#### Abstract

The focus of this paper is to analyze the influence of thermal radiation on some unsteady magnetohydrodynamic (MHD) free convection flows of an incompressible Brinkman type fluid past a vertical flat plate embedded in a porous medium with the Newtonian heating boundary condition. The fluid is considered as a gray absorbing-emitting but nonscattering medium and the Rosseland approximation in the energy equations is used to describe the radiative heat flux for optically thick fluid. For a detailed analysis of the problem, four important situations of flow due to (i) impulsive motion of the plate (ii) uniform acceleration of the plate (iii) nonuniform acceleration of the plate, and (iv) highly nonuniform acceleration of the plate are considered. The governing equations are first transformed into a system of dimensionless equations and then solved analytically using the Laplace transform technique. Numerical results for temperature and velocity are shown graphically, while skin friction and Nusselt number are computed in tables. The results show that temperature and velocity increase on increasing radiation and Newtonian heating parameters. However, the results of magnetic and porosity parameters on velocity are found quite opposite.

#### 1. Introduction

Generally speaking, a free convection flow field is a self-sustained flow driven by the presence of a temperature gradient. Extensive research work has been published on free convection flow past a vertical plate for different boundary conditions (see [1–10] and the references therein). Perhaps, it is due to their numerous applications in engineering, geophysical and astrophysical environments, geothermal energy, oil reservoir modelling, building insulation, food processing, and grain storage. Most of these existing studies have used the usual boundary conditions of constant wall temperature or constant heat flux. However, in many practical situations it was realized that the convection flow can also be set up by the Newtonian heating from the surface known as conjugate convective flow [11]. The Newtonian heating situation occurs in many important engineering devices such as in heat exchanger where the conduction in solid tube wall is greatly influenced by the convection in the fluid flowing over it and in conjugate heat transfer around fins where the conduction within the fin and the convection in the fluid surrounding it are simultaneously analyzed in order to obtain the vital design information [12]. In view of these applications various researchers are involved in dealing with the Newtonian heating problems via numerical or analytical techniques.

Chaudhary and Jain [13, 14] investigated the problem of unsteady free convection flow of an incompressible viscous fluid for different motions of the plate with the Newtonian heating condition and obtained the exact solutions using the Laplace transform technique. Recently, Guchhait et al. [15] reinvestigated the problem of Chaudhary and Jain [14] numerically by using the Crank-Nicolson implicit finite-difference scheme. Mebine and Adigio [16] obtained an exact solution for unsteady free convection flow of an incompressible viscous optically thin fluid past an impulsively started vertical porous plate with the Newtonian heating. Salleh et al. [17, 18] analyzed numerically the steady boundary layer flow and heat transfer over a horizontal circular cylinder and over a stretching sheet with the Newtonian heating. Narahari and Ishak [19] investigated the influence of thermal radiation on unsteady free convection flow past a moving vertical plate with Newtonian heating. Recently Akbar and Nadeem [20] studied mixed convective MHD peristaltic flow of a Jeffrey nanofluid with the Newtonian heating. Narahari and Yunus [21] performed an analytical study of free convection flow near an impulsively started infinite vertical plate with Newtonian heating in the presence of thermal radiation and constant mass diffusion. The closed-form exact solutions were obtained by the Laplace transform method.

On the other hand, the study of MHD flows has important technological and geothermal applications in cooling of nuclear reactors, liquid metals fluid, power generation system, aerodynamics, and engineering problems. The study of MHD flow with heat transfer also plays an important role in biological sciences. In addition, the flow through a porous medium is of great physical interest because of its various applications found in hydrology for the sand structures in earth, in filter beds for purification of drinking water and sewage, in chemical engineering for the permeation chromatography, filtering of gases, and liquids chromatography, and in petroleum technology for the production of petroleum and gases. Having such motivation, Hayat et al. [22] analyzed MHD stagnation point flow of a Jeffery fluid past a stretching surface with the Newtonian heating. Lesnic et al. [23, 24] analyzed the free convection flow in a porous medium generated by Newtonian heating. In another study, Lesnic et al. [25] considered the steady free convection boundary layer flow along a semi-infinite plate, slightly inclined to the horizontal and embedded in a porous medium with the flow generated by Newtonian heating. Furthermore, radiative convective flows are frequently encountered in many environmental and scientific processes such as aeronautics, fire research, and heating and cooling of channels. Radiation and thermal diffusion effects on MHD free convection flow of an incompressible viscous fluid near an oscillating plate embedded in a porous medium were considered by Khan et al. [26]. Radiation and porosity effects on the MHD flow past an oscillating vertical plate with uniform heat flux were presented by Samiulhaq et al. [27]. Mohammed et al. [28] discussed analytical solutions of MHD natural convection transients flow near an oscillating plate emerged in a porous medium.

Most of the above studies with Newtonian heating conditions are performed for viscous fluids and numerical or exact solutions are obtained. However, such studies for non-Newtonian fluids are very few, especially those that are solved for exact solutions. Non-Newtonian fluids problems on the other hand are of great and increasing interest for the last five decades. Such fluids differ from the Newtonian fluids in that the relationship between the shear stress and flow field is more complicated. Examples of the non-Newtonian fluids are coal water, food products, inks, glues, soaps, and polymer solutions [29–33]. Therefore, in the present investigation we have chosen Brinkman type fluid as a non-Newtonian fluid. This model was first used by Brinkman in [34, 35]. Later, several authors have used the Brinkman model in describing several types of flows through a porous medium. Among them, Hsu and Cheng [36] studied the natural convection about a semi-infinite vertical flat plate in a porous medium using Brinkman model. Varma and Babu [37], Gorla et al. [38], and Rajagopal [39] also used Brinkman model in their studies. By using Brinkman type fluid, Fetecǎu et al. [40] reported some interesting results. Theyused Fourier sine transform and obtained exact solutions for some unsteady motions. Recently, this problem was reinvestigated by Ali et al. [41] and they established some new types of exact solutions using Laplace transform.

Motivated by above studies, the present research aims to analyze the influence of thermal radiation on unsteady MHD free convection flow of Brinkman type fluid past a vertical plate embedded in a porous medium with Newtonian heating. To the best of authors’ knowledge such study is not available in the literature and the present work claims to fill this space. Moreover, for a detailed analysis of the problem four different types of plate motion are considered. As a result, the following flow situations known as (i) flow induced by an impulsive motion of the plate, (ii) flow due to uniform acceleration of the plate, (iii) flow due to nonuniform acceleration of the plate, and (iv) flow due to highly nonuniform acceleration of the plate are discussed. Exact solutions of momentum and energy equations are obtained by using Laplace transform technique. They satisfy all imposed initial and boundary conditions and can easily be reduced to the similar solutions for hydrodynamic Brinkman type and Newtonian fluids. Analytical as well as numerical results for skin friction and Nusselt number are provided. Graphical results are presented and discussed for various physical parameters. Applications of the present problem are diverse. Because this work presents a new research on free convection flow of Brinkman type fluid for some fundamental fluid motions, therefore this problem can be extended for more complicated studies on Brinkman type fluid in various porous media. Furthermore, the exact solutions obtained in this study will assist scientist and engineers as they can use them as a check of correctness for numerical or approximate solutions of complicated flow problems.

#### 2. Mathematical Formulation and Solution of the Problem

Let us consider unsteady free convection flow of an incompressible and electrically conducting fluid of Brinkman type [40].It is assumed that the plate occupies the half porous spaceover an infinite vertical flat plate situated in theplane of a Cartesian coordinate system, andThe-axis is taken along the vertical plate and the-axis is taken normal to the plate. A uniform magnetic field of strengthis transversely applied to the plate in the outward direction. The applied magnetic field is assumed to be strong enough so that the induced magnetic field due to the fluid motion is weak and can be neglected. As mentioned by Cramer and Pai [42], this assumption is physically justified for partially ionized fluids and metallic liquids because of their small magnetic Reynolds number. Since there is no applied or polarization voltage imposed on the flow field, therefore the electric field due to polarization of charges is zero. Initially, both the plate and fluid are at rest with constant temperatureAfter time, the flow is caused by translating the plate in its plane with time-dependent velocityagainst the gravitational field. It is assumed that the rate of heat transfer from the surface is directly proportional to the local surface temperature. Due to the shear, the fluid is gradually moved and its velocity is of the form whereis the unit vector in the flow direction. The physical model is shown in Figure 1.

In addition to the above assumptions, we assume that the internal dissipation is absent and the usual Boussinesq approximation is taken into consideration. Moreover, the pressure gradient in the flow direction is compensated by the gradient of the hydrostatic pressure gradient of the fluid. As a result the governing equations of momentum and energy are derived as follows: whereis the temperature of the fluid,(is the viscosity andis the constant density of the fluid) is the kinematic viscosity,is the electrical conductivity of the fluid,(is the drag coefficient that is usually assumed to be positive),is the porosity parameter,is the permeability parameter,is the volumetric coefficient of thermal expansion,is the acceleration due to gravity,is the free stream temperature,is the specific heat of the fluid at a constant pressure,is the thermal conductivity, andis the radiative flux along the-axis.

The initial and boundary conditions relevant to the present flow situation are whereis constant andis the heat transfer parameter for Newtonian heating. The radiative heat flux term in (3) is simplified using the Rosseland approximation as whereis the Stefan-Boltzmann constant andis the mean absorption coefficient. It is assumed that the temperature differences inside the flow are sufficiently small such thatmay be expressed as a linear function of the temperature, and then by expanding aboutusing Taylor series and neglecting the higher order terms, we get

Now (3) after using (5) and (6) takes the following form:

By introducing the following dimensionless variables into (2)–(4) and (7), we obtain the following dimensionless system: whereis the Newtonian heating parameter. Using the Laplace transform technique, (9)–(11) give the following transformed solutions: where

The Laplace inverse of (13) results in

For a detailed analysis of (12), we are considering the following four cases.

#### 3. Flow Induced by an Impulsive Motion of the Plate

For an impulsive motion of the plate (), (12) yields

The solution of (16), after taking the Laplace inverse, is given by

#### 4. Flow due to Uniform Acceleration of the Plate

For the uniform acceleration of the plate (), (12) yields

The solution of (18), after taking the Laplace inverse, is given by

#### 5. Flow due to Nonuniform Acceleration of the Plate

For the nonuniform acceleration of the plate (),(12) yields

The solution of (20), after taking the Laplace inverse, is given by

#### 6. Flow due to Highly Nonuniform Acceleration of the Plate

In this case,and (12) yields

The solution of (22), after taking the Laplace inverse, is given by where

#### 7. Special Cases

The following solutions from the literature appear as the limiting cases of our general solutions.

##### 7.1. Absence of Thermal Effects

###### 7.1.1. Case I: Impulsive Motion of the Plate

By eliminating the convectional termtogether with MHD and porosity effects (and that is,in (17), we get which is in quite agreement with [40, Equation ] whenFurthermore by putting, (25) reduces to which is the well-known solution of Newtonian fluid [40, Equation ] for the impulsive motion of the plate obtained by a different technique.

###### 7.1.2. Case II: Accelerated Motion of the Plate

In this case when convectional termwithand, (19) reduces to which is in complete agreement with [41, Equation ] for accelerated motion of the plate.

*Skin Friction*. The expressions for skin frictionsandevaluated from (17) and (19), respectively, are given by
where

*Nusselt Number*. The rate of heat transfer evaluated from (15) is given by

#### 8. Graphical Results and Discussion

In this paper, the influence of thermal radiation on unsteady free convection flow of MHD Brinkman type fluid past a vertical plate in a porous medium is studied with Newtonian heating condition. Four different types of motions in terms of different boundary conditions on velocity are discussed. Exact solutions are obtained by means of Laplace transforms. Straightforward computations show that the velocitiesgiven by (17), (19), (21), and (23) and temperature (15), respectively, satisfy both the governing equations and all imposed initial and boundary conditions. As a special case when, the solutions that have been established reduce to the well-known solutions for Newtonian fluids. It is worthy pointing out that solutions corresponding to hydrodynamic motion of Brinkman fluid in a nonporous medium can also be obtained as a limiting case. In order to study the physical aspects of the problem, the analytical results for velocitiesanddue to impulsive motion and uniform acceleration as well as for temperature distribution are plotted graphically in Figures 2–12, whereas the numerical results of skin friction and Nusselt number for different flow parameters are shown in Tables 1 and 2. The parameters entering into the problem are Prandtl number, magnetic parameter, porosity parameter, Grashof number, radiation parameter, Newtonian heating parameter, dimensionless time, and Brinkman parameter.

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Figure 2 is plotted to see the effect ofon velocity in cases of impulsive motionand uniform accelerationof the plate. Three different values of, , andare chosen such that physically they correspond to air, electrolytic solution, and water, respectively. It is clear from this figure that velocity decreases for both impulsive motion and accelerated motion of the plate asincreases. Physically, it is true due to the fact that an increase in Prandtl number increases viscosity of the fluid. Hence, the fluid becomes thick and consequently velocity decreases. Figure 3 is prepared to see the effects of the magnetic parameter on velocity profiles for impulsive and accelerated motions of the plate. It is observed that velocity in both cases decreases when is increased. This physical interpretation of the velocity is an indication that the increasing values of magnetic parametermake the resistive forces strong enough so that they can oppose the fluid motion and as a result velocity decreases. This figure also shows the comparison of hydrodynamic velocity to hydromagnetic velocity of Brinkman fluid. In the absence of magnetic effects the velocity is maximum. Furthermore, it is observed that the momentum boundary layer thickness for the impulsive motion is greater compared to uniformly accelerated motion. However, the magnitude of velocity in case of uniformly accelerated motion is greater than that of impulsive motion.

Figure 4 is plotted to show the effects of porosity parameteron the velocitiesand. It appears from this figure that with increasing values of, velocity increases for both impulsive and uniformly accelerated motions. Physically, it is true because increasingreduces the drag force and causes the velocity profiles to increase. Thus increasing value of the porosity parameteryields an effect opposite to that of the magnetic parameter. Figure 5 shows the effect of Grashof numberon the velocities for the cases of impulsive and uniformly accelerated motions. It is observed that an increase in Grashof number leads to increasing both types of velocitiesanddue to enhancement in buoyancy force. It is true because thermal Grashof number signifies the relative effect of thermal buoyancy force to viscous hydrodynamic force. Increase of Grashof numbermeans increase of temperature gradient () due to which the contribution from the buoyancy near the plate becomes significant and hence a short rise in the velocity near the plate is observed. Physically,corresponds to the absence of free convection current andmeans heating of the fluid or cooling of the plate by natural convection. For the positive values of, heat is conducted away from the vertical plate into the fluid which increases the temperature and thereby enhances the buoyancy force.

The graphs showing the variations of radiation parameter on the velocitiesandare displayed in Figure 6. We found that in both cases the velocity increases asincreases. Physically, it is true, as higher radiation occurs when temperature is higher and ultimately velocity rises. Figure 7 is plotted to show the influence of conjugate parameteron velocitiesand. We observe that velocity increases in both figures asincreases. The influence of dimensionless timeon velocity is shown in Figure 8. It is seen from this figure that velocity increases whenincreases. Furthermore, both of these velocitiesandsatisfy the given boundary conditions (see (11)). This proves the accuracy of our obtained results. The effects of Brinkman parameteron the flow due to impulsive and uniformly accelerated motions of the plate are studied in Figure 9. We observe that velocity decreases asincreases. Physically, it is true in the sense that the Brinkman type fluid can move slowly due to the drag properties of the fluid.

Figure 10 is sketched to show the effects of Prandtl numberon temperature profiles. Three different values of, andthat physically correspond to air, electrolytic solution, and water are chosen. It is observed that increasing values oflead to a fall in the temperature. A similar behavior was also expected due to the fact that at smaller values offluids possess high thermal conductivity and heat diffuses away from the surface faster than at higher values of. Thus the boundary layer becomes thicker and consequently temperature decreases whenis increased.

Figure 11 is plotted to show the influence of radiation parameteron the temperature profiles. It is found that temperature shows an increasing behavior asincreases. Physically, a similar effect was also expected because the radiation parametersignifies the relative contribution of conduction heat transfer to thermal radiation transfer. The temperature profiles for different values of the Newtonian heating parameterare plotted in Figure 12. We observe that temperature profiles increase whenincreases. It is due to the fact that, as we increase, it results in an increase in the heat transfer rate from the surface which raises the temperature.

The numerical results for skin friction and Nusselt are shown in Tables 1 and 2. Table 1 is prepared to show the effects of,, , , , ,,, andon the skin frictions andcorresponding to impulsive and uniformly accelerated motions of the plate, respectively. It is observed that bothandincrease when the values of, andare increased. However, this behavior is quite opposite for radiation parameter, conjugate parameter, permeability parameterand Grashof numberTable 2 represents the numerical results for the rate of heat transfer in terms of Nusselt number Nu for different values of Prandtl number, radiation parameter, and conjugate parameter. It is found that the rate of heat transfer decreases with the increasing values ofand increases when , andare increased.

#### 9. Conclusions

The exact solutions for the unsteady free MHD convection flow of an incompressible Brinkman fluid past a vertical plate in a porous medium with radiation effects and Newtonian heating condition are obtained using the Laplace transform technique. For a detailed discussion of the problem, four important flow situations are discussed. It was observed from the analytical and graphical results that the obtained solutions satisfy the imposed initial and boundary conditions as well as the basic equations of the present problem. The analytical results of velocity corresponding to the impulsive and uniformly accelerated motions of the plate together with temperature are plotted graphically, whereas the numerical results for skin friction and Nusselt number are provided in tables. From the present analysis, we observed the following:(i)The effects ofandon the velocities are quite opposite.(ii)The magnitude of velocity for uniformly accelerated motion is greater than the magnitude for the impulsive motion of the plate.(iii)In the absence of free convection velocity shows the highest amplitude.(iv)Increasing thermal radiation increases velocity.(v)Velocities increase for large values of the Newtonian heating parameter.(vi)Velocities are increasing functions of time.(vii)Increasing non-Newtonian fluid parameterdecreases fluid velocity.(viii)The temperature increases for large values ofand.(ix)The solutions in [40, 41] appear as the limiting cases of the present solutions.

#### Acknowledgment

The authors would like to acknowledge the Research Management Centre, UTM, for the financial support through vote nos. 4F109 and 04h27 for this research.