Unsteady MHD Decelerating Flow over a Wedge with Heat Generation/Absorption
G. Ashwini1, *, A. T. Eswara2
1Department of Mathematics, Govt. College for Women’s, Mandya, India
2Department of Mathematics, GSSS Institute of Engineering and Technology for Women, Mysore, India
This paper deals with the study of unsteady, MHD laminar boundary layer forced flow of an incompressible electrically conducting fluid over a wedge in the presence of heat generation/absorption. Similarity transformation is used to convert the governing nonlinear boundary-layer equations to non-linear ordinary differential equations and later, these equations are solved numerically using Keller-box method to obtain self-similar solutions. The results are obtained for local skin friction coefficient and Nusselt number for different governing flow parameters. It is found that dual solutions exist up to a critical value of the unsteady parameter beyond which, the boundary layer separates from the surface. Further, it is established that application of the magnetic field delays the boundary layer separation.
MHD Decelerating Flow, Self-Similar Solution, Skin Friction, Heat Transfer, Heat Generation/Absorption
Received: July 4, 2015
Accepted: July 20, 2015
Published online: July 31, 2015
@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/
In their innovative work of 1931, Falkner and Skan  have explored two-dimensional incompressible flow past a wedge. Since then it has been studied by many researchers [2-14] employing various methods suitable for flow and heat transfer phenomena.
The hydro-magnetic boundary-layer flow and heat transfer in a incompressible electrically conducting fluid over a wedge is a significant type of flow having wide applications in industries and engineering. This type of flow has attracted the interest of many researchers due to its applications in MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. Indeed, Vajravelu and Nayfeh  examined the hydromagnetic convection at a cone and a wedge. Kafoussias and Nanousis  have considered the magneto hydrodynamic Falkner-Skan flow with suction or injection. K.A.Yih  studied MHD forced convection flow adjacent to a nonisothermal wedge while, Seddeek. et al.  presented similarity solutions for a steady MHD Falkner-Skan flow and heat transfer over a wedge.
Of late, Ashwini and Eswara  have presented the effect of internal heat generation or absorption on the Falkner-Skan boundary layer flow with an applied magnetic field.
In all the above studies the effect of unsteadiness were not studied. Hence, the present study is aimed at analyzing the unsteady, MHD decelerating self-similar Falkner-Skan flow with internal heat generation or absorption where unsteadiness in the flow is due to the time-dependent free stream velocity. The governing non-linear partial differential equations are reduced to a system of non-linear ordinary differential equations by applying similarity transformations. Later, they are solved numerically using an implicit finite difference scheme viz., Keller-box method, to obtain self-similar solutions. Similarity solutions are very useful in the sense that they reduce the independent variables of the problem.
It is remarked here that the counter part of the present paper pertaining to accelerating flow has been studied recently by Ashwini and Eswara .
2. Mathematical Formulation
Consider the flow of an unsteady laminar incompressible viscous fluid over a wedge (Figure 1) where is measured along the surface of the wedge and is normal to it. The unsteadiness in the flow field is introduced by the free stream velocity, varying inversely with time. The temperature of the wall is constant and greater than that of free stream temperature. A transverse magnetic field is applied in the direction normal to the wedge surface and, it is assumed that the magnetic Reynolds number is small, so that the induced magnetic field can be neglected. No electric field is assumed to exist and Hall effect is negligible. The fluid properties are assumed to be constant and the viscous dissipation has been neglected in the energy equation. Under the above assumptions, the boundary layer equations governing the unsteady, MHD forced convection flow past a wedge is given by:
The boundary conditions are given by
Equations (1)-(3) are a system of partial differential equations with three independent variables and. It is possible that these partial differential equations can be reduced to a system of ordinary differential equations, if the free stream velocity varies inversely as a linear function of time viz., where is the parameter characterizing unsteadiness in the free stream and is the non dimensional time variable (Eswara and Nath ; Schlichting ). This similarity property permits a decrease in the number of independent variables from three to one and yields treatment using ordinary differential equations instead of partial differential equations, to obtain self similar solutions for the nonzero values of. In fact, the flow is accelerating if and the flow is decelerating if. In this paper results pertaining to decelerating flow are displayed and discussed, since results of accelerating flow are already been published in a standard journal. Assuming the value zero to, the problem reduces to the steady case.
Introducing the following transformations:
to Eqns. (1), (2) and (3), we see that the continuity Eqn. (1) is identically satisfied and Eqns.(2) and (3) reduce, respectively, to
Further, we note that in (6) and (7), the parameter is connected with the apex angle by the relation.
The transformed boundary conditions are
The heat generation or absorption parameter appearing in Eqn. (7) is the non-dimensional parameter based on the amount of heat generated or absorbed per unit volume given by with being constant coefficient that may take either positive or negative values. The source term represents the heat generation that is distributed everywhere when is positive () and the heat absorption when is negative (); is zero, in case no heat generation or absorption.
It is worth mentioning here that when, Eqns. (6) and (7) reduce to:
The physical quantities of engineering interest here are the skin friction coefficient and heat transfer coefficient in the form of Nusselt number; and they are defined, respectively, as
and are shear stress and heat flux along the surface of the wedge, whereandare dynamic viscosity and thermal conductivity, respectively. Using Eqns. (5) and (8) in (12) the skin friction and Nusselt number can be written as
where , the local Reynolds number.
3. Results and Discussion
An accurate and very efficient implicit finite difference scheme called Keller-box method  is used to solve the coupled ordinary differential equations (6) and (7) along with boundary conditions (9). To assess the accuracy of the method which we have used, the skin frictionand heat transfer parameter results have been compared with those of Watanabe  for the range of, [See Table I]. Further, the steady state heat transfer results for with a magnetic field () are compared with those of Ashwini and Eswara  [See Fig.2]. There is an excellent agreement between the present results and the above mentioned studies.
The effect of magnetic fieldon skin friction  and heat transfer  coefficients when (60o), (no heat generation/absorption and thermal radiation) for (decelerating flow) are illustrated in Fig.3. It is observed that as magnetic field increases both and increases quantitatively. The percentage of increase in is about 95.65% and percentage of increase in is around 2.83% in the range when . It is interesting to view the existence of dual solutions for both and , in the range of (0), and no solution for , where is a critical value of Therefore, solution exist up to a critical value , beyond which the boundary layer separates from the wedge surface and the solution based upon the boundary layer approximations are not possible. Based on our computation, the values of are and , respectively for [Fig. 3(a)] and[Fig.3(b)] in the absence of the magnetic field . Further, the values are seen to be pushed further by the application of the magnetic field and, in fact, they are and , respectively for and . This vindicates the significant role played by the transverse magnetic field in the stabilization process of the laminar boundary layer flow during this short span of time.
Figure 4 shows the consequential velocity and temperature profiles of the corresponding first and second solutions, when. It is clearly seen that both momentum and thermal boundary layer thicknesses are found to decrease with the increase of the magnetic field. Further, these profiles satisfy the far field boundary conditions asymptotically, which support the obtained numerical results. It is remarked that the upper branch solutions (first solution) are stable and physically realizable, while the lower branch solutions (second solution) are not. Although such solutions are deprived of physical significance, they are nevertheless of mathematical interest .
Figure 5 displays the effects of heat generation or absorption parameter () on heat transfer coefficient  and temperature profiles () for (decelerating flow) at the wedge angle and . It is observed that decreases as increases (). The percentage of decrease in is 23.56% at = -1.5.
Fig.5 (b) shows the corresponding temperature profiles, where thermal boundary layer thickness decreases when heat is absorbed, whereas opposite trend is observed for the heat generation. This is due to the fact that heat generation increases the thermal state of the fluid leading to enhancement of the heat transfer rate.
The effect of magnetic field and heat generation/absorption on the behavior of unsteady decelerating flow of an electrically conducting fluid flow past a wedge has been numerically investigated. Dual solutions are found to exist in forced convection decelerating flow regime. The magnetic field stabilizes the flow which in turn delays the boundary layer separation from the wedge surface. Heat generation increases thermal boundary layer thickness due to decrease in heat transfer coefficient and opposite trend is observed during heat absorption.
The authors are thankful to the reviewer for their comments and suggestions. Further, the authors express their gratitude to PETRF and the Department of Mathematics, P.E.S. College of Engineering, Mandya-571 401, for providing finest facilities for research work.
|Free stream velocity|
|Free stream temperature|
|Transverse magnetic field|
|Cartesian co-ordinates along and normal to surface, respectively|
Velocity components along and-directions, respectively
|Density of the fluid|
|Heat generation or absorption coefficient|
|Length of the wedge|
|Falkner – Skan power law Parameter|
|Non-dimensional time variable|
|Dimensional stream functions|
|Dimensionless stream functions|
|Dimensionless magnetic parameter|
|Local Reynolds number|
|Hartree pressure gradient parameter|
|Conditions at the free stream|
|Conditions at the edge|
|Conditions at the wall|
|(’)||Partial derivatives with respect to η|