#### Abstract

This study investigates the effects of an arbitrary wall shear stress on unsteady magnetohydrodynamic (MHD) flow of a Newtonian fluid with conjugate effects of heat and mass transfer. The fluid is considered in a porous medium over an inclined plate with ramped temperature. The influence of thermal radiation in the energy equations is also considered. The coupled partial differential equations governing the flow are solved by using the Laplace transform technique. Exact solutions for velocity and temperature in case of both ramped and constant wall temperature as well as for concentration fields are obtained. It is found that velocity solutions are more general and can produce a huge number of exact solutions correlative to various fluid motions. Graphical results are provided for various embedded flow parameters and discussed in detail.

#### 1. Introduction

The study of the natural convection heat transfer from different geometries of the surfaces has received much attention in recent years both analytically and experimentally. Amongst them a very little attention has been given to the problem of natural convection over an inclined plate as shown in Figure 1. Free convection heat transfer from an inclined surface has important role in making engineering devices and natural environment. Natural convection over an inclined plate was first studied experimentally by Rich [1]. A solution for the boundary layer on a horizontal plate showing that if the plate is heated and faces downwards or is cooled and faces upwards was presented by Stewartson [2]. Free convection heat transfer from an isothermal plate with arbitrary inclination was investigated by [3]. Chen et al. [4] have obtained a numerical solution for the problem of natural convection over an inclined plate with variable surface temperature. Ganesan and Ekambavanan [5] solved the unsteady natural convection boundary layer flow over a semi-infinite inclined plate with the wall temperature varying as the axial coordinate using an implicit finite difference scheme. The results of an experimental study on natural convection between inclined plates were presented by [6]. MHD natural convection from a nonisothermal inclined surface with multiple suction/injection slots embedded in a thermally stratified high porosity medium has been studied by [7]. A numerical solution of the transient free convection MHD flow of an incompressible viscous fluid past a semi-infinite inclined plate with variable surface heat and mass flux is presented by [8]. Unsteady free convection flow of water at its maximum density past a semi-infinite inclined plate with variable surface temperature of the plate was studied by [9]. An investigation deals with study of laminar natural convection flow of a viscous fluid over a semi-infinite flat plate inclined at a small angle to the horizontal with internal heat generation and variable viscosity was presented by [10]. Pressure effects on unsteady free convection and heat transfer flow of an incompressible fluid past a semi-infinite inclined plate with impulsive and uniformly accelerated motion were analyzed by [11]. Chemical reaction effects on MHD free convection flow through a porous medium bounded by an inclined surface were studied by [12]. An analysis to study the heat and mass transfer characteristics of natural convection flow along horizontal and inclined plates with variable surface temperature/concentration or heat/mass flux under the combined buoyancy effects of thermal and mass diffusion was presented by [13]. A steady two dimensional MHD free convection and mass transfer flow past an inclined semi-infinite surface in the presence of heat generation has been studied numerically in [14]. A two-dimensional steady MHD mixed convection and mass transfer flow over a semi-infinite porous inclined plate in the presence of thermal radiation with variable suction and thermophoresis has been analyzed numerically by [15]. Heat and mass transfer of an incompressible viscous fluid past a semi-infinite inclined surface with first-order homogeneous chemical reaction by Lie group analysis had been presented by [16]. Recently, the exact analysis of combined effects of radiation and chemical reaction on the MHD free convection flow of an electrically conducting incompressible viscous fluid over an inclined plate embedded in a porous medium was studied by [17]. The inherent irreversibility and thermal stability in a gravity driven temperature dependent variable viscosity thin liquid film along an inclined heated plate with convective cooling were investigated by Makinde [18]. The inherent irreversibility in hydromagnetic boundary layer flow of variable viscosity fluid over a semi-infinite flat plate under the influence of thermal radiation and Newtonian heating was analyzed by [19]. Saha et al. [20] investigated that a natural convection boundary layer adjacent to an inclined semi-infinite plate subject to a temperature boundary condition which follows a ramped function up until some specified time and then remains constant. Saha et al. [21] also performed a scaling analysis for the transient boundary layer established adjacent to an inclined flat plate following a ramp cooling boundary condition. Recently, Ismail et al. [22] conducted the combined effects of heat and mass transfer on unsteady MHD free convection flow in a porous medium past an infinite inclined plate with ramped wall temperature. Fetecau et al. [23] investigated free convection flow near a vertical plate that applies arbitrary shear stress to the fluid when the thermal radiation and porosity effects are taken into consideration. However, from the literature, it is found that no study has been presented to investigate the unsteady MHD conjugate flow of an incompressible viscous fluid in a porous medium past over an inclined plate with ramped temperature in the presence of radiation under the boundary condition of wall shear stress.

Therefore, the aim of the present investigation is to provide exact solutions for MHD conjugate flow of a Newtonian fluid past an infinite plate that applies arbitrary shear stress to the fluid. More exactly, we consider the inclined plate situated in the plane of a Cartesian coordinate system , the domain of the flow is the porous half-space , and the arbitrary shear stress on the plate is given by , where is an arbitrary function and is the viscosity. Closed form solutions of the initial and boundary value problems that govern the flow are obtained by means of the Laplace transform. Some special cases are extracted from the general solutions together with some limiting solutions in the literature. The results for velocity, temperature, and concentration profiles are plotted graphically and discussed for the embedded flow parameters.

#### 2. Mathematical Formulation

Let us consider the unsteady MHD free convection flow of an incompressible viscous fluid over an infinite inclined plate. The physical configuration of the problem is shown in Figure 1. The -axis is taken along the plate and the -axis is taken normal to it. Initially, both the plate and fluid are at stationary conditions with the constant temperature and concentration . After time , the plate applies a time dependent shear stress to the fluid along the -axis. Meanwhile, the temperature of the plate is raised or lowered to when , and, thereafter, for , is maintained at constant temperature and concentration is raised to . The radiation term is also considered in the energy equation. However, the radiative heat flux is considered negligible in -direction compared to -direction. We assume that the flow is laminar and the fluid is grey absorbing-emitting radiation but no scattering medium. Using Boussinesqâ€™s approximation and neglecting the viscous dissipation, the equations governing the flow are given by [22] where , , , , , , , , , , , , , , and are the velocity of the fluid in -direction, its temperature and concentration, the kinematic viscosity, the constant density, the gravitational acceleration, the heat transfer coefficient, the mass transfer coefficient, the permeability of the porous medium, the electric conductivity of the fluid, the applied magnetic field, the heat capacity at constant pressure, the thermal conductivity, the radiative heat flux, and mass diffusivity.

The corresponding initial and boundary conditions are The radiation heat flux under Rosseland approximation for optically thick fluid [24â€“26] is given by where and is the Stefan-Boltzman constant and the mean absorption coefficient, respectively. We can see from (5) that the radiation term is nonlinear. Recently David Maxim Gururaj and Anjali Devi [27] used nonlinear radiation effects and studied MHD boundary layer flow with forced convection past a nonlinearly stretching surface with variable temperature. Therefore, we follow David Maxim Gururaj and Anjali Devi [27] and assume that the temperature differences within the flow are sufficiently small; that is, the difference between the fluid temperature and the free stream temperature is negligible, so that (5) can be linearized by expanding into the Taylor series about , which after neglecting higher order terms takes the form Substituting (6) into (5) and then putting the obtained result in (2), we get where , , and are defined by By introducing the following dimensionless variables into (1), (3) and (7) and eleminating the star notations, we get where is the effective Prandtl number [24, ] and are the Grashof number, modified Grashof number, magnetic parameter, Schmidt number, the inverse permeability parameter for the porous medium, and the characteristic time, respectively.

The corresponding dimensionless initial and boundary conditions are

#### 3. Solution of the Problem

In order to solve (10)â€“(12) under conditions (14), we use the Laplace transform technique and get the following differential equations: with boundary conditions Solving (16) in view of (18), we get which upon inverse Laplace transform gives where is the corresponding heat transfer rate also known as Nusselt number. Here and denote the error function and complementary error function of Gauss.

Solution of (17) using boundary conditions from (18) yields which upon inverse Laplace transform gives that is the corresponding mass transfer rate also known as Sherwood number.

The solution of (15) under boundary conditions (18) gives which upon inverse Laplace transform gives where correspond to the convective and mechanical parts of velocity.

It is noted from (20) and (28) that is valid for all positive values of while the is not valid for . Therefore, to get when the effective Prandtl number is not equal to one, we make into (11), use a similar procedure as discussed above, and obtain By taking inverse Laplace transform, we find that where

#### 4. Plate with Constant Temperature

Equations (20) and (27) give analytical expressions for the temperature and velocity near an inclined plate with ramped temperature. In order to highlight the effect of the ramped temperature distribution of the boundary on the flow, it is important to compare such a flow with the one near a plate with constant temperature. It can be shown that the temperature, rate of heat transfer, and velocity for the flow near an isothermal plate are As previously, (36) is not valid for . Therefore we calculate separately solution for velocity by taking into (11) and finally get

#### 5. Limiting Cases

In this section, we discuss few limiting cases of our general solutions.

##### 5.1. Solution in the Absence of Porous Effects for Ramped and Constant Wall Temperature ()

Consider

##### 5.2. Solution in the Absence of Thermal Radiation ()

In the absence of thermal radiation, the corresponding solutions for ramped and constant wall temperature are directly obtained from the general solutions (20), (22), (27), and (33)â€“(36) by taking and replacing Pr_{eff} by ; that is,
where

##### 5.3. Solutions in the Absence of Free Convection

Let us assume that the flow is caused only due to bounding plate and the corresponding buoyancy forces are zero equivalently it shows the absence of free convection due to the differences in temperature and mass gradients that is, the terms and are zero. This shows that the convective parts of velocities are zero in both cases of ramped wall and constant temperature and the flow is only governed by the mechanical part of velocitiesgiven by (29) and (36).

##### 5.4. Solutions in the Absence of Mechanical Effects

In this case we assume that the infinite plate is in static position at every time; that is, the function is zero for all values of and the mechanical parts for both ramped and constant wall temperature are equivalently zero. In such a situation, the motion in the fluid is induced only due to the free convection. Therefore, the velocities of the fluid in both cases of ramped and constant wall temperature are only represented by their convective parts given by (28) and (36).

##### 5.5. Solution in the Absence of Magnetic Parameter ()

It is clear from (20) and (24) that the temperature and concentration distributions are not influenced by the magnetic parameter , and the velocities with for both ramped and constant wall temperature are given by

#### 6. Special Cases

We noted that the solutions for velocity obtained in Section 3 are more general. Therefore, we want to discuss some special cases of the present solutions together with some limiting solutions in order to know more about the physical insight of the problem. Hence, we discuss the following important special cases in the case of ramped wall temperature whose technical relevance is well known in the literature. Similarly we can discuss some special cases of constant wall temperature solutions.

##### 6.1. Case-I:

In this first case we take the arbitrary function where is a dimensionless constant and denotes the unit step function. After time the infinite inclined plate applies a constant shear stress to the fluid. The convective part of the velocity remains unchanged while the mechanical part takes the following form: equivalently for , . Moreover, if we take , (42) reduces to the form which is equivalent to [28, Equation ] with the correction of .

Furthermore, in the absence of both and , (42) is identical with [23, Equation ]

##### 6.2. Case-II:

In the second case, we take the arbitrary function of the form in which the plate applies an oscillating shear stress to the fluid. Here denotes the dimensionless frequency of the shear stress. As previously, the convective part of velocity remains the same whereas the mechanical part takes the form It can be further written as a sum of the steady-state and transient solutions where By taking , the steady-state component reduces to [28, Equation ] In addition when physically it corresponds to the absence of porous effects and (49) results in which can be written in simplified form as equivalent to [23, Equation ].

##### 6.3. Case-III:

In the final case, we take in which the plate applies an accelerating shear stress to the fluid where the mechanical part takes the following form: The corresponding solution for , namely, is identical with [28, Equation ].

Additionally, if we take (53) yields

#### 7. Results and Discussion

In order to understand the physical aspects of the problem, the numerical results for velocity, temperature, and concentration are computed and plotted for various parameters of interest such as magnetic parameter , porosity parameter , effective Prandtl number Grashof number , modified Grashof number , dimensionless time , Schmidt number , and shear stress . The graphs for velocity are shown in Figures 2â€“17, where corresponds to isothermal velocity and is for ramped velocity. Figures 18â€“21 are plotted to show the temperature variations for two types of boundary conditions, namely, ramped and constant wall temperatures. Furthermore, Figure 22 is displayed to show variations in fluid concentration. Figures 2 and 3 illustrate the influence of Grashof number on the velocity. It is observed that velocity increases with increasing . This implies that thermal buoyancy force tends to accelerate velocity for both ramped temperature and isothermal plates. In Figures 4 and 5, the velocity profiles for different values modified Grashof number are shown. It is found that velocity increases on increasing for both ramped temperature and isothermal plate. Further, it can be observed that the velocity and boundary layer thickness decrease along with increasing distance from the leading edge. Moreover, from Figures 4 and 5, we observed that the amplitude of velocity in case of isothermal plate is greater and converges slowly as compared to ramped velocity. In Figures 6 and 7, the velocity profiles are shown for different values of Schmidt number . Here the values of are chosen , , and . to represent the presence of species by hydrogen, water vapor, and carbon dioxide respectively. It is observed that the velocity decreases with increasing Schmidt number. Physically, this refers to the phenomenon that increasing Schmidt number implies the dominance of the viscous forces over the diffusional effects. As a result, the flow will be therefore decelerated with a rise in Schmidt number. The velocity profiles for different values of magnetic parameter are shown in Figures 8 and 9. The range of magnetic field is taken from 0 to 2. It is found that the velocity is decreasing with increasing values of in both cases of ramped and isothermal plates. Physically, it is true due to the fact that increasing values of causes the frictional force to increase which tends to resist the fluid flow, thus reducing its velocity. It is further observed that when the magnetic field imposed on the flow is zero (), the MHD effect vanishes and the flow is termed as hydrodynamic flow. Physically, it is true due to the fact that increasing values of causes the frictional force to increase which tends to resist the fluid flow, thus reducing its velocity. Figures 10 and 11 are plotted to see the difference between the ramped and isothermal plate velocities. The values of correspond to ramp velocity whereas is for isothermal plate. It is found that ramp velocity is less than isothermal plate and converges faster. Further velocity in both cases increases with increasing time. The effects of inverse permeability parameter on the velocity profiles are presented in Figures 12 and 13. It is found that velocity decreases with increasing in both cases of ramp and isothermal plate. Physically, it is due to the fact that increasing permeability of the porous medium increases the resistance and consequently velocity decreases. This observation is an excellent agreement with the previous study [28, Figure 3]. The effects of the shear stress induced by the bounding plate on the nondimensional velocity profiles are shown in Figures 14 and 15. The velocity of fluid is found to decrease with increasing in both cases of ramped velocity and isothermal plate. Graphical results to show the influence of the effective Prandtl number on velocity profiles are presented in Figures 16 and 17. It is observed that the velocity is a decreasing function with respect to . These graphical results are in accordance with [28, Figure 2]. The temperature variations against for various values of effective Prandtl number are highlighted in Figures 18 and 19. The significant decrease of the temperature is found as a result of an increase of the effective Prandtl number. The fluid temperature decreases from maximum at the boundary to a minimum value as far from the plate in both cases of ramped and constant temperature. In Figures 20 and 21, we have shown the temperature variations for two types of boundary conditions ramped and constant plate temperatures. It is noted that the fluid temperature is greater in the case of isothermal plate than in the case of ramped temperature at the plate. This should be expected since, in the latter case, the heating of the fluid takes place more gradually than in the isothermal case [29]. Moreover, with increasing time, the temperature is found to increase in both cases of ramped and constant wall temperature. The concentration profiles for different values of Schmidt number are shown in Figure 22. It is clear from this figure that the concentration profiles and the concentration boundary layer thickness decrease with increasing values of . Physically, it is true, since increase of means decrease of molecular diffusivity which results in a decrease of concentration boundary layer.