Multiple Slip Effects on MHD Unsteady Flow Heat and Mass Transfer ImpingingonPermeable StretchingSheetwithRadiation

-is paper reports multiple slip effects on MHD unsteady flow heat and mass transfer over a stretching sheet with Soret effect; suction/injection and thermal radiation are numerically analyzed. We consider a time-dependent applied magnetic field and stretching sheet which moves with nonuniform velocity. Suitable similarity variables are used to transform governing partial differential equations into a system of coupled nonlinear ordinary differential equations. -e transformed equations are then solved numerically by applying an implicit finite difference method with quasi-linearization technique. -e influences of the various parameters on the velocity temperature and concentration profiles as well as on the skin friction coefficient and Sherwood and Nusselt numbers are discussed by the aid of graphs and tables.


Introduction
e Navier-Stokes theory is centered on the central idea of no-slip condition. Many authors have obtained both numerical and analytical solutions by applying no-slip boundary conditions to study velocity and temperature profiles. e importance of slip conditions in microchannel or nanochannels has stimulated much interest on the study of vibrating values [1]. It is now known that a slip can occur if the working fluids contain concentrated suspensions. Soltani and Yilmazor [2] performed using a parallel disk rheometer with emphasis on a wall slip phenomenon on the rheological characterization of highly filled suspensions consisting of a Newtonian matrix, mixed with two different sizes of aluminum power and two different sizes of glass beads. When the fluid is particulates such as suspensions, emulsions foams, and polymer solutions, a partial velocity slip may occur on the stretching boundary. Slip effects can arise in various industrial processes at boundaries of pipes, walls, and/or curved surfaces. A Navier velocity slip condition is a usual approach in studying slip phenomena. Mahanthesh et al. [3] examined three-dimensional flow of nanofluid for the effects of partial slip and chemical reaction towards an exponential stretching sheet. Hayat et al. [4] examined simultaneous effects of slip and heat transfer on peristaltic flow. Motsa and Shateyi [5] studied the problem of a nonlinear boundary value problem arising in rotating disk flow under the effects of a partial slip, thermal diffusion, and diffusion thermo. Shateyi and Mabood [6] have illustrated the impact of slip and viscous dissipation on MHD mixed convection stagnation-point flow over a nonlinear stretching sheet. Khan et al. [7] carried out a study on a two-dimensional flow of an incompressible Williamson fluid of Cattaneo-Christov heat flux type over a linearly stretched surface with the influence of magnetic field, thermal radiation-diffusion, heat generation, and viscous dissipation.
Due to the many applications in engineering and industries, the magnetohydrodynamic fluid flows on a stretching sheet have achieved much importance nowadays [8]. Such applications include the liquid coating on photographic films, the boundary layer through the liquid film in the concentration process, and aerodynamic extrusion of plastic sheets. In addition, a wide range of applications on MHD flow can be found in numerous fields such as electronic cooling, boilers, heat insulation and metal extrusion, liquid metal uid oil reservoirs, geothermal systems, nuclear process, micro-MHD pumps, high temperature plasmas, groundwater systems, energy storage units, biological transportation, and thermal energy storage devices . Mabood et al. [9] examined the e ects of heat source and chemical reaction on MHD rotating uid towards a vertical plate in uenced by a porous medium. Kumar et al. [10] investigated the impact of frictional heating on MHD ferrouid with radiation. Heat transfer, determined by thermal radiation, has vast applications in di erent technological processes, including missiles, nuclear power plants, satellites and space vehicles, gas turbines, and the numerous propulsion devices for aircraft. Linear radiation is not analytically valid for immense temperature di erence. Abbas et al. [11] discussed the e ects of radiation in the presence of uniform magnetic eld for a nano uid on a curved stretching surface by incorporating slip e ect. Recently, Makinde et al. [12] developed a numerical study of radiation e ects on chemically reacting MHD nano uid in uenced by heat source/sink and combined heat and mass transfer analysis for mixed convection ow over vertical surface with radiation and chemical reaction illustrated by Ibrahim et al. [13]. Prasannakumara et al. [14] studied the e ects of velocity slip, temperature jump, solutal slip and thermal radiation on a steady ow, heat and mass transfer of an incompressible Je rey nano uid over a horizontal stretching surface. Imtiaz et al. [15] examined unsteady MHD ow of curved stretching surface. Some interesting investigations relevant to ow and heat transfer can be viewed in [16][17][18][19][20][21][22][23].
To the author's knowledge, no studies have this far been communicated with regard to the multiple slips on hydromagnetic unsteady ow and heat and mass transfer in uenced by radiation in permeable frame of reference. Numerical solutions are provided for some special cases, while the physical interpretation for the various parameters is discussed with the help of graphs.

Governing Equations
A two-dimensional MHD ow of an incompressible electrically conducting uid over a permeable stretching surface in the presence of thermal radiation is considered. A coordinate system is chosen in such a way that x-axis is measured along the sheet, and y-axis is normal to it as shown in Figure 1. e sheet is moving with nonuniform velocity U(x, t) ax/(1 − λt) along x-axis where a is the stretching rate and λ is the positive constant with the property λt < 1. A transverse magnetic eld that is assumed to be the function of distance from origin is de ned as B(x) B 0 x −1/2 with B 0 ≠ 0, where x is the coordinate along the surface and B 0 is the magnetic eld strength. e induced magnetic eld is negligible as compared to the applied magnetic eld. Let T ∞ and C ∞ are the free stream temperature and free mass concentration. e governing equations for the continuity, momentum, energy, and concentration can be written: zT zt where x and y are the coordinates along and normal to the sheet; u and v are the components of the velocity in the x and y directions, respectively; ρ is the density of the uid; ] is the kinematic viscosity of the uid; σ is the electrical conductivity; g is the acceleration due to gravity; β T is the thermal expansion coe cient; β C is the concentration expansion coe cient; α is the thermal di usivity; T is the temperature; C is the concentration; D M is the molecular di usivity; D T is the thermal di usivity; σ * is the Stefan-Boltzmann constant; and k * is the mean absorption coe cient. e boundary conditions for the above described model are: where v w v 0 / x √ is the suction/injection velocity. e temperature of the sheet T w (x, t) and the concentration C w (x, t) at the surface are assumed as of the following form: where T 0 , and C 0 are the reference temperature and reference concentration, respectively, such that 0 ≤ T 0 ≤ T w and 0 As per usual, the stream function ψ is defined as u � zψ/zy and v � −zψ/zx so that equation (1) is satisfied. Let us introduce the following dimensionless functions f, θ, ϕ, and similarity variable η as follows: Now substituting equation (7) into equations (2)-(4), we get the following system of nonlinear ordinary differential equations: 1 Pr e transformed boundary conditions of the problem are where δ � λ/a is the unsteadiness parameter; λ 1 � gβ T T 0 /a] and λ 2 � gβ C C 0 /a] are the buoyancy parameters; Pr � ]/α is the Prandtl number; R � 16T 3 ∞ σ * /3k * κ is the thermal radiation parameter; M � ���� σ/ρaB 0 is the magnetic field param- ) is the suction/ injection parameter. Here, f w � 0 represents the impermeable surface, f w > 0 represents the suction, and f w < 0 represents the injection of the fluid through permeable surface. e physical quantities of interest are local skin friction coefficient C f , local Nusselt number Nu, and local Sherwood number Sh are defined as Substitute equation (7) into equations (12)- (14) to obtain the final dimensionless form: where Re x � U w x/] is the local Reynolds number, Cfr is the reduced skin friction, Nur is the reduced Nusselt number, and Shr is the reduced Sherwood number.

Numerical Method
e reduced ordinary differential equations (9)-(11) subject to the boundary conditions (12) were solved numerically using an implicit finite difference method with quasilinearization technique. e details of the proposed method can be found in the studies of Mabood et al. [21], Inoyue and Tate [22], and Bellman and Kalaba [23]. e effects of the various parameters on the velocity, temperature, concentration, and heat and mass transfer rates are investigated.
e step size and convergence criteria were taken as Δη � 0.001 and 10 −6 , respectively. e asymptotic boundary conditions in equation (12) were approximated by using a value of 10 for the similarity variable η max as follows: e choice of η max � 10 ensures that all numerical solutions approached the asymptotic values correctly.

Results and Discussion
In order to validate our current results and to determine the accuracy of the present analysis, comparisons with available results of the skin friction coefficient −f ″ (0) for the unsteady flow of viscous incompressible Newtonian fluid are done. In Table 1, we compare our results of the skin friction coefficient generated by Chamkha et al. [16]. In this table, we observe that there is an excellent agreement between our results, and those observe in this table that the skin friction coefficient increases with the increasing values of the stretching parameter. In Table 2, we observe that there is an excellent agreement between our present results with those previously obtained by Mabood and Das [8]. We also observe in this table that, as the magnetic parameter M increases, the skin friction significantly increases due to the Lorentz drag force caused by electromagnetism increases. Lastly, Table 3 depicts the comparison of our present results to those obtained by Ali [17] when investigating the effect of the Prandtl number on the heat transfer rate. We observe that the heat transfer rate on the surface is greatly affected by the Prandtl number. Increasing the Prandtl number increases the rate of heat transfer on the stretching surface. Figure 2 depicts the effect of increasing the magnetic field parameter M on the velocity profiles with (Figure 2(b)) and without (Figure 2(a)) hydrodynamic slip. We observe in both cases that the velocity profiles are reduced with increasing values of the magnetic parameter. Physically, the presence of the transverse magnetic field in the fluid flow produces a drag-like force called the Lorentz which in turn decelerates the fluid motion. However, the existence of the hydrodynamic slip as can be clearly seen in Figure 2 increases the velocity boundary layer. We also observe in Figure 2 that suction reduces the velocity boundary layer.
us, suction can be used as a stabilizing mechanism to delay the transition from the laminar boundary layer flow to turbulent flow. e effect of varying the thermal buoyancy parameter on the velocity distribution is depicted in Figure 3 and that of solutal buoyancy parameter is shown in Figure 4. Increasing values of the buoyancy parameters leads to the increase in the temperature and solutal gradients. Physically, this explains why the velocity profiles increase as more forces are added with these increases in buoyancy parameters δ. We also observe in Figure 3 that the velocity profiles decrease as the unsteadiness parameter increases. Also increasing the values of thermal radiation parameter R causes the profiles of the velocity to increase (Figure 4). e influence of the magnetic parameter M and thermal radiation R on the temperature profiles is shown on Figure 5. Reduction of the flow velocity due to the increase of the magnetic field strength causes the temperature profiles ( Figure 5) and concentration profiles ( Figure 6) to increase. Physically, applying the magnetic field heats up the fluid and thus reduces the heat and mass transfer rates from the wall causing increases in fluid temperature and concentration distributions. We also observe in Figure 5 that the fluid temperature increases with increasing values of the thermal radiation parameter R. e effect of radiation in the thermal boundary layer (equation (3)) is equivalent with an increased thermal diffusivity. Figure 7 depicts the influence of thermal buoyancy parameter on the dimensionless temperature profiles. It is clearly observed that an increase in the thermal Grashof number causes a decrease in the thermal boundary layer thickness and consequently the fluid temperature decreases due to the buoyancy effect. We also observe in Figure 7 that the temperature profiles are lower for suction f w > 0. Figure 8 shows the effect of the stretching parameter δ and the Prandtl number on the temperature distribution. We observe in this figure that the temperature profiles are reduced with increasing values of the Prandtl number. It is also observed in this figure that the temperature profiles are reduced with increasing values of the stretching parameter. Figure 6 displays the effect of varying the Schmidt number and the magnetic field parameter on the concentration profiles. e Schmidt number represents the relative ease of the molecular momentum and mass transfer, and it is very important in multiphase flows. e effect of increasing values of the Schmidt number is to reduce the momentum boundary layer, and this leads to the thinning of the diffusion layer. Figure 9 depicts the effects of magnetic, unsteadiness, slip velocity, and suction/injection parameters on the skin friction coefficient. We observe in this figure that |f ″ (0)| decreases with increasing values of the slip velocity S f and buoyancy parameters. But it increases with increasing values of the magnetic and suction parameters. Figure 10 exhibits the nature of the heat transfer coefficient with the thermal radiation, thermal slips, buoyancy, magnetic, and suction and injection parameters. We observe in this figure that increasing the magnetic field strength M strongly suppresses the Nusselt numbers. Physically, an extra work done by dragging the fluid against the action of transverse magnetic field is then dissipated on heat energy in the boundary layer.
is leads to the heating up of the boundary layer resulting in more heat transferred to the fluid flow. us heat transfer to the wall is reduced. e rate of heat transfer on the stretching surface is significantly enhanced by the increasing values of the thermal radiation parameter, suction parameter, and solutal buoyancy parameter. But opposite         Modelling and Simulation in Engineering responses are observed with increasing values of thermal buoyancy and injection parameters. Finally, Figures 11(a) and 11(b) present the Sherwood distributions with variation of the Schmidt number, magnetic parameter, solutal slip parameter, Soret number, sectional injection, and unsteadiness parameter. We observe in these gures that the mass transfer rate decreases with magnetic and solutal slip parameters. We can clearly see in these gures that the rate of mass transfer increases with the Schmidt number and reduces the solutal boundary layer thickness. Finally, we observe that the Sherwood number is signi cantly elevated with increasing values of the unsteadiness parameter.

Conclusions
In this article, a mathematical model has been developed to simulate an unsteady two-dimensional magnetohydrodynamic ow of an incompressible electrically conducting uid over a permeable stretching surface in the presence of multiple slip e ects, Soret e ect, and thermal radiation. Suitable similarity variables have been used to transform the boundary layer conservation equations into a nondimensional, coupled, and nonlinear system of ordinary di erential equations.
e resultant system of ordinary di erential equations was then numerically solved using an implicit nite di erence method with quasi-linearization technique. We also used special cases of the present model to verify our results with previous studies. A parametric study was performed to explore the e ects of various governing parameters on the ow and heat and mass transfer characteristics.
e following conclusions can be drawn from the present study: (i) Increasing the values of the magnetic eld parameter, suction parameter, slip parameters, and unsteady parameter leads to the deceleration of the uid velocity near the boundary layer region. (ii) e velocity pro les are increased with increasing values of the injection parameter, buoyancy parameters, and thermal radiation. (iii) e velocity pro les are found to be reduced with increasing values of the magnetic parameter, suction parameter, and slip velocity parameter. (iv) Increments in thermal radiation, magnetic eld parameter, and slip parameters yield enhancement in the uid temperature. is in turn leads to rapid reductions in the heat transfer rate. However, the opposite e ect is experienced with increasing values of thermal buoyancy parameter, Prandtl number, and unsteadiness parameter.