Slip MHD flow over permeable stretching surface with chemical reaction

In this paper the magneto hydrodynamic (MHD) slip flow over a permeable stretching surface has been evaluated in the presence of a chemical reaction. Slip flow comes about if the characteristic size of the flow regime is small or the flow pressure is very low. By using appropriate similarity variables, the fundamental equations of the boundary layer are transformed to ordinary differential equations containing the Schmidt number, nonlinearity velocity of the surface and magnetic parameter which for a fixed values of slip coefficient (K) at the boundary conditions, local similarity solution would be valid. The ordinary differential equations of the problem are solved numerically using an explicit Runge-Kutta (4, 5) formula, the DormandPrince pair and shooting method. The velocity and concentration profiles in addition to the local skin-friction and the local Sherwood number for the various values of the involved parameters of the problem are presented and discussed in details. Introduction Microfluidics as a young research field plays a great role to develop control accuracy of small devices. In no-slip-flow, as a requirement of continuum physics, the flow velocity is zero at a solid-fluid interface. But in the existence of slip-flow, the flow velocity at the solid walls is nonzero [1]. Even if the separation of individual molecules is obvious at the nanoscales, it is still possible to explain the main transport phenomena in nanofluidic systems with a theory based on continuum and mean-field approaches [2, 3 and 4]. It is well known that flow past a permeable surface has practical applications especially in geophysical fluid dynamics. Examples of natural porous media are wood, beach sand, sandstone, limestone, the human lung and in small blood vessels. The magnetohydrodynamic (MHD) flow of a fluid in a micro/nanochannel is of interest in connection with certain problems of the movement of conductive physiological fluids, e.g., the blood, blood pump machines and with the need for theoretical research on the problem of the slip MHD flow along permeable surfaces. Thus, the micro-nano magnetohydrodynamic effects are recognized as a tool for controlling the micro-nanostructure of materials. Numerous investigations have been done analytically regarding to the slip flow regime. Martin and Boyd [5] have analyzed Blasius boundary layer problem with slip flow. Their results demonstrated that the boundary layer equations can be used to study flow at the MEMS scale and provide useful information to study the effects of rarefaction on the shear stress and structure of the flow. In another task [6] they have analyzed momentum and heat transfer in a laminar boundary layer with slip flow at constant wall temperature. Based on the boundary layer theory, non-equilibrium effects will cause a reduction in drag on airfoils. According to their studies of liquids over flat plate at constant wall temperature boundary conditions, there is no temperature jump. Recently, Matthews and Hill [7] have studied the effect of replacing the standard no-slip boundary condition with a nonlinear Navier boundary condition for the boundary layer equations. In another task they have investigated Newtonian flow with nonlinear Navier boundary condition for three simple pressure-driven flows through a pipe, a channel and an annulus [8]. The axisymmetric flow of a Newtonian fluid due to a stretching sheet with partial slip boundary condition has been investigated by Ariel [9]. Yazdi et al. [10] have investigated friction and heat transfer in the slip flow boundary layer at constant heat flux boundary conditions. In another task [11] they have studied liquid fluid past embedded open parallel microchannels within the surface. Wang [12] has studied the viscous flow due to a stretching sheet with partial slip and suction. Recently Yazdi et al [13] have analyzed convective heat transfer of the slip liquid flow past horizontal surface within the porous media at constant heat flux boundary conditions. Their results suggest that slip liquid flow can successfully reduce wall friction through slip-flow boundary conditions in convective heat transfer problems and increase heat transfer rate. It has been found that suction makes a significant effect on the velocity adjacent to the wall in the presence of slip. On the topic of MHD flow modeling, the boundary-layer equation of flow over a nonlinearly stretching sheet in the presence of a chemical reaction and a magnetic field has been investigated by Kechil and Hashim [14]. Recently Fang et al [15] have studied analytically hydrodynamic boundary layer of slip MHD viscous flow over a stretching sheet. Their investigation shows the velocity and shear stress profiles are influenced by the slip, magnetic and mass transfer parameters. They have illustrated that wall drag force increases with the increase of magnetic parameter. There have been many theoretical models developed to describe slip flow along the surface. To the best of our knowledge, no investigation has been made yet to analyze the slip MHD flow over permeable stretching surface with chemical reaction. Mathematical formulation For modeling fluid transport in slip boundary layer, the assumptions made for the derivation of the full Navier–Stokes equations have been examined. These assumptions are the fluid is assumed to be a continuum, the fluid is Newtonian. In addition, the fluid can be assumed to be incompressible. We will study the 2-D, steady, laminar flow in the presence of a transverse magnetic field with strength B(x) which is applied in the vertical direction, given by the special form.