Unsteady Mixed Convection Slip Flow around a Stretching Sheet in Porous Medium

The heat and mass transfer of unsteady MHD two-dimensional mixed convection boundary layer flow over an exponentially porous stretching sheet is presented in this paper. Multiple slip conditions, radiation, suction or blowing, heat generation or absorption along with magnetism and porous medium are incorporated. We reduce the leading equations which are partial differential equations into a family of ordinary differential equations that are non-linear using a set of similarity transformations. The resulting equations with coupled boundary conditions are solved numerically with the aid of bvp4c solver with MATLAB package. The impacts of several non-dimensional governing parameters on the flow fields such as velocity, temperature and concentration profiles along with friction coefficient, temperature gradient and concentration gradient are portrayed graphically and discussed in detail. The result in-dicates that the magnetic parameter decreases the skin friction coefficient. Thermal boundary layer thickness reduces with increasing radiation parameters and enhances with increasing Prandtl number. It is also observed that the thermal slip parameter depreciates the heat transfer rate and the mass slip parameter diminishes the mass transfer rate. A comparison has been made between the current results and the numerical results of previous studies and observed a very close good agreement.


Introduction
If natural convection is combined with forced convection, then the resulting convection is known as mixed convection, which is frequently used to get the desired result when the forced convection can't dissipate all the heat in the devices involving high voltage. There are several industrial and technical applications of mixed convection including the cooling of nuclear reactors during an emergency shutdown, a heat exchanger placed in a low-velocity environment, solar collectors, electronic devices cooled by fans, etc. [1]. Srinivasacharya and RamReddy [2] investigated the problem of mixed convection boundary flow, heat and mass transfer over an exponentially stretching sheet in the presence of Soret and Dufour. Aman and Ishak [3] studied the mixed convection boundary layer flow and heat transfer over an impermeable vertical plate, taking convective thermal conditions at the boundary. Isa et al. [4] described joule heating effects on MHD mixed convection flow past an exponentially stretching vertical sheet with viscous dissipation, heat generation and convective boundary condition. Patil et al. [5] obtained a non-similar solution of mixed convection boundary layer flow, heat and mass transfer over a vertical semi-infinite impermeable exponentially stretching surface with first order chemical reaction. Ali et al. [6] described the effects of thermal radiation on two-dimensional mixed convection stagnation point flow and heat transfer over a vertical stretching sheet in the presence of the external magnetic field.
The study of porous media with heat transfer has attracted the attention of researchers due to its wide applications in applied science, soil mechanics, rock mechanics, petroleum engineering, construction engineering, geoscience and many more. For example, food processing, cooling of nuclear reactors, underground disposal of nuclear waste, underground water resources and seepage of water in river beds in agricultural engineering, petroleum reservoir operations, filtration and purification in chemical engineering, building insulation, and casting and welding in manufacturing processes, etc. [7] [8]. Several investigators studied the combined effects of heat and mass transfer in mixed convection through porous medium under different conditions. Afify and Elgazery [9] studied the combined effects of thermal radiation, melting and double dispersion on the non-Darcy mixed convective boundary layer flow, heat and mass transfer over a vertical porous surface using Chebyshev pseudo spectral method. Tripathy et al. [10] described the effects of a non-uniform heat source and chemical reaction on the convected flow, heat and mass transfer of a micropolar fluid along a stretching sheet embedded in a porous medium. Nithyadevi et al. [11] presented the problem of MHD mixed convection flow of a nano fluid numerically in a vertically lid-driven porous enclosure with a center heater. Krishna et al. [12] studied the combined effects of chemical reaction, hall and ion slip on unsteady MHD free convective rotating flow, heat and mass transfer over an exponentially accelerated inclined plate in a porous medium.
Nowadays, researchers have considered unsteady two-dimensional boundary layer flow under different conditions due to its importance in many engineering processes. The flow fields become unsteady as a result of a sudden change in the surface temperature, and impulsive stretching of the surface or external stream [13]. Journal of Applied Mathematics and Physics The study of heat generation or absorption is important in heat and mass transfer in certain porous media applications. The temperature distribution may be changed with the effects of heat generation. Therefore, the particle deposition rate is affected in nuclear reactors, electronic chips, and semiconductor wafers [14]. Moreover, radiative heat transfer is significant in processes involving high temperatures.
The problem of the slip flow regime is extremely important in this area of recent science, technology and immense extending automation. The fluid slippage phenomenon at the solid boundaries appears in many applications like micro channels or Nano channels. It is also used where a thin film of light oils is involved in the moving plates or when the surfaces are coated with special coating to lessen the friction between them [15]. The flow field can be changed significantly through the bounding surface with the suction or blowing of a fluid.
Generally, suction has a tendency to increase skin friction, whereas the reverse occurs with blowing. Blowing or removal of fluid through a porous bounding wall is of general attention in practical problems including boundary layer control applications such as film cooling, polymer fibre coating, and coating of wires [16]. Mukhopadhyay [17] analyzed the slip effects on unsteady mixed convection flow over a stretching surface with suction. Sreelakshmi and Nagendramma [18] studied the influence of thermophoresis, viscous dissipation and temperature dependent heat source on the unsteady boundary layer flow of a viscous fluid over an exponentially stretching sheet with chemical reaction. Mabood and Shateyi [19] described the combined effect of Soret and thermal radiation on unsteady MHD mixed convection flow over a stretching sheet with suction and slip boundary conditions. Through porous medium, unsteady MHD boundary layer slip flow over an exponentially stretching sheet including radiation, heat generation, and suction was analyzed numerically by Islam et al. [20].
As far as is known no studies have been described on the unsteady MHD mixed convection heat and mass transfer flow over a porous radiative sheet which stretches exponentially with heat generation or absorption, suction and slip boundary conditions by considering L = 1 and α = 1 in the similarity and dimensionless variables. We convert the governing partial differential equations to a group of nonlinear ordinary differential equations using similarity transformations. Numerical solutions of those equations are found using the bvp4c function in MATLAB software.

Formulation of the Problem
We study an unsteady, laminar, two-dimensional, incompressible and electrically conducting MHD mixed convection boundary layer flow in a porous medium with slip boundary conditions. We select a cartesian co-ordinate system such that the x-axis is along the sheet and y-axis is normal to it. The surface is stretched with the velocity C denote the reference velocity, temperature and concentration respectively, T ∞ and C ∞ represent the ambient temperature and concentration respectively with w T T ∞ > and w C C ∞ > .
It is worth mentioning that we take 1 L = and 1 α = in similarity and dimensionless variables. The flow is produced by the sheet stretching in the exponential form.
The equations of conservation of mass, momentum, energy and species concentration governing the mixed convection boundary layer flow over a porous exponential stretching sheet can be written as: with the corresponding boundary conditions: and 0, , as where the components of velocity in x and y directions are represented by u, v respectively, t denotes the time, ρ stands for the density and kinematic viscosity, gravitational acceleration, thermal expansion coefficient, the temperature of the fluid, mass expansion coefficient, concentration and electric conductivity of the fluid are indicated by , , , , , ν β β and σ respectively. Moreover, κ is for the thermal conductivity, C p represents the specific heat at constant pressure, the radiative heat flux is expressed by q r and the co-efficient of heat generation and mass diffusivity are denoted by Q 0 and D m respectively. Further, the uniform magnetic field, the permeability of the porous medium, the suction or injection velocity and velocity, thermal and mass slip factors are defined as follows:

of Applied Mathematics and Physics
Here B 0 is the intensity of magnetic field, K 0 is a constant, V 0 is the strength of suction or blowing, N 1 stands for the initial values of velocity slip factor, E 1 denotes the initial values of thermal slip factor and F 1 characterizes the initial values of mass slip factor. We can attain the condition of the no-slip condition con- The q r , known as radiative heat flux through Rosseland approximation [21], is as follows: Here σ * signifies the Stefan-Boltzman constant and K * stand for the absorption coefficient. If the difference in the temperature within the flow is sufficiently small then we can expand 4 T in Taylor's series about T ∞ . If we neglect the higher order terms except the first degree in ( ) Then the Equation (3) can be written as follows: ( ) Introducing the dimensionless variables as: and upon replacement of (8) in the Equations (2), (3) and (4), a family of nonlinear ordinary differential equations are obtained as follows: and the Equation (5) representing boundary conditions, is simplified as follows: where X x = is the dimensionless co-ordinate, ( ) ( )

Result and Discussions
In this study, an investigation is carried out to illustrate graphically the effects of different non-dimensional physical parameters on the velocity, temperature and concentration profiles taking X = 2, η = 2, Gr = Gc = K = Q H = 0.1, M = S = 0.5, Pr = 5, Sc = 1.5, R = A = δ = γ = 0.3 and λ = 0.05. In Table 1, we compare the value of Skin-friction co-efficient f(0) for unsteadiness parameter A with the results of Mabood and Shateyi [19] and find a close good agreement. In Table 2, a comparison of the value of Skin-friction co-efficient f(0) for some values of magnetic parameter M with the results of Mabood and Shateyi [19] is presented and it is observed that an excellent agreement exists. We also see from Table 2 that skin-friction co-efficient reduces for large values of magnetic parameter M.       Figures 7-9 describes the effects of magnetic parameter on velocity, temperature and concentration profiles respectively. It is seen from Figure 7 that fluid velocity reduces with the rise in magnetic parameters. This happens because of a resistive force named Lorenz force which is produced due to the imposed magnetic field. On the other hand, temperature and concentration profiles are seen to enhance the rising values of the magnetic parameter. Journal of Applied Mathematics and Physics    For various values of porosity parameter (K), the velocity, temperature and concentration distribution are plotted in Figures 10-12. It is understood from Figure 10 that increasing values of porosity parameter decline the fluid velocity and therefore decrease the momentum boundary layer thickness. As like magnetic parameter, temperature and concentration distribution (Figure 11 & Figure 12) decrease as porosity parameter (K) increases. Figures 13-15 clarify the effects of the unsteadiness parameter on velocity, temperature and concentration profiles respectively. As the unsteadiness parameter increases fluid velocity, temperature and concentration profiles increase respectively. Figures 16-18 elucidate the effects of velocity slip parameter on velocity, temperature and concentration profiles respectively. It is noticed from Figure 16 that velocity profile reduces as the velocity slip parameter rises. With the increasing values of velocity slip parameter, the slip velocity increases and so       Figure 20 presents that fluid temperature increases due to increase in radiation parameter.
Because heat transfer increases with the rise of radiation parameter (R). Figure   21 exhibits the influence of heat generation and absorption parameter on temperature profile. It is seen that heat generation (Q H > 0) parameter increases the fluid temperature by increasing the thermal boundary layer thickness whereas             but increases after a certain distance η from the sheet in the presence of thermal Grashof number. The effect of radiation parameter on temperature gradient is portrayed in Figure 29, which shows that rate of heat transfer diminishes initially but enhances after a certain distance for η from the sheet. On the other hand, an opposite effect is observed for Prandtl number in Figure 30; that is temperature gradient accelerates initially but decelerates after a certain distance for η from the sheet. Figure 31 highlights the effect of Schmidt number on concentration gradient against suction parameter. There exists intercept among the curves. It is found that Sherwood number increases initially but decreases for both large values of Schmidt number and suction parameter.

Conclusions
The problem of unsteady MHD mixed convective radiative flow over a porous sheet stretching exponentially with the effects of suction or blowing, multiple slips and internal heat generation or absorption is studied numerically. Some important results of the existing study are given below: • The thermal Grashof number enhances the velocity and depreciates the temperature and concentration of the fluid. The mass Grashof number has the same effects as like thermal Grashof number on velocity, temperature and concentration profile respectively. • The velocity temperature and concentration profile are increased with the rise in the unsteadiness parameter. • The porosity and magnetic parameters have the same effects on fluid temperature and concentration profile as like velocity graph. • The velocity slip parameter has the ability to reduce the fluid velocity and enhance the temperature and concentration of the fluid. • Suction parameter has an inclination to reduce the fluid velocity, temperature profile and concentration distribution; however blowing parameter acts reversely.