Analytical Analysis of Effects of Buoyancy, Internal Heat Generation,MagneticField, andThermalRadiationonaBoundary 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. *e 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.*e 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.


Introduction
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 [1] 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). is leads to a condensed set of equations known as the boundary layer equations. e study on the boundary layer flow concerning a motionless plate was first done by Blasius [2].
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. e 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][4][5][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 [7]. 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][9][10][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 [7]. e first and leading work regarding boundary layer behaviour in moving surfaces in a quiescent fluid was executed by Sakiadis [12]. Consequently, many researchers [13][14][15][16][17] worked on the problem of moving or stretching plates under different situations.
e heat transfer analysis of boundary-layer flows with radiation is also vital in electrical power generation, astrophysical flows, solar power technology, space vehicle reentry, and other industrial areas [7]. Raptis et al. [18] investigated the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past a semi-infinite stationary plate. Hayat et al. [19] 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 [20] 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. [23] 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 [24] investigated the buoyancy effects on a thermal boundary layer over a vertical plate with a convective surface boundary condition. ey also investigated the effect of buoyancy force, thermal radiation, and internal heat generation by the numerical method [7]. 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).

Basic Concepts of the Optimal Homotopy
Asymptotic Method e OHAM was introduced and developed by Marinca et al. [25][26][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 condition where A is a differential operator, B is a boundary operator, w(x) is an unknown function, D is the problem domain, and b(x) is a known analytic function. e 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 ψ(x; p): where p ∈ [0, 1] is an embedding parameter, H(p) is a nonzero auxiliary function for p ≠ 0, and H(0) � 0. When e auxiliary function H(p) is chosen in the form e convergence of the solution greatly depends on this function. e function H(p) also adjusts the convergence domain and controls the convergence region. To obtain an approximate solution, Substituting equation (6) into equation (3) and equating the coefficient of like powers of p, the zero-order problem is obtained as given in equation (4). e first-and secondorder problems are, respectively, given by equations (7) and (8): 2 International Journal of Differential Equations Hence, the general governing equations for w j (x) are given by where N m (w 0 (x); w 1 (x); w 2 (x); . . . ; u m (x)) is the coefficient of p m in the expansion of N(ψ(x; p, c 1 , c 2 , c 3 , ...)) with respect to the embedding parameter p as follows: where ψ(x; p, c 1 , c 2 , c 3 , . . .) is given in equation (6). It should be underlined that w j for j ≥ 0 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, c 1 , c 2 , c 3 , c 4 , . . . If the series converges at p � 1, one has Substituting equation (12) in equation (1), we get the following expression for the residual error: IfR(x, c i ) � 0, then w m (x, c i ) is the exact solution of the given equation, but this does not happen, especially in nonlinear problems.
e optimal values of the convergence-control parameters (c i ) 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 c 1 , c 2 , c 3 , c 4 , . . . by the least square method, we first construct functional and then minimize it, where a and b are in the domain of the given problem and R is given by equation (13). e unknown convergence-control parameters c i , i � 1, 2, 3, . . . , m, can be identified from the following condition: With these parameters known, the m th -order approximate solution given by equation (12) is well determined. e 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][29][30].

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 T ∞ which moved over the right surface of the plate with a uniform velocity U ∞ , whereas the left surface of the plate was heated by convection from a hot fluid at temperature T f which provided a heat transfer coefficient h f , is considered. e 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). e continuity, momentum, and energy equations describing the flow are summarized by the following boundary value problem [21,24]: International Journal of Differential Equations (18) with the boundary conditions defined as follows: where u and v are the velocity components along the flow direction (x-direction) and normal to the flow direction (ydirection), υ is the kinematic viscosity, β is the thermal volumetric expansion coefficient, g is the gravitational acceleration, k is the thermal conductivity, c p is the specific heat of the fluid at constant pressure, T is the temperature of the fluid inside the thermal boundary layer, ρ is the density, q r is the radiative heat flux in the y-direction, u ∞ is a constant free-stream velocity, δ is the electrical conductivity of the base fluid, B 0 is the imposed magnetic field, T ∞ is the constant temperature of the ambient fluid, Q is the heat released per unit per mass, and h f 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. e radiative heat flux q r is described by Rosseland approximation as where k ′ and σ * are the mean absorption coefficient and the Stefan-Boltzmann constant, respectively. Suppose the temperature variations within the flow are adequately small so that T 4 can be expressed as a linear function after using Taylor series to expand T 4 about the free stream temperature T ∞ and ignoring higher-order terms. is results in the following approximation: On using equation (22) in equation (21), we have Plugging equation (23) into equation (18), we get the following equation: where α � k/ρc p 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 N R � kk ′ /4σ * T 3 ∞ as the radiation parameter in equation (24), the equation becomes where 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 η and dimensionless stream function f(η) are introduced as follows: Differentiating u and v with respect to x and y, respectively, we get where f ′ � df/dη and u ∞ 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: e nondimensional form of the temperature θ(η) is defined as follows: where T w is a constant temperature of the wall. Substituting equations (27)-(30) into equation (17), we obtain where 4 International Journal of Differential Equations Gr x is the local Grashof number, and M is the magnetic parameter.
From equation (30), we have Substituting equations (27)-(30) and (33) into equation (25), we get with the boundary conditions where Equations (19) and (27) give the transformed boundary conditions for velocity fields as follows: For the momentum and energy equations to have a similarity solution, the parameters Gr x , λ x , M 2 , and Bi x must be constants and not functions of x as in equations (32) and (36).
is condition can be met if the heat transfer coefficient h f is proportional to 1/ � � x √ and the thermal expansion coefficient is proportional to 1/x. We therefore assume where c, t, d, and m are constants.

Solution of the Problem
Applying the OHAM on equation (40), we can construct the following homotopy: where p ∈ [0, 1] and the primes denote differentiation of the function f with respect to η; f, θ , H 1 , and H 2 are considered as follows: International Journal of Differential Equations Using equation (43) in equation (42), simplifying, and equating the coefficients of the same powers of p, we get the following set of zeroth-, first-, second-, and third-order problems.
e zeroth-order problems are subjected to boundary conditions e first-order problems are subjected to boundary conditions e second-order problems are subjected to boundary conditions e third-order problems are subjected to boundary conditions Solving in-sequence equations (44), (46), (48), and (50) with boundary condition equations (45), (47), (49), and (51) and using the OHAM for p � 1, we obtain the following solutions: 6 International Journal of Differential Equations , International Journal of Differential Equations 7 Adding up solution components (52)-(56), the fourterm solution obtained by the OHAM for p � 1 is For the calculations of the unknown convergence-control parameters in equation (56) Table 1 depicts the validity of the method for the considered boundary layer flow. In Table 1, a comparison of the International Journal of Differential Equations 11 values for the Nusselt number (-θ'(0)) and surface temperature (θ(0)) for Pr � 0.72 and Gr � λ � Ra � M � 0 is presented, and it is in an excellent agreement with the corresponding values reported by Olanrewaju et al. [7], Aziz [22], Ishak [20], and Makinde and Oladapo Olanrewaju [24]. Table 2 shows a comparison of the values of the skin-friction   Aziz [22] Ishak [20] Makinde and Oladapo Olanrewaju [24] -θ'(0)   coefficient f ″ (0) and the local Nusselt number (−θ ′ (0)) obtained by the OHAM in this study and the numerical method reported by Olanrewaju et al. [7] for various values of embedded parameters and shows an excellent agreement. As illustrated in Table 2, the skin friction and the rate of heat transfer at the plate surface increased with an increase in the local Grashof number, the internal heat generation parameter, the Biot number, and the radiation absorption parameter. However, an increase in the Prandtl number reduced the skin friction but increased the rate of heat transfer at the plate surface. Figures 1-6 exhibit the effects of various parameters on the nondimensional velocity profiles within the boundary layer. Generally, the fluid velocity is zero at the plate surface and rises slowly away from the plate towards the free stream value satisfying the boundary condition. e velocity profile for different values of the Prandtl number, thermal radiation, and internal heat generation is described in Figures 1, 3, and 4, respectively. It is evident from the figures that the variation of values of the Prandtl number, internal heat generation, and radiation parameter had little effect or immaterial effect on the fluid velocity. Even though the velocity profile shows a bit decreasing trend as the value of the Prandtl number increases, it is insignificant. Similarly, from Figures 3 and 4, we see that the variation of values of the radiation parameter and internal heat generation has no significant effect even if the velocity profile shows a little increasing tendency in both cases. e effect of the Biot number and Grashof number on the fluid velocity is    International Journal of Differential Equations 13 depicted in Figures 2 and 5, respectively. As one can observe from the figures, increase in the Biot number and Grashof number enhances the fluid velocity. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the plate as the Grashof number increases, and it is understood that the rate of heat transfer at the plate surface increases with the increase of the Grashof number. Figure 6 reveals that an increase in the magnetic parameter results in the decrease of the fluid velocity. In reality, the existence of the magnetic field in an electrically conducting fluid introduces a force called Lorentz force which acts contrary to the flow if the magnetic field is applied normal to the fluid flow. is type of resistive force leads to slowing down of the flow field. us, our result and physical reality are consistent. e influences of various embedded parameters on the fluid temperature within the boundary layer are illustrated in Figures 7-10. e figures illustrate the behaviour of the temperature profiles within the boundary layer. Commonly, at the plat surface, the fluid temperature is high and exponentially decreases to zero far away from the plate, satisfying boundary conditions. Figures 7 and 8, respectively, demonstrate the behaviour of dimensionless temperature profiles for different values of internal heat generation λ and the thermal radiation parameter, and it is seen that the temperature profile increases with the increase of some values of both internal heat generation and thermal radiation parameter. erefore, the internal heat generation parameter and radiation parameter enhance       the thermal diffusion within the boundary layer. It is seen from Figure 9 that the temperature profile decrease for increasing values of the Prandtl number at all points which indicates that thermal boundary layer thickness is reduced under the influence of the Prandtl number. Figure 10 illustrates that the Biot number causes the fluid temperature to increase. Generally, from these figures, it is notable that the thermal boundary layer thickness increased with an increase in the Biot number, internal heat generation, and thermal radiation but decreased with increasing values of the Grashof number, magnetic parameter, and Prandtl number. us, the convective surface heat transfer, the internal heat generation parameter, and the radiation parameter improved thermal diffusion, whereas an increase in the Prandtl number and the intensity of buoyancy force slowed down the rate of thermal diffusion within the boundary layer.

Conclusion
In this study, the optimal homotopy asymptotic method was successfully applied to investigate the effects of thermal radiation, internal heat generation, Prandtl number, Grashof number, and Biot number on the laminar boundary layer flow over a vertical plate in the presence of a convective surface boundary condition. e present analysis leads to the following conclusions: (1) With the increase in the Grashof number, the fluid velocity and the values of the skin-friction coefficient increase, but the fluid temperature and the value of the Nusselt number reduce with the increase in the Grashof number. (2) With the increase in the Prandtl number, the fluid temperature, the coefficient of skin friction, and heat transfer rate decrease, but the significant effect is not observed on the fluid velocity. (3) e fluid velocity, temperature profile, and skinfriction coefficient increase with the increase in the Biot number. (4) e fluid temperature, the values of the skin-friction coefficient, and the heat transfer rate increase with the increase of internal heat generation. (5) With the increase of the radiation absorption parameter, the fluid temperature, the coefficient of skin friction, and the heat transfer rate enhance. (6) Both fluid velocity and temperature profile fall with the increase in the magnetic parameter. (7) e results confirm that the optimal homotopy asymptotic method is powerful in solving nonlinear differential problems.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.