Influence of Thermal Radiation on Unsteady Free Convection MHD Flow of Brinkman Type Fluid in a Porous Medium with Newtonian Heating

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 space𝑦>0over an infinite vertical flat plate situated in the(𝑥,𝑧)plane of a Cartesian coordinate system𝑥,𝑦, and𝑧.The𝑥-axis is taken along the vertical plate and the𝑦-axis is taken normal to the plate. A uniform magnetic field of strength𝐵₀is 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 temperature𝑇_∞.After time𝑡=0⁺, the flow is caused by translating the plate in its plane with time-dependent velocity𝐴𝑡^𝑛against the gravitational field. It is assumed that the rate of heat transfer from the surface is directly proportional to the local surface temperature𝑇. Due to the shear, the fluid is gradually moved and its velocity is of the form 𝐯=𝐯𝑦,𝑡=𝑢𝑦,𝑡𝐢, (1) where𝐢is the unit vector in the flow direction. The physical model is shown in Figure 1.

Figure 1 

Physical model and coordinates system.

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: 𝜕𝑢𝑦,𝑡𝜕𝜕𝑡+𝛽𝑢𝑦,𝑡=𝜈²𝑢𝑦,𝑡𝜕𝑦²−𝜎𝐵²₀𝑢𝑦,𝑡𝜌−𝜈𝜙𝑢𝑦,𝑡𝑘+𝑔𝛽_𝑇𝑇−𝑇_∞,𝜌𝑐_𝑝𝜕𝑇𝜕𝑡=𝑘₁𝜕²𝑇𝜕𝑦²−𝜕𝑞_𝑟,𝜕𝑦 (2)(3) where𝑇is the temperature of the fluid,𝜈=𝜇/𝜌(𝜇is the viscosity and𝜌is the constant density of the fluid) is the kinematic viscosity,𝜎is the electrical conductivity of the fluid,𝛽=𝛼/𝜌(𝛼is the drag coefficient that is usually assumed to be positive),𝜙is the porosity parameter,𝑘is the permeability parameter,𝛽_𝑇is the volumetric coefficient of thermal expansion,𝑔is the acceleration due to gravity,𝑇_∞is the free stream temperature,𝑐_𝑝is the specific heat of the fluid at a constant pressure,𝑘₁is the thermal conductivity, and𝑞_𝑟is the radiative flux along the𝑦-axis.

The initial and boundary conditions relevant to the present flow situation are 𝑡≤0∶𝑢𝑦,𝑡=0,𝑇𝑦,𝑡=𝑇_∞();𝑦>0,𝑡>0∶𝑢0,𝑡=𝐴𝑡^𝑛,𝜕𝑇(0,𝑡)𝜕𝑦=−ℎ_𝑠𝑢()()𝑇,∞,𝑡=0,𝑇∞,𝑡=𝑇_∞, (4) where𝑛≥0is constant andℎ_𝑠is the heat transfer parameter for Newtonian heating. The radiative heat flux term in (3) is simplified using the Rosseland approximation as 𝑞_𝑟=−4𝜎₁3𝑘₂𝜕𝑇⁴,𝜕𝑦 (5) where𝜎₁is the Stefan-Boltzmann constant and𝑘₂is the mean absorption coefficient. It is assumed that the temperature differences inside the flow are sufficiently small such that𝑇⁴may be expressed as a linear function of the temperature, and then by expanding 𝑇⁴about𝑇_∞using Taylor series and neglecting the higher order terms, we get 𝑇⁴≅4𝑇³_∞𝑇−3𝑇⁴_∞. (6)

Now (3) after using (5) and (6) takes the following form: 𝜌𝑐_𝑝𝜕𝑇𝜕𝑡=𝑘₁1+16𝜎𝑇³_∞3𝑘₁𝑘₂𝜕²𝑇𝜕𝑦². (7)

By introducing the following dimensionless variables 𝑣_𝑛=𝑢(𝜈^𝑛𝐴)^1/(2𝑛+1)𝐴,𝜉=𝑦𝜈^{𝑛+11/(2𝑛+1)},𝐴𝜏=𝑡²𝜈^1/(2𝑛+1),𝛽_𝑛𝜈=𝛽𝐴^21/(2𝑛+1),𝜃=𝑇−𝑇_∞𝑇_∞, (8) into (2)–(4) and (7), we obtain the following dimensionless system: 𝜕𝑣_𝑛(𝜉,𝜏)=𝜕𝜕𝜏²𝑣_𝑛(𝜉,𝜏)𝜕𝜉²−𝛽_𝑛𝑣_𝑣(𝜉,𝜏)−𝐻𝑣_𝑛(𝜉,𝜏)+Gr𝜃(𝜉,𝜏),Pr𝜕𝜃=()𝜕𝜕𝜏1+𝑅²𝜃𝜕𝜉²,𝜏≤0∶𝑣_𝑛(𝜉,𝜏)=0,𝜃(𝜉,𝜏)=0;𝜉>0,𝜏>0∶𝑣_𝑛(0,𝜏)=𝜏^𝑛,𝜕𝜃(0,𝜏)(),𝑣𝜕𝜉=−𝛾1+𝜃_𝑛(∞,𝜏)=0,𝜃(∞,𝜏)=0, (9)(10)(11) where𝛾is the Newtonian heating parameter. Using the Laplace transform technique, (9)–(11) give the following transformed solutions: 𝑣_𝑛=Γ()𝜉,𝑞𝑛+1𝑞^𝑛+1𝑒^{−𝜉𝑞+𝐻}+𝑎₁𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑑+𝑎₂𝑒^{−𝜉𝑞+𝐻}𝑞+𝑎₃𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑎₄𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₅√𝑞𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₁𝑒^{√−𝜉𝑞𝑎}√𝑞−𝑑−𝑎₂𝑒^{√−𝜉𝑞𝑎}𝑞−𝑎₃𝑒^{√−𝜉𝑞𝑎}√𝑞+𝑎₄𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁+𝑎₅√𝑞𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁,𝜃=𝑑𝜉,𝑞𝑞√𝑒𝑞−𝑑^{√−𝜉𝑞𝑎}, (12)(13) where 𝛽_𝑛𝜈=𝛽𝐴^21/(2𝑛+1),𝑀²=𝜎𝐵²₀𝜌𝜈𝐴^21/(2𝑛+1),1𝐾=𝜈𝜙𝑘𝜈𝐴^21/(2𝑛+1),Gr=𝑔𝛽_𝑇𝑇_∞𝐴³𝜈^{𝑛−11/(2𝑛+1)},𝛾=ℎ_𝑠𝜈^𝑛+1𝐴^1/(2𝑛+1),Pr=𝜇𝑐_𝑝𝑘₁,𝑅=16𝜎𝑇³_∞3𝑘₁𝑘₂,𝐻=𝑀²+1𝐾,𝐻₁=𝐻,𝑎−1𝑎=Pr𝛾1+𝑅,𝑑=√𝑎,Gr₁=Gr𝑑𝑎−1,𝑎₁=Gr₁𝑑²𝑑²−𝐻₁,𝑎₂=Gr₁𝑑𝐻₁,𝑎₃=Gr₁𝑑²𝐻₁,𝑎₄=Gr₁𝑑𝐻₁𝑑²−𝐻₁,𝑎₅=Gr₁𝐻₁𝑑²−𝐻₁. (14)

The Laplace inverse of (13) results in 𝜃(𝜉,𝜏)=exp𝑑²𝜏−𝑦√𝑦√𝑎𝑑erf𝑐𝑎2√𝜏−𝑑√𝜏𝑦√−erf𝑐𝑎2√𝜏. (15)

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 (𝑛=0), (12) yields 𝑣₀=1𝜉,𝑞𝑞𝑒^{−𝜉𝑞+𝐻}+𝑎₁𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑑+𝑎₂𝑒^{−𝜉𝑞+𝐻}𝑞+𝑎₃𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑎₄𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₅√𝑞𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₁𝑒^{√−𝜉𝑞𝑎}√𝑞−𝑑−𝑎₂𝑒^{√−𝜉𝑞𝑎}𝑞−𝑎₃𝑒^{√−𝜉𝑞𝑎}√𝑞+𝑎₄𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁+𝑎₅√𝑞𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁. (16)

The solution of (16), after taking the Laplace inverse, is given by 𝑣₀(𝜉,𝜏)=𝑓₁(𝜉,𝜏,𝐻)+𝑎₂𝑓₁(𝜉,𝜏,𝐻)−𝑎₂𝑓₁(𝜉,𝜏,0)−𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉,𝜏,𝐻+𝐻₁+𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉√𝑎,𝜏,𝐻₁−𝑎₁𝑓₂𝜉√𝑎,𝜏,𝑑+2𝑓₆(𝜉,𝑧,𝜏,𝐻)+2𝑑𝑓₇(𝜉,𝑧,𝜏,𝐻)+2𝑓₈(𝜉,𝑧,𝜏,𝐻)−2𝑓₉(𝜉,𝑧,𝜏,𝐻)−2𝐻₁𝑓₁₀(𝜉,𝑧,𝜏,𝐻)−𝑓₁₁(𝜉,𝑧,𝜏,𝐻)+2𝑓₁₂(𝜉,𝑧,𝜏,𝐻)+2𝐻₁𝑓₁₃(𝜉,𝑧,𝜏,𝐻). (17)

4. Flow due to Uniform Acceleration of the Plate

For the uniform acceleration of the plate (𝑛=1), (12) yields 𝑣₁=1𝜉,𝑞𝑞²𝑒^{−𝜉𝑞+𝐻}+𝑎₁𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑑+𝑎₂𝑒^{−𝜉𝑞+𝐻}𝑞+𝑎₃𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑎₄𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₅√𝑞𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₁𝑒^{√−𝜉𝑞𝑎}√𝑞−𝑑−𝑎₂𝑒^{√−𝜉𝑞𝑎}𝑞−𝑎₃𝑒^{√−𝜉𝑞𝑎}√𝑞+𝑎₄𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁+𝑎₅√𝑞𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁. (18)

The solution of (18), after taking the Laplace inverse, is given by 𝑣₁(𝜉,𝜏)=𝑎₂𝑓₁(𝜉,𝜏,𝐻)−𝑎₂𝑓₁𝜉√𝑎,𝜏,0−𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉,𝜏,𝐻+𝐻₁+𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉√𝑎,𝜏,𝐻₁−𝑎₁𝑓₂𝜉√𝑎,𝜏,𝑑+𝑓₃(𝜉,𝜏,𝐻)+2𝑓₆(𝜉,𝑧,𝜏,𝐻)+2𝑑𝑓₇(𝜉,𝑧,𝜏,𝐻)+2𝑓₈(𝜉,𝑧,𝜏,𝐻)−2𝑓₉(𝜉,𝑧,𝜏,𝐻)−2𝐻₁𝑓₁₀(𝜉,𝑧,𝜏,𝐻)−𝑓₁₁(𝜉,𝑧,𝜏,𝐻)+2𝑓₁₂(𝜉,𝑧,𝜏,𝐻)+2𝐻₁𝑓₁₃(𝜉,𝑧,𝜏,𝐻). (19)

5. Flow due to Nonuniform Acceleration of the Plate

For the nonuniform acceleration of the plate (𝑛=2),(12) yields 𝑣₂=1𝜉,𝑞𝑞³𝑒^{−𝜉𝑞+𝐻}+𝑎₁𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑑+𝑎₂𝑒^{−𝜉𝑞+𝐻}𝑞+𝑎₃𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑎₄𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₅√𝑞𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₁𝑒^{√−𝜉𝑞𝑎}√𝑞−𝑑−𝑎₂𝑒^{√−𝜉𝑞𝑎}𝑞−𝑎₃𝑒^{√−𝜉𝑞𝑎}√𝑞+𝑎₄𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁+𝑎₅√𝑞𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁. (20)

The solution of (20), after taking the Laplace inverse, is given by 𝑣₂(𝜉,𝜏)=𝑎₂𝑓₁(𝜉,𝜏,𝐻)−𝑎₂𝑓₁𝜉√𝑎,𝜏,0−𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉,𝜏,𝐻+𝐻₁+𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉√𝑎,𝜏,𝐻₁−𝑎₁𝑓₂𝜉√𝑎,𝜏,𝑑+𝑓₄(𝜉,𝜏,𝐻)+2𝑓₆(𝜉,𝑧,𝜏,𝐻)+2𝑑𝑓₇(𝜉,𝑧,𝜏,𝐻)+2𝑓₈(𝜉,𝑧,𝜏,𝐻)−2𝑓₉(𝜉,𝑧,𝜏,𝐻)−2𝐻₁𝑓₁₀(𝜉,𝑧,𝜏,𝐻)−𝑓₁₁(𝜉,𝑧,𝜏,𝐻)+2𝑓₁₂(𝜉,𝑧,𝜏,𝐻)+2𝐻₁𝑓₁₃(𝜉,𝑧,𝜏,𝐻). (21)

6. Flow due to Highly Nonuniform Acceleration of the Plate

In this case,𝑛=3and (12) yields 𝑣₃=1𝜉,𝑞𝑞⁴𝑒^{−𝜉𝑞+𝐻}+𝑎₁𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑑+𝑎₂𝑒^{−𝜉𝑞+𝐻}𝑞+𝑎₃𝑒^{−𝜉𝑞+𝐻}√𝑞−𝑎₄𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₅√𝑞𝑒^{−𝜉𝑞+𝐻}𝑞−𝐻₁−𝑎₁𝑒^{√−𝜉𝑞𝑎}√𝑞−𝑑−𝑎₂𝑒^{√−𝜉𝑞𝑎}𝑞−𝑎₃𝑒^{√−𝜉𝑞𝑎}√𝑞+𝑎₄𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁+𝑎₅√𝑞𝑒^{√−𝜉𝑞𝑎}𝑞−𝐻₁. (22)

The solution of (22), after taking the Laplace inverse, is given by 𝑣₃(𝜉,𝜏)=𝑎₂𝑓₁(𝜉,𝜏,𝐻)−𝑎₂𝑓₁𝜉√𝑎,𝜏,0−𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉,𝜏,𝐻+𝐻₁+𝑎₄𝑒^𝐻1𝑡𝑓₁𝜉√𝑎,𝜏,𝐻₁−𝑎₁𝑓₂𝜉√𝑎,𝜏,𝑑+𝑓₅(𝜉,𝜏,𝐻)+2𝑓₆(𝜉,𝑧,𝜏,𝐻)+2𝑑𝑓₇(𝜉,𝑧,𝜏,𝐻)+2𝑓₈(𝜉,𝑧,𝜏,𝐻)−2𝑓₉(𝜉,𝑧,𝜏,𝐻)−2𝐻₁𝑓₁₀(𝜉,𝑧,𝜏,𝐻)−𝑓₁₁(𝜉,𝑧,𝜏,𝐻)+2𝑓₁₂+2𝐻₁𝑓₁₃, (23) where 𝑓₁=1𝑦,𝑡,𝑤2𝑒^{√−𝑦𝑤}𝑦erf𝑐2√𝑡−√𝑤𝑡+𝑒^𝑦√𝑤𝑦erf𝑐2√𝑡+√,𝑓𝑤𝑡₂=1𝑦,𝑡,𝑤√−𝑦𝜋𝑡exp²4𝑡+𝑤exp−𝑦𝑤+𝑤²𝑡𝑦erf𝑐2√𝑡−𝑤√𝑡,𝑓₃=1𝑦,𝑡,𝑤2𝑦𝑡−2√𝑤𝑒^{√−𝜉𝑤}𝑦erf𝑐2√𝑡−√+𝑦𝑤𝑡𝑡+2√𝑤𝑒^𝑦√𝑤𝑦×erf𝑐2√𝑡+√,𝑓𝑤𝑡₄=1𝑦,𝑡,𝑤4𝑡²−𝑦𝑡√𝑤+𝑦²+𝑦4𝑤4𝑤√𝑤×𝑒^{√−𝑦𝑤}𝑦erf𝑐2√𝑡−√+𝑡𝑤𝑡₂+𝑦𝑡√𝑤+𝑦²−𝑦4𝑤4𝑤√𝑤×𝑒^𝑦√𝑤𝑦erf𝑐2√𝑡+√,𝑓𝑤𝑡₅=1𝑦,𝑡,𝑤63𝑦4𝑤²−𝑦𝑡𝑤𝑡𝑤𝑦exp²+−𝑦4𝑡−𝑤𝑡₂32𝑤²−𝑦²𝑡+𝑡16𝑤1³2×𝑒^{√−𝑦𝑤}𝑦erf𝑐2√𝑡−√𝑤𝑡+𝑒^𝑦√𝑤𝑦erf𝑐2√𝑡+√×−𝑦𝑤𝑡32𝑤²√𝑤−𝑦²96𝑤√𝑤+𝑦𝑡16𝑤√𝑤−𝑦𝑡²8√𝑤×𝑒^{√−𝑦𝑤}𝑦erf𝑐2√𝑡−√𝑤𝑡+𝑒^𝑦√𝑤𝑦erf𝑐2√𝑡+√,𝑓𝑤𝑡₆()=𝑎𝜉,𝑧,𝜏,𝐻₁𝜋^∞_{√𝜉/2𝜏}exp−𝑧²−𝐻𝜉²/4𝑧²𝜏−𝜉²/4𝑧²𝑓𝑑𝑧,₇()=𝑎𝜉,𝑧,𝜏,𝐻¹√𝜋^∞_{√𝜉/2𝜏}exp−𝑧²𝜉−𝐻²4𝑧²+𝑑²𝜏−𝑑²𝜉²4𝑧²,×erf−𝑑𝜉𝜏−²4𝑧²𝑓𝑑𝑧₈()=𝑎𝜉,𝑧,𝜏,𝐻³𝜋^∞_{√𝜉/2𝜏}exp−𝑧²−𝐻𝜉²/4𝑧²𝜏−𝜉²/4𝑧²𝑓𝑑𝑧,₉()=𝑎𝜉,𝑧,𝜏,𝐻⁵𝜋^∞_{√𝜉/2𝜏}exp−𝑧²−𝐻𝜉²/4𝑧²𝜏−𝜉²/4𝑧²𝑓𝑑𝑧,₁₀()=𝑎𝜉,𝑧,𝜏,𝐻⁵√𝜋^∞_{√𝜉/2𝜏}exp−𝑧²𝜉−𝐻²4𝑧²+𝐻₁𝜏−𝐻₁𝜉²4𝑧²×erf𝐻₁𝜉𝜏−²4𝑧²𝑓𝑑𝑧,₁₁()𝜉𝜉,𝑧,𝜏,𝐻=exp²𝑎,𝑓4𝜏₁₂()=𝑎𝜉,𝑧,𝜏,𝐻⁵√𝑎𝜋^∞_{√𝜉/2𝜏}exp−𝑧²𝑎𝜏−𝜉²/4𝑧²𝑓𝑑𝑧,₁₃()=𝑎𝜉,𝑧,𝜏,𝐻⁵√𝑎√𝜋^∞_{√𝜉/2𝜏}exp−𝑧²𝑎+𝐻₁𝜏−𝐻₁𝜉²4𝑧²×erf𝐻₁𝜉𝜏−²4𝑧²𝑑𝑧. (24)

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 termGr=0together with MHD and porosity effects (𝑀=0and 𝐾→∞;that is,𝐻=𝛽₀)in (17), we get 𝑣₀=𝑓₁𝜉,𝜏,𝛽₀, (25) which is in quite agreement with [40, Equation (2.7)] when𝜈=𝑈=1.Furthermore by putting𝛽₀=0, (25) reduces to 𝑣_0𝑁=𝑓₁(𝜉,𝜏,0), (26) which is the well-known solution of Newtonian fluid [40, Equation (2.8)] 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 termGr=0with𝑀=0and𝐾→∞, (19) reduces to 𝑣₁=𝑓₃𝜉,𝜏,𝛽₁, (27) which is in complete agreement with [41, Equation (15)] for accelerated motion of the plate.

Skin Friction. The expressions for skin frictions𝜏₀and𝜏₁evaluated from (17) and (19), respectively, are given by 𝜏₀=−𝜕𝑣₀(𝜉,𝜏)𝜕𝜉_𝜉=0=−𝑔₁(𝜏,𝐻)−𝑎₂𝑔₁(𝜏,𝐻)+𝑎₂𝑔₁(𝜏,0)+𝑎₄𝑒^𝐻1𝜏𝑔₁𝜏,𝐻+𝐻₁−𝑎₄√𝑎𝑒^𝐻1𝜏𝑔₁𝜏,𝐻₁+𝑎₁√𝑎𝑔₂(),𝜏𝜏,𝑑₁=−𝜕𝑣₁(𝜉,𝜏)𝜕𝜉_𝜉=0=−𝑎₂𝑔₁(𝜏,𝐻)+𝑎₂𝑔₁(𝜏,0)+𝑎₄𝑒^𝐻1𝜏𝑔₁𝜏,𝐻+𝐻₁−𝑎₄√𝑎𝑒^𝐻1𝜏𝑔₁𝜏,𝐻₁+𝑎₁√𝑎𝑔₂(𝜏,𝑑)−𝑔₃(𝜏,𝐻), (28) where 𝑔₁(𝑡,𝑤)=𝜕𝑓₁𝜕𝑦𝑦,𝑡,𝑤_𝑦=0𝑒=−^−𝑤𝑡1+𝑒^𝑤𝑡√𝑤√𝜋𝑡erf√𝑤𝑡√,𝑔𝜋𝑡₂()=𝑡,𝑤𝜕𝑓₂𝜕𝑦𝑦,𝑡,𝑤_𝑦=0=𝑎₁√𝑎𝑑−1−2𝑑𝑒^𝑑2𝑡√𝜋𝑡+𝑑𝑒^𝑑2𝑡√𝜋𝑡erf𝑐𝑑√𝑡√,𝑔𝜋𝑡₃(𝑡,𝑤)=𝜕𝑓₃𝜕𝑦𝑦,𝑡,𝑤_𝑦=0𝑒=−^−𝑤𝑡√𝑡√𝜋−(1+2𝑡𝑤)erf√𝑤𝑡2√𝑤. (29)

Nusselt Number. The rate of heat transfer evaluated from (15) is given by ()Nu=−𝜕𝜃𝜉,𝜏𝜕𝜉_𝜉=01=𝛾𝜃()10,𝜏+1=𝛾𝑒^𝑑2𝜏1+erf𝑑√𝜏.−1+1 (30)

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 velocities𝑣₀(𝑦,𝑡),𝑣₁(𝑦,𝑡),𝑣₂(𝑦,𝑡),and𝑣₃(𝑦,𝑡)given 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 when𝛽_𝑛(𝑛=0--3)→0, 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 velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏)due 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 numberPr, magnetic parameter𝑀, porosity parameter𝐾, Grashof numberGr, radiation parameter𝑅, Newtonian heating parameter𝛾, dimensionless time𝜏, and Brinkman parameter𝛽_𝑛(𝑛=0,1).

Table 1 

Variation in skin friction 𝜏₀ and 𝜏₁.

Table 2 

Variation in Nusselt number Nu.

(a)

(b)

Figure 2 

Velocity profiles for different values ofPrwhere𝑀=2, 𝐾=1, Gr=0.5, 𝑅=1, 𝛾=1, 𝛽0=𝛽1=1, and𝜏=1.

(a)

(b)

Figure 3 

Velocity profiles for different values of𝑀where Pr=0.71, 𝐾=1, Gr=0.5, 𝑅=1, 𝛾=1, 𝛽0=𝛽1=1, and𝜏=1.

(a)

(b)

Figure 4 

Velocity profiles for different values of𝐾where𝑀=2, Pr=0.71, Gr=0.5, 𝑅=1, 𝛾=1, 𝛽0=𝛽1=1, and𝜏=1.

(a)

(b)

Figure 5 

Velocity profiles for different values of Gr where𝑀=2, 𝐾=1, Pr=0.71, 𝑅=1, 𝛾=1, 𝛽0=𝛽1=1, and𝜏=1.

(a)

(b)

Figure 6 

Velocity profiles for different values of 𝑅 where 𝑀=2, 𝐾=1, Gr=0.5, Pr=0.71, 𝛾=1, 𝛽0=𝛽1=1, and𝜏=1.

(a)

(b)

Figure 7 

Velocity profiles for different values of 𝛾 where 𝑀=2, 𝐾=1, Gr=0.5,𝑅=1,Pr=0.71,𝛽0=𝛽1=1, and 𝜏=1.

(a)

(b)

Figure 8 

Velocity profiles for different values of 𝜏 where 𝑀=2, 𝐾=1, Gr=0.5, 𝑅=1,𝛾=1,𝛽0=𝛽1=1, and Pr=0.71.

(a)

(b)

Figure 9 

Velocity profiles for different values of 𝛽0 and 𝛽1 where𝑀=2, 𝐾=1, Gr=0.5,𝑅=1,𝛾=1,Pr=0.71, and𝜏=1.

Figure 10 

Temperature profiles for different values ofPrwhere𝑅=1,𝛾=1, and 𝜏=1.

Figure 11 

Temperature profiles for different values of𝑅wherePr=0.71,𝛾=1, and𝜏=1.

Figure 12 

Temperature profiles for different values of𝛾where𝑅=1,Pr=0.71, and𝜏=1.

Figure 2 is plotted to see the effect ofPron velocity in cases of impulsive motion𝑣₀(𝜉,𝜏)and uniform acceleration𝑣₁(𝜉,𝜏)of the plate. Three different values ofPr=0.71, 1, and7are 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 asPrincreases. Physically, it is true due to the fact that an increase in Prandtl numberPr 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 parameter𝑀 on velocity profiles for impulsive and accelerated motions of the plate. It is observed that velocity in both cases decreases when𝑀 is increased. This physical interpretation of the velocity is an indication that the increasing values of magnetic parameter𝑀make 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 parameter𝐾on the velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏). It appears from this figure that with increasing values of𝐾, velocity increases for both impulsive and uniformly accelerated motions. Physically, it is true because increasing𝐾reduces the drag force and causes the velocity profiles to increase. Thus increasing value of the porosity parameter𝐾yields an effect opposite to that of the magnetic parameter𝑀. Figure 5 shows the effect of Grashof numberGron the velocities for the cases of impulsive and uniformly accelerated motions. It is observed that an increase in Grashof numberGr leads to increasing both types of velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏)due 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 numberGrmeans increase of temperature gradient (𝑇_𝑤−𝑇_∞) 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,Gr=0corresponds to the absence of free convection current andGr>0means heating of the fluid or cooling of the plate by natural convection. For the positive values ofGr, 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 parameter𝑅 on the velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏)are displayed in Figure 6. We found that in both cases the velocity increases as𝑅increases. 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 parameter𝛾on velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏). We observe that velocity increases in both figures as𝛾increases. The influence of dimensionless time𝜏on velocity is shown in Figure 8. It is seen from this figure that velocity increases when𝜏increases. Furthermore, both of these velocities𝑣₀(𝜉,𝜏)and𝑣₁(𝜉,𝜏)satisfy the given boundary conditions (see (11)). This proves the accuracy of our obtained results. The effects of Brinkman parameter𝛽_𝑛(𝑛=0,1)on the flow due to impulsive and uniformly accelerated motions of the plate are studied in Figure 9. We observe that velocity decreases as𝛽increases. 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 numberPron temperature profiles. Three different values ofPr=0.71,1, and7that physically correspond to air, electrolytic solution, and water are chosen. It is observed that increasing values ofPrlead to a fall in the temperature. A similar behavior was also expected due to the fact that at smaller values ofPrfluids possess high thermal conductivity and heat diffuses away from the surface faster than at higher values ofPr. Thus the boundary layer becomes thicker and consequently temperature decreases whenPris increased.

Figure 11 is plotted to show the influence of radiation parameter𝑅on the temperature profiles. It is found that temperature shows an increasing behavior as𝑅increases. Physically, a similar effect was also expected because the radiation parameter𝑅signifies the relative contribution of conduction heat transfer to thermal radiation transfer. The temperature profiles for different values of the Newtonian heating parameter𝛾are plotted in Figure 12. We observe that temperature profiles increase when𝛾increases. It is due to the fact that, as we increase𝛾, 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 ofPr,𝑀, 𝐾, Gr, 𝑅, 𝛾,𝛽₀,𝛽₁, and𝜏on the skin frictions𝜏₀ and𝜏₁corresponding to impulsive and uniformly accelerated motions of the plate, respectively. It is observed that both𝜏₀and𝜏₁increase when the values ofPr,𝑀,𝜏, and𝛽are increased. However, this behavior is quite opposite for radiation parameter𝑅, conjugate parameter𝛾, permeability parameter𝐾,and Grashof numberGr.Table 2 represents the numerical results for the rate of heat transfer in terms of Nusselt number Nu for different values of Prandtl numberPr, radiation parameter𝑅, and conjugate parameter𝛾. It is found that the rate of heat transfer decreases with the increasing values ofPrand increases when 𝑅,𝛾, and𝜏are 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 of𝑀and𝐾on 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 parameter𝛽decreases fluid velocity.(viii)The temperature increases for large values of𝑅and𝛾.(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.