Research Article | Open Access
Solomon Bati Kejela, Mitiku Daba Firdi, "Analytical Analysis of Effects of Buoyancy, Internal Heat Generation, Magnetic Field, and Thermal Radiation on a Boundary Layer over a Vertical Plate with a Convective Surface Boundary Condition", International Journal of Differential Equations, vol. 2020, Article ID 8890510, 16 pages, 2020. https://doi.org/10.1155/2020/8890510
Analytical Analysis of Effects of Buoyancy, Internal Heat Generation, Magnetic Field, and Thermal Radiation on a Boundary Layer over a Vertical Plate with a Convective Surface Boundary Condition
In this paper, the effects of magnetic field, thermal radiation, buoyancy force, and internal heat generation on the laminar boundary layer flow about a vertical plate in the presence of a convective surface boundary condition have been investigated. In the analysis, it is assumed that the left surface of the plate is in contact with a hot fluid, whereas a stream of cold fluid flows steadily over the right surface, and the heat source decays exponentially outwards from the surface of the plate. The governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations with the help of similarity transformation which were solved analytically by applying the optimal homotopy asymptotic method. The variations of fluid velocity and surface temperature for different values of the Grashof number, magnetic parameter, Prandtl number, internal heat generation parameter, Biot number, and radiation absorption parameter are tabulated, graphed, and interpreted in physical terms. A comparison with previously published results on similar special cases of the problem shows an excellent agreement.
In physics and fluid mechanics, a Blasius boundary layer defines the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is apprehended parallel to a constant unidirectional flow. By means of scaling arguments, Anderson  argued that about half of the terms in the Navier–Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate). This leads to a condensed set of equations known as the boundary layer equations. The study on the boundary layer flow concerning a motionless plate was first done by Blasius .
Blasius used the similarity transformation method in the governing equation to diminish the Navier–Stoke equation for the viscous incompressible steady laminar flow over a solid boundary from the PDE to the ODE. Blasius obtained a laminar boundary layer equation (Blasius equation) which is a third-order nonlinear ODE. The notion of the similarity solution formulated by Blasius for the boundary layer flow of a Newtonian fluid over a flat surface forms the foundation for numerous consequent studies. Later, it has been extended by various researchers [3–6] to explore the similar solutions for thermal boundary layer flows over a flat plate under altered flow configurations and boundary conditions.
In the boundary layer theory, similarity solutions were found to be valuable in the interpretation of certain fluid motions at big Reynolds numbers. Similarity solutions frequently exist for the flow over semi-infinite plates and stagnation point flow for two-dimensional, irregular, and three-dimensional bodies . In exceptional cases, when there is no similarity solution, one has to solve a system of nonlinear partial differential equations. For similarity boundary-layer flows, velocity profiles are alike, but this kind of similarity is missing for nonsimilarity flows [8–11].
Boundary-layer flows over a moving or stretching plate are of great importance in view of their relevance to an extensive variety of technical applications, especially in the manufacture of fibers in glass and polymer industries . The first and leading work regarding boundary layer behaviour in moving surfaces in a quiescent fluid was executed by Sakiadis . Consequently, many researchers [13–17] worked on the problem of moving or stretching plates under different situations.
The heat transfer analysis of boundary-layer flows with radiation is also vital in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, and other industrial areas . Raptis et al.  investigated the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past a semi-infinite stationary plate. Hayat et al.  prolonged the analysis of Raptis et al. for a second-grade fluid.
Convective heat transfer studies are very substantial in processes involving high temperatures, such as gas turbines, thermal energy storage, and nuclear plants. Ishak  examined the similarity solutions for flow and heat transfer over a permeable surface with a convective boundary condition. Aziz [21, 22] investigated a similarity solution for the laminar thermal boundary layer over a flat plate with a convective surface boundary condition and also studied hydrodynamic and thermal slip flow boundary layers over a flat plate with a constant heat flux boundary condition. Garg et al.  studied a similarity solution for the laminar thermal boundary layer over a flat plate with internal heat generation and a convective surface boundary condition.
Moreover, Makinde and Oladapo Olanrewaju  investigated the buoyancy effects on a thermal boundary layer over a vertical plate with a convective surface boundary condition. They also investigated the effect of buoyancy force, thermal radiation, and internal heat generation by the numerical method . However, to the best of our knowledge, no investigation has been made yet to analyze the effects of buoyancy, internal heat generation, thermal radiation, and magnetic parameter on a boundary layer flow over a vertical plate with a convective surface boundary condition by the analytical or semianalytical method.
Hence, the objective of this study was to investigate the combined effects of buoyancy force, internal heat generation, thermal radiation, and magnetic field on a boundary layer over a vertical plate with a convective surface boundary condition via optimal homotopy asymptotic method (OHAM).
2. Basic Concepts of the Optimal Homotopy Asymptotic Method
The OHAM was introduced and developed by Marinca et al. [25–27]. OHAM is a modification of the homotopy asymptotic method which is based on reducing the residual error. In this method, the control and adjustment of the convergence region are provided in a suitable way. To illustrate the basic concepts of the optimal homotopy asymptotic method, the following nonlinear differential equation is considered:with boundary conditionwhere A is a differential operator, B is a boundary operator, is an unknown function, D is the problem domain, and b(x) is a known analytic function. The operator A can be decomposed as A = L + N, where L is a linear operator and N is a nonlinear operator.
According to the OHAM, one can construct an optimal homotopy which satisfieswhere is an embedding parameter, H(p) is a nonzero auxiliary function for , and H(0) = 0. When and , we get , respectively. Thus, as p increases from 0 to 1, the solution varies from to, where is obtained from the homotopy (3). For , we have
The auxiliary function H(p) is chosen in the form
The convergence of the solution greatly depends on this function. The function H(p) also adjusts the convergence domain and controls the convergence region. To obtain an approximate solution, is expanded in a series about as
Substituting equation (6) into equation (3) and equating the coefficient of like powers of , the zero-order problem is obtained as given in equation (4). The first- and second-order problems are, respectively, given by equations (7) and (8):
Hence, the general governing equations for are given bywhere is the coefficient of in the expansion of with respect to the embedding parameter p as follows:where is given in equation (6). It should be underlined that are governed by the linear equations with linear boundary conditions that come from the original problem, which can be smoothly solved. Moreover, the convergence of the series given in equation (6) depends upon the convergence-control parameters,
If the series converges at , one has
The general solution of equation (1) can be determined approximately in the form
If, then is the exact solution of the given equation, but this does not happen, especially in nonlinear problems.
The optimal values of the convergence-control parameters can be identified via various methods, such as the least square method, the Galerkin method, the collocation method, the Ritz method, and the Kantorovich method. In order to find the optimal values of by the least square method, we first construct functionaland then minimize it, where a and b are in the domain of the given problem and R is given by equation (13). The unknown convergence-control parameters , can be identified from the following condition:
With these parameters known, the mth-order approximate solution given by equation (12) is well determined. The least square method is a powerful technique and has been used in many other methods such as the optimal auxiliary function method (OAFM), optimal homotopy perturbation method (OHPM), and optimal homotopy asymptotic method (OHAM) with Daftardar–Jeffery polynomials (OHAM-DJ) for calculating the optimum values of arbitrary constants [28–30].
3. Mathematical Formulation
A two-dimensional steady incompressible fluid flow coupled with heat transfer by convection over a vertical plate in a stream of cold fluid at temperature which moved over the right surface of the plate with a uniform velocity , whereas the left surface of the plate was heated by convection from a hot fluid at temperature which provided a heat transfer coefficient , is considered. The density difference as a result of buoyancy force influences and the effect of the magnetic field were taken into account in the momentum equation; the internal heat generation and the thermal radiation effects were taken into description in the energy equation (the Boussinesq approximation). The continuity, momentum, and energy equations describing the flow are summarized by the following boundary value problem [21, 24]:with the boundary conditions defined as follows:where u and are the velocity components along the flow direction (x-direction) and normal to the flow direction (y-direction), is the kinematic viscosity, is the thermal volumetric expansion coefficient, is the gravitational acceleration, k is the thermal conductivity, is the specific heat of the fluid at constant pressure, T is the temperature of the fluid inside the thermal boundary layer, is the density, is the radiative heat flux in the y-direction, is a constant free-stream velocity, is the electrical conductivity of the base fluid, is the imposed magnetic field, is the constant temperature of the ambient fluid, Q is the heat released per unit per mass, and is the heat transfer coefficient.
It is assumed that the viscous dissipation is neglected, the physical properties of the fluid are constant, and the Boussinesq and boundary layer approximation may be adopted for steady laminar flow. The radiative heat flux is described by Rosseland approximation aswhere and are the mean absorption coefficient and the Stefan–Boltzmann constant, respectively. Suppose the temperature variations within the flow are adequately small so that can be expressed as a linear function after using Taylor series to expand about the free stream temperature and ignoring higher-order terms. This results in the following approximation:
Plugging equation (23) into equation (18), we get the following equation:where is the thermal diffusivity; from this equation, it is clearly seen that the effect of radiation is to enhance the thermal diffusivity. If we take as the radiation parameter in equation (24), the equation becomeswhere . It is clear that the classical solution for the energy equation without the thermal radiation influence can be obtained from equation (25), which reduces to the following equation:
A similarity variable and dimensionless stream function are introduced as follows:
Differentiating with respect to , respectively, we getwhere and is a constant free-stream velocity. Substituting equation (28) into equation (15), the equation of continuity is satisfied identically. From equation (27), we can also obtain:
The nondimensional form of the temperature is defined as follows:where is a constant temperature of the wall.
is the local Grashof number, and M is the magnetic parameter.
From equation (30), we have
For the momentum and energy equations to have a similarity solution, the parameters , , M2, and must be constants and not functions of x as in equations (32) and (36). This condition can be met if the heat transfer coefficient is proportional to and the thermal expansion coefficient is proportional to . We therefore assumewhere c, t, d, and m are constants.
Putting equation (38) into equations (32) and (36), we getwith Gr, , and M being defined by equation (39); the solutions of equation (40) with boundary condition equation (41) produce similarity solutions. Equations (20) and (27) suggested that .
4. Solution of the Problem
Applying the OHAM on equation (40), we can construct the following homotopy:where and the primes denote differentiation of the function f with respect to ; are considered as follows:
The zeroth-order problems aresubjected to boundary conditions
The first-order problems aresubjected to boundary conditions
The second-order problems aresubjected to boundary conditions
The third-order problems aresubjected to boundary conditions