Effects of Viscous Dissipation on MHD Natural Convection Flow along a Vertical Wavy Surface with Heat Generation

The effect of external magnetic field and internal heat generation or absorption on a steady two-dimensional natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface has been investigated. The governing boundary layer equations are first transformed into a non-dimensional form using suitable set of dimensionless variables. The transformed boundary layer equations are solved numerically using the implicit finite difference method, known as Keller-box scheme. Numerical results for velocity, temperature, skin friction, the rate of heat transfer are obtained for different values of the selected parameters, such as viscous dissipation parameter (Vd), heat generation parameter (Q), magnetic parameter (M) and presented graphically and discussed. Streamlines and isotherms are presented for selected values of heat generation parameter and explained.


Introduction
The natural convection boundary layer flow about a heated vertical wavy surface has received a great deal of attention due to its relation to practical applications of complex geometries. There is also a model problem for the investigation of heat transfer from roughened surfaces in order to understand heat transfer enhancement. The natural convection along a vertical wavy surface was first studied by Yao [1] and used an extended Prantdl's transposition theorem and a finite-difference scheme. He proposed a simple transformation to study the natural convection heat transfer from isothermal vertical wavy surfaces, such as sinusoidal surface. Moulic and Yao [2] also investigated mixed convection heat transfer along a vertical wavy surface. Alam et al. [3] have also studied the problem of free convection from a wavy vertical surface in presence of a transverse magnetic field. The combined effects of thermal and mass diffusion on the natural convection flow of a viscous incompressible fluid along a vertical wavy surface have been investigated by Hossain and Rees [4]. Wang and Chen [5] investigated tran-sient force and free convection along a vertical wavy surface in micropolar fluid. Hossain et al. [6] have studied the problem of natural convection of fluid with temperature dependent viscosity along a heated vertical wavy surface. Natural and mixed convection heat and mass transfer along a vertical wavy surface have been investigated by Jang [7,8]. Recently, Molla et al. [9] have studied natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation/absorption. Tashtoush and Al-Odat [10] investigated magnetic field effect on heat and fluid flow over a wavy surface with a variable heat flux. Hossain [11] investigated the natural convection flow past a permeable wedge for the fluid having temperature dependent viscosity and thermal conductivity. Very recently, Parveen and Alim [12] investigated Joule heating effect on Magnetohydrodynamic natural convection flow along a vertical wavy surface with viscosity dependent on temperature. The present study is to incorporate the effects of the viscous dissipation on MHD natural convection flow along a uniformly heated vertical wavy surface with heat generation.
Numerical results have been obtained in terms of local skin friction coefficient and the rate of heat transfer in terms of local Nusselt number, and the velocities as well as the temperature profiles for a selection of relevant physical parameters are shown graphically.

Formulation of the Problem
Steady two dimensional laminar free convection boundary layer flow of a viscous incompressible and electrically conducting fluid along a vertical wavy surface in presence of uniform transverse magnetic field is considered. It is assumed that the wavy surface is electrically insulated and is maintained at a uniform temperature T w . The fluid is stationary above the wavy plate and is kept at a temperature T  . The surface temperature T w is greater than the ambient temperature T  that is, T w > T  . The flow configuration of the wavy surface and the two-dimensional cartesian coordinate system are shown in Figure 1.
The boundary layer analysis outlined below allows   X  being arbitrary, but our detailed numerical work assumed that the surface exhibits sinusoidal deformations. The wavy surface may be defined by where  is the amplitude and L is the wave length associated with the wavy surface. The governing equations of such flow of magnetic field in presence of heat generation/absorption with viscosity variation along a vertical wavy surface under the usual Boussinesq approximations can be written in a dimensional form as: Continuity Equation where   Using Prandtl's transposition theorem to transform the irregular wavy surface into a flat surface as extended by Yao [1] and boundary layer approximation, the following dimensionless variables are introduced for non-dimensionalizing the governing equations where θ is the dimensionless temperature function and   It is worth noting that the σ x and σ xx indicate the first and second derivatives of σ with respect to x, therefore, In the above equations Pr, Vd, M and Q are respectively known as the Prandtl number, viscous dissipation parameter, magnetic parameter and heat generation parameter which are defined as: For the present problem the pressure gradient  0 p x     is zero. Thus, the elimination of p y   from Equations (9) and (10) The corresponding boundary conditions for the present problem then turn into 0, 1, at 0 0, as Now we introduce the following transformations to reduce the governing equations to a convenient form: ,  where  ,  f x  is the dimensionless stream function, η is the dimensionless similarity variable and ψ is the stream function that satisfies the continuity Equation (8) and is related to the velocity components in the usual way as Introducing the transformations given in Equation (15) and using (16) into Equations (13) and (11) The boundary conditions (14) now take the following form: Here prime denote the differentiation with respect to η. However, once we know the values of the functions f and  and their derivatives, it is important to calculate the values of the rate of heat transfer in terms of local Nusselt number Nu x and the shearing stress  w in terms of the local skin friction coefficient C fx from the following relations: where    

Results and Discussion
We have investigated the effects of viscous dissipation on natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface. Although there are five parameters of interest in the present problem, the effects of Prandtl number Pr, viscous dissipation Vd, magnetic parameter M, the heat generation parameter Q and the amplitude of the wavy surface  on the surface shear stress in terms of local skin friction coefficient, the rate of heat transfer in terms of the local Nusselt number, the velocity and temperature profiles, the streamlines and the isotherms. Numerical values of local shearing stress and the rate of heat transfer are calculated from Equations (22)  and v have been presented. From Figure 5(a) it is found that skin friction decreases significantly for greater magnetic field strength. This is physically realizable as the magnetic field retards the velocity field and consequently reduces the frictional force at the wall. However rate of heat transfer opposite pattern due to the higher values of magnetic parameter M which are presented in Figure  5  therms) distribution shows that temperature decreases significantly as the values of the heat generation parameter Q increases which have been presented in Figure 8(b).
The value of isotherm is 1.0 at the wall and isotherms decreases slowly along the y-direction and finally aproach to zero. p

Conclusions
The effects of the Prandtl number Pr, the magnetic parameter M, the viscous dissipation parameter Vd, the heat generation parameter Q and the amplitude of wavy surface  on MHD natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface have been studied. From the present investigation the following conclusions may be drawn: The temperature and the rate of heat transfer coefficient increase for increasing values of magnetic parameter. The velocity decreases and at the position of 5.5   becoming constant that is velocity profile meets at the point and then crosses the side and increases with magnetic parameter. The local skin friction coefficient decreases due to the increased value of magnetic parameter.
The velocity and the temperature rise up and the local skin friction coefficient increase due to the higher values of viscous dissipation parameter Vd which cause reduction of the rate of heat transfer.
The velocity, temperature and the skin friction coefficient enhance for higher values of internal heat generation parameter Q but for the same reason the rate of heat transfer reduces.