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radiation effects

surface velocity is proportional to the distance from a fixed point. Since then, extensive research has been done capturing the various physical conditions and rheology of the fluids with different conditions, see for example Refs. [3–10]. Flow of an electrically-conducting fluid subject to a

magnetic field has important applications, such as cooling nuclear reactors and magnetohydrodynamic (MHD) gen- erators, plasma studies, oil exploration, geothermal energy extraction and boundary layer control in the field of aerodynamics [11]. In metallurgical processes, such as drawing, annealing and thinning of copper wires which involve cooling of continuous strips or filaments, the MHD effect is believed to improve the rate of cooling and hence, the properties of the final products. Mansur and Ishak [12] studied numerically magnetohydrodynamic (MHD) bound- ary layer flow of a nanofluid past a stretching/shrinking sheet with velocity, thermal, and solutal slip boundary conditions. Siddheshwar and Mahabaleshwar [13] exam- ined analytically MHD flow of micropolar fluid over linear stretching sheet using regular perturbation technique and Ahmed et al. [14] applied the successive linearization method to study the effects of radiation and viscous dissipation on MHD boundary layer convective heat transfer with low pressure gradient in porous media. Other studies on the MHD flow in different fluids as well as different physical situations were considered for example in Refs. [15–22]. Due to its great range of applications in various fields, the

investigation of convective heat transfer in fluid-saturated porous media has become a subject of interest, especially in geothermal energy recovery, food processing, fibre and

granular insulation, design of packed bed reactors and dispersion of chemical contaminants in various processes in the chemical industry and environment [23]. Compre- hensive studies can be found in Vafai [24], Nield and Bejan [25] and Vadasz [26]. There is an abundance of literature available which discusses fluid flow over stretching surfaces in porous medium. Some of them are Gbadeyan et al. [27] who investigated the effects of thermal diffusion and diffusion thermos effects on combined heat and mass transfer on mixed convection boundary layer flow over a stretching vertical sheet in a porous medium filled with a viscoelastic fluid in the presence of magnetic field, Imran et al. [28] studied the analysis of an unsteady mixed convection flow of a fluid saturated porous medium adjacent to heated/cooled semi-infinite stretching vertical sheet in the presence of heat source and Aly and Ebaid [29] investigated the mixed convection boundary-layer nano- fluids flow along an inclined plate embedded in a porous medium using both analytical and numerical approaches. Dessie and Kishan [30] examined the MHD boundary layer flow and heat transfer of a fluid with variable viscosity through a porous medium towards a stretching sheet along with viscous dissipation and heat source/sink effects. Narayana [31] carried out a study on the effects of radiation and first-order chemical reaction on unsteady mixed con- vection flow of a viscous incompressible electrically con- ducting fluid through a porous medium of variable permeability between two long vertical non conducting wavy channels in the presence of heat generation, and to name a few.

Jeffrey fluid is a type of non-Newtonian fluid that uses a relatively simpler linear model using time derivatives

instead of convected derivatives, which are used by most fluid models. Recently, this model of fluid has prompted active discussion. Some of the studies can be found in Shehzad et al. [32], Nallapu and Radhakrishnamacharya [33], Ahmad and Ishak [34] and Prasad et al. [35]. In view of the above discussions, the aim of this paper is to investigate the effects of MHD Jeffrey fluid flow embedded in porous medium over vertical stretching sheet. The model of the Jeffrey fluid flow is presented mathematically and has been solved numerically using a finite difference scheme.

2. Analysis

Consider the unsteady two-dimensional incompressible Jeffrey fluid in a porous medium over a vertical stretching sheet coinciding with the plane y¼0, with the flow being confined to y40. The surface is assumed to stretch with velocity uw¼ax, where a is stretching constant. Here, the x- axis is chosen parallel to the vertical surface and the y-axis is taken normal to it. The plate temperature is Tw¼T∞þbx, where Tw is the surface temperature, T∞ is the ambient fluid temperature and b is constant. Tw4T∞ and TwoT∞ are for heated surface (assisting flow) and cooled surface (opposing flow), respectively. A uniform transverse magnetic field of strength B0 is applied parallel to the y-axis. By invoking the boundary layer and Boussinesq approximations, the gov- erning boundary layer equations for this problem can be written as

∂u ∂x

þ ∂v ∂y

¼ 0 ð1Þ

u ∂u ∂x

þ v ∂u ∂y

¼ ν 1þ λ1

� ∂ 2u

∂y2 þ λ2 u

∂3u ∂x∂y2

þ v ∂ 3u

∂y3 þ ∂u

∂y ∂2u ∂x∂y

− ∂u ∂x

∂2u ∂y2

� �� �

þg βT T−T∞ð Þ− ν

ε u−

σB02u

ρ ð2Þ

u ∂T ∂x

þ v ∂T ∂y

¼ α ∂ 2T

∂y2 ð3Þ

subject to the boundary conditions

u¼ uw; v¼ 0; T ¼ Tw at y¼ 0 u→0; ∂u∂y→0; T→T∞ as y→∞

ð4Þ

where u and v are the velocity components in the x and y directions, respectively. λ1 is the ratio of the relaxation and retardation times; λ2 is the relaxation time and T is the fluid temperature. ν¼ μρ is the kinematic viscosity, where μ is the coefficient of fluid viscosity and ρ is the fluid density. g, βT, ε and σ are gravitational acceleration, thermal expansion coefficient, permeability coefficient of porous medium and fluid electrical conductivity, respectively.