Boundary Layer Stagnation-Point Slip Flow and Heat Transfer towards a Shrinking/Stretching Cylinder over a Permeable Surface ()
1. Introduction
The problem of flow and heat transfer induced by continuous shrinking or stretching surfaces has received considerable attention in recent years because it is an important type of flow occurring in many engineering disciplines. The stagnation point flow problems have been extensively studied by several authors. The problem of flow due to a shrinking or stretching sheet has been later extended to many flow situations. Crane [1] was the first studied the steady boundary layer flow of a viscous fluid due to a linearly stretching surface. Later, Chiam [2] extended the works of Crane [1] to stagnation-point flow over a stretching sheet. Wang [3] first investigated the stagnation-point flow towards a shrinking sheet for both two-dimensional and axiymmetric cases. He found that problem have the dual solutions as well as unique solution for a specific values of the ratio of shrinking.
The problem of unsteady stagnation-point flow of a viscous and incompressible fluid by considering both the stretching and shrinking sheet situations have been investigated by Fan et al. [4] . On the other hand, Bachok et al. [5] discussed the effect of melting on boundary layer stagnation-point flow towards a stretching or shrinking sheet. Ahmad et al. [6] investigated the behaviour of the steady boundary layer flow and heat transfer of a micropolar fluid near the stagnation point on a stretching vertical surface with prescribed skin friction. Lok et al. [7] studied the steady axisymmetric stagnation point flow of a viscous and incompressible fluid over a shrinking circular cylinder with mass transfer (suction). Bhattacharyya et al. [8] analyzed the effects of partial slip on the steady boundary layer stagnation-point flow of an incompressible fluid and heat transfer towards a shrinking sheet. This investigation explores the conditions of the non-existence, existence, uniqueness and duality of the solutions of self-similar equations numerically. They also studied the same case but under the condition of unsteady-state towards a stretching. Stagnation-point flow and heat transfer over an exponentially shrinking sheet was analyzed by Bhattacharyya and Vajravelu [9] . They obtained dual solutions for the velocity and the temperature fields and also they observed that their boundary layers are thinner for the first solution.
Therefore, the present investigation deals with the effects of partial slip on the boundary layer stagnation- point flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface.
2. Problem Formulation
Consider the steady stagnation-point flow towards a horizontal linearly stretching/shrinking cylinder with radius
placed in an incompressible viscous fluid of constant temperature
. Using boundary layer approximation, the equations of motion and temperature distribution may be written in usual notations as below
(1)
(2)
(3)
where
and
are coordinates measured along the surface of the cylinder and in the radial direction, respectively, with
and
being the corresponding velocity components and
is the straining velocity of the stagnation-point flow. Further,
is the temperature in the boundary layer,
is the kinematic viscosity coefficient and
is the thermal diffusivity.
The appropriate boundary conditions for the velocity components with partial slip condition at the wall and for the temperature are given by
(4)
(5)
where
and
are the shrinking/stretching rate (of the surface) and the straining rate parameters, respectively, with
for a shrinking cylinder,
for a stretching cylinder and
is the velocity slip factor,
is the thermal slip factor,
is the mass transfer velocity with
for suction and
for injection
is the surface temperature and
is the free stream temperature, both are assumed to be constant with
.
Next, we introduce the transformation as follows:
(7)
where
is the similarity variable and is defined as
and
is the stream function define as
and
, which identically satisfies Equation (1). Since this problem have effects
of suction and injection, we have consider that
in Equation (4) has the following expression,
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where
is a non-dimensional constant determines the transpiration rate, with
is the constant suction parameter,
is the constant injection parameter and
for an impermeable surface. Physically permeable surface (suction/injection) allows fluid to pass through either in or out. Using (7) we obtain the following self-similar equations
, (8)
(9)
subject to the boundary conditions (5) and (6) which become
(10)
where
is the curvature parameter and
is the Prandtl number defined respectively as
(11)
and
is the velocity ratio parameter,
is the velocity slip parameter and
is the thermal slip parameter.
The main physical quantities of interest are the value of
, being a measure of the skin friction coefficient (shear stress at the surface), and
, the local Nusselt number (rate of heat transfer at the surface). Our main aim is to find how the values of
and
vary in terms of the parameters
,
,
,
and
. The present problem reduces to those considered by Bhattacharyya et al. [8] when
(flat plate),
and
.
3. Results and Discussion
Equations (8) and (9) with the boundary conditions (10) are solved numerically using the shooting method. Table 1 shows the variations of
(critical values of
) with parameters
and
when
,
(without thermal slip) and
. In order to verify the accuracy of the present method, we have com-
pared the present results obtained with those of Bhattacharyya et al. [8] for
(flat plate),
. This comparison is given in Table 1 and we can see that there is a very good agreement between these results, and thus we are confident that the numerical results obtained are accurate. Table 2 presents the numerical values of
with respect to
,
and
when
,
(with thermal slip) and
. From this table it is observed that with increasing
, the values of
increase for several values of
. Figure 1 and Figure 2 show the variations of
and
with
, respectively, for various values of
when
,
,
and
. As the curvature parameter
increases, the skin friction coefficient
and the local Nusselt number
increase. On the other hand,
increase with the
increasing
. It is show that for increasing slip at the sheet, the vorticity generated due to the shrinking. velocity is slightly reduced and hence with the same straining, vorticity remains confined within the boundary layer for larger shrinking velocity also and consequently the steady solution is also possible for some large value of
. Figure 3 and Figure 4 represent the variations of
and
for various values of
when
,
,
and
. It is shown that the velocity slip parameter
decreases the shear stress rate but increases the heat transfer rate at the surface. It is seen that the region of non-unique solutions exist for
and the solution is unique for
. The value of
is zero when
for all values of
and
considered and
is the Hiemenz flow while
is a degenerate inviscid flow where the stretching cylinder matches the conditions at infinity.
Figure 5 shows the variations of
for various values of
(thermal slip parameter) for
,
,
and
. As
increases, the heat transfer rate measured by
increases too, while it decreases with the thermal slip parameter,
. It should be pointed out that the effect of thermal slip parameter
on the shear stress is not significant so it is not presented here. Figure 6 and Figure 7 show the variations of
and
with
, respectively, for various values of
when
,
,
and
. The values of
and
increase with
for the first solution and for the second solution the effect is completely the opposite. Further, as
increases,
increases too. Figure 8 shows the variations of
with
for various values of
when
,
,
and
. The values of
increase with
for the first and the second solutions. It is also noted that
is decreasing as
increases.
Figure 9 and Figure 10 represent the variations of
and
with
for different values of
when
,
,
and
. It is found that the values of
and
increase with
for the first solution and for the second solution the effect is completely the opposite. The behavior are
similar as
increases. Figure 11 and Figure 12 show the velocity profiles
and the temperature profiles
for various values of
when
,
,
,
and
. Figure 13 and Figure 14 represent the velocity profiles
and temperature profiles
for various values of
when
,
,
,
and
. It is shown that all the curves approach the far
field boundary conditions asymptotically. Further, these figures also support the existence of dual nature of the solutions as presented in the earlier figures where the boundary layer thickness for the first solution is smaller than that of the second solution.
4. Conclusion
The study of boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface has been investigated numerically using the shooting method. Comparisons of the present results with those of Bhattacharyya et al. [8] are in excellent agreement. The velocity slip parameter
increases the heat transfer rate at the surface, while the thermal slip parameter
decreases it. On the other hand, it is found that dual solutions exist for the shrinking cylinder when
. Finally, as the curvature parameter
increases, the values of
and
increase too, similar with the effect of suction/injection parameter
for the first solution.
Acknowledgements
The authors gratefully acknowledged the financial support received in the form of a FRGS research grant from the Ministry of Education, Malaysia.
NOTES
*Corresponding author.