is known only at a finite number of points in the physical domain. The number of those points can be selected by the user of the numerical method. In general, increasing the number of points not only increases the resolution but also the accuracy of the numerical solution. The discrete approximation results in a set of algebraic equations that are evaluated (or solved) for the values of the discrete unknowns. The mesh is the set of locations where the discrete solution is computed. These points are called nodes and if one were to draw lines between adjacent nodes in the domain the resulting image would resemble a net or mesh.
When time-accurate solutions are important, the Nicolson scheme has significant advantages. The Crank-Nicolson scheme is not significantly more difficult to implement and it has a temporal truncation error that is
2
( )
O as explained by (Recktenwald, 2011). The Crank-Nicolson scheme is implicit, it is also unconditional stable (Ames, 1992, Isaacson and Keller,1994 and Burden and Faires, 1997). In order to solve the equations (5) and (6) under the initial and boundary conditions (7), an implicit finite difference scheme of Crank-Nicolson's type has been employed. The right hand side of the equations (5) and (6) is approximated with the average of the central difference scheme evaluated at the current and the previous time step. The finite difference equation corresponding to equations (5) and (6) are as follows:
, 1 , 1, ,
i j i j i j i j
u u u u S
= 1, 2 , 1, 1,21 2 , 1 1, 1 , 1 , , 2 2( )
i j i j i j i j i j i j i j i j
u u u u u u
Gr
(9)
and i j, 1 i j, S ui 1,j ui j,
1, , 1, 1, 1 , 1 1, 1
2
2 2
1
= ,
2( )
i j i j i j i j i j i j
Pr (10)
The boundary conditions (7) become
,0 0, ,0 0 for all 0,
i i
u i
1, 0,
0, 1, (1 0, ),
j j
j j
u
(11)
, 0, , 0,
N j N j
u
where N corresponds to . Here the suffix i corresponds to and j corresponds to . Also = j1j and = i1i. Knowing the values of ,u at a time we can calculate the values at a time as follows . We substitute = 1, 2,...,i N1, in equation (10) which constitute a tri-diagonal system of equations, the system can be solved by Thomas algorithm as discussed in (Carnahan et al., 1969). Thus is known for all values of at time . Then knowing the values of and applying the same procedure with the boundary conditions, we calculate, u from equation (9). This procedure is continued to obtain the solution till desired time . The Crank-Nicolson scheme has a truncation error of
2 2O O , i.e. the temporal truncation error is significantly smaller.
Figure 2: Finite Difference Grids
The implicit method gives stable solutions and requires matrix inversions which we did at step forward in time because this problem is an initial - boundary value problem with a finite number of spatial grid points. Though, the corresponding difference equations do not automatically guarantee the convergence of the mesh 0. To achieve maximum numerical efficiency, we used the tridiagonal procedure to solve the two point conditions governing the main coupled governing equations of momentum and energy. The convergence (consistency) of the process was quite satisfactory and the numerical stability of the method was guaranteed by the implicit nature of the numerical scheme. Hence, the scheme is consistent. Stability and consistency ensure convergence.
Results and Discussion
We have presented the non-dimensional fluid velocity u and fluid temperature for several values of Prandtl number Pr, Grashof number Gr, suction parameter S and time in Figs.3-9. It is found from Fig.3 that the fluid velocity u decreases with an increase in Prandtl number Pr. Fig.4 reveals that the fluid velocity u
fluid velocity u increases with an increase in time in the boundary layer region which means that there is an enhancement in the fluid velocity as time progresses. The suction parameter produces an inward flow of heating that accelerates the flux of heat to the plate and hence the suction causes an increase in heat transfer. It is observed from Figs.7-9 that the temperature of the fluid is maximum at the surface of the plate and it decreases with an increase boundary layer coordinates y. It is seen from Fig.7 that the fluid temperature decreases with increases in Pr in the boundary layer region as Pr measures the relative effects of viscosity to thermal conductively. This means that thermal diffusion tends to increase the fluid temperature in the boundary layer region. It is observed from Figs. 8 that the fluid temperature increases with an increase in suction parameter
S. Further, Fig.9 illustrates that the fluid temperature increases with an increase in time in the boundary layer region which means that there is an enhancement in fluid temperature as the time progresses.
Figure 4: Velocity profiles for different S when Pr= 7, Gr= 10 and = 0.2
Figure 5: Velocity profiles for different Gr when S= 1, Pr= 7 and = 0.2
Figure 6: Velocity profiles for different time when Pr= 7, Gr= 10 and S= 1
Figure 8: Temperature profiles for different S when Pr= 0.71 and = 0.2
Figure 9: Temperature profiles for different time when Pr= 0.71 and S= 1
=0 , x du d
(12)
=0
and d 1 (0, ).
d
(13)
Numerical values of the non-dimensional shear stress x and the rate of heat transfer
=0 d d
at the plate
= 0
due to the flow are presented in Tables 1 and 2 respectively for several values of Prandtl number Pr, suction parameter S and time with Gr= 15. It is seen from Table 1 that the shear stress x at the plate
= 0
increases with an increase in either Prandtl number Pr or suction parameter S or time . An increase in the Prandtl number lead to an increase in shear stress in the presence of suction. As time progresses there is a rise in the shear stress. Further, it is seen from Table 2 that the rate of heat transfer
=0 d d
at the plate
increases with an increase in either Prandtl number Pr or suction parameter S or time . This may be explained by the fact that frictional forces become dominant with increasing values of Pr and hence yield greater heat transfer rate. As time progresses there is a rise in the rate of heat transfer.
Table 1. Shear stress 10 3x
at the plate = 0 when Gr= 5
Pr with S= 1 S with Pr= 0.71
0.71 2 5 7 0.5 1.0 1.5 2.0 0.02 0.04 0.06 0.08 0.04634 0.04919 0.05247 0.05625 0.04897 0.05827 0.07178 0.09157 0.05067 0.07186 0.13022 0.30194 0.05264 0.09865 0.30574 1.24078 0.04506 0.04634 0.04771 0.04919 0.04511 0.04835 0.05204 0.05623 0.04680 0.05266 0.05974 0.06830 0.04912 0.05793 0.06929 0.08400Table 2. Rate of heat transfer 1
=0 10 d d
at the plate = 0
Pr with S= 1 S with Pr= 0.71
0.71 1 2 3 0.5 1.0 1.5 2.0 0.02 0.04 0.06 0.08 0.11526 0.12214 0.14918 0.18221 0.13284 0.14918 0.22255 0.33201 0.15311 0.18221 0.33201 0.60496 0.17647 0.22255 0.49530 1.10232 0.10736 0.11526 0.12374 0.13284 0.11526 0.13284 0.15311 0.17647 0.12374 0.15311 0.18946 0.23443 0.13284 0.17647 0.23443 0.31143 ConclusionThe unsteady free convection flow past a moving vertical porous plate with Newtonian heating is investigated. It is found that both the velocity u as well as the temperature of the fluid increase with an increase in Prandtl number Pr or suction parameter S. An increase in Grashof number Gr leads to rise the fluid velocity. Further, it is observed that the shear stress x and the rate of heat transfer at the moving plate ( = 0) increases with an
increase in either Prandtl number Pr or suction parameter S or time .
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