Heat transfer near the stagnation point of an unsteady three-dimensional ﬂow

In this paper, the unsteady laminar three-dimensional ﬂow of an incompressible viscous ﬂuid in the neighbourhood of a stagnation point is studied. The magnetic ﬁeld is applied normal to the surface and the effects of viscous dissipation and Ohmic heating are taken into account. The unsteadiness in the ﬂow is caused by the external free stream varying arbitrarily with time. The governing equations are solved both analytically and numerically. An approximate analytical solution has been obtained for ﬂow and heat transfer in the form of series solution using Homotopy Analysis Method while the numerical solutions are computed using the Runge–Kutta–Fehlberg Method with a shooting technique. The inﬂuence of various parameters such as the viscous dissipation, unsteadiness, ratio of the velocity gradient and magnetic ﬁeld effects on ﬂow and heat transfer parameters are studied. A detailed error analysis is done to compute the averaged square residual errors for velocity and temperature. The optimal values of the convergence control parameter are computed for the ﬂow which is used for obtaining all the other results presented.

wings. The unsteadiness increases the complexities of the problem because the dimension of the problem is increased. During recent years, the subject of unsteady boundary layer studies near stagnation region has received considerable attention.
The three-dimensional laminar incompressible boundary-layer flow in the neighbourhood of the nodal point region (i.e., 0 ≤ c ≤ 1) was first studied by Howarth [1]. The corresponding flow near the saddle point region (i.e., −1 ≤ c < 0) was considered by Davey [2] where he has observed the appreciable changes in the similarity solution of boundary layer equations compared to the solution near the nodal point region. The heat transfer in the three-dimensional stagnation flow at nodal point region has been analysed by Cheng et al. [3] and a self-similar solution has been obtained.
The unsteady incompressible flow of a viscous fluid near the stagnation region of a three-dimensional body was investigated by Kumari and Nath [4]. The unsteady heat flux at a general three-dimensional body when the velocity is varying as a linear function of time was considered by Teipel [5]. A stagnation point solution near nodal and saddle point regions of an incompressible micro polar fluid was studied by Guram et al. [6]. The effect of the magnetic field on the stagnation point flow and heat transfer of a viscous fluid was studied by Kumari et al. [7]. The unsteady stagnation point flow of a stretching surface in a three-dimensional flow was studied by Rajeswari et al. [8]. Various aspects of this problem were studied by the authors [9,10]. The unsteady laminar MHD flow of a viscous fluid with variation of Ohmic heating and viscous dissipation was studied by Osalusi et al. [11], as well as [12,13].
The importance of studying the unsteady boundary layer flow near stagnation region is that the heat transfer is maximum near the stagnation point. When the free stream velocity varies arbitrarily with time, unsteadiness is caused in the flow field. This unsteadiness influences the flow and heat transfer to a great extent. Most of the studies cited above present the numerical solutions of flow and heat transfer near the stagnation point. Our aim is to obtain an approximate analytical solution for the flow velocity and the temperature in the form of a power series solution. Since the approximate analytical solution for the three-dimensional stagnation point boundary layer flow has not been studied, we obtain the same using Homotopy Analysis Method (HAM). Since this method is described in detail in [14,15], it is not explained here for the sake of brevity.
Generally, HAM solutions are always represented as a power series form in which the coefficients contain the study parameters. In our analysis the various study parameters considered are magnetic field strength (M); unsteadiness in the flow (λ); Prandtl number (Pr); the ratio of the velocity gradient c and the viscous dissipation parameters α and β. We have also computed the numerical solutions for our governing equations using the Runge-Kutta-Fehlberg (RKF) method with shooting technique.

Problem formulation and governing equation
We consider the unsteady incompressible viscous flow of an electrically conducting fluid in the neighbourhood of a stagnation point of a three-dimensional surface. Let us consider the orthogonal set of coordinates (x, y, z) with the origin "O" at the forward stagnation region, with x and y coordinates along the body surface and z is the coordinate perpendicular to the body surface at origin as shown in Fig. 1. The unsteadiness in the flow field is imparted by the free stream varying arbitrarily with time.
Uniform magnetic field of strength B is applied normal to the surface in the positive z direction. Since the magnetic Reynolds number is considered to be small, the induced magnetic field can be neglected. The effects of viscous dissipation and Ohmic heating are taken into account. The Navier-Stokes equations governing the three dimensional flow and heat transfer are given by The initial conditions when t = 0 are The boundary conditions are Here T is the temperature; κ is the thermal conductivity; p is the static pressure; ρ is the density; ν is the kinematic viscosity; C ρ is the specific heat at a constant pressure; p 0 is the stagnation pressure; B is the magnetic field; U , V and W are the components of velocity in the potential flow; the subscripts t, x, y and z denote the derivatives with respect to t, x, y and z, respectively.
The velocity components in x-and y-directions in the potential flow vary directly as a linear function of distance and inversely as a linear function time.
The velocity components in z-direction is obtained from Eq. (1) so that the continuity equation is satisfied exactly.
The magnetic field B for the unsteady flow and the wall temperature T w varies as where T w 0 is the value of T w when t = 0. The expression of pressure p can be obtained from Bernoulli equation as Since the governing equations are a system of partial differential Eqs. (1)(2)(3)(4)(5) with four independent variables (x, y, z, t), we apply a suitable similarity transformation by which the partial differential equations can be converted into a system of nonlinear ordinary differential equations.
The similarity transformations are where M = σ B 2 0 /ρa. Here s, and θ used in temperature represent the dimensionless temperatures without viscous dissipation, with viscous dissipation and with Ohmic heating, respectively.
The boundary conditions are Here f and g are the dimensionless velocity components in x-and y-directions, respectively; λ is the unsteadiness parameter; η is the similarity variable; M is the magnetic parameter; Pr is the Prandtl number; a and b are the velocity gradient along x-and y-directions in the potential flow; c = b/a is ratio of the velocity gradient; α and β are the viscous dissipation parameters; L = (ν/a) 1/2 is the characteristic length and prime denotes derivative with respect to η.
Equations (12-16) with λ = 0 (steady-state case) and without Ohmic heating (i.e., magnetic parameter M = 0) are the same as those of [1] for the flow in the nodal point region (0 ≤ c ≤ 1) and to those of [2] in the saddle point region (−1 ≤ c < 0). Further may be noted that Eqs. (12-16) without the last term in (14) for M = 0 are same to those of [5].
The heat transfer coefficient in terms of Nusselt number and local Skin friction coefficients in x and y directions are expressed as where Re x = U x/ν.

HAM solution
The main features of the HAM procedure are selecting suitable initial profiles satisfying the boundary conditions of the problem concerned; choosing an appropriate linear operator so that its solutions are simpler to evaluate analytically. The initial profiles that satisfy the initial and boundary conditions of the flow are (19) The Linear Operator chosen for this set of equations are in which C i are the arbitrary constants. The nonlinear operator is directly written from the governing equation of the problem. We choose the set of base functions and initial guesses, auxiliary linear operators as follows: Then, we can write where X here denotes our study variables and X = f, g, s, , θ and A k ,n are the coefficients.

Zeroth and higher-order deformation problems
To obtain the HAM solution for the governing Eqs. (12)(13)(14)(15)(16), let τ ∈ [0, 1] be an embedding parameter and c f , c g and c s , c , c θ be the basic convergence control parameters. Then the zeroth order deformation equation and the non-linear operators take the following form: where X = f , g, s, and θ ; X 0 = f 0 , g 0 , s 0 , 0 and θ 0 .
with appropriate boundary conditions from Eq. (18) we have For th-order deformation equations, we first differentiate Eq. (23) -times with respect to τ, dividing them by ! and then set τ = 0. Following this we have with boundary conditions where the remainder term R X (η) are expanded and rewritten from the governing Eqs. (12)(13)(14)(15)(16) as follows: where defines as Using Taylor's series, X (η; τ ) can be expanded in terms of τ as follows: The auxiliary parameters are selected as τ = 0 and τ = 1 from Eq. (23), one may write Thus as τ increases from 0 to 1 and X (η; τ ) varies from the initial guess X 0 (η) to the solution X (η) of the governing equations respectively. The auxiliary parameters are selected so that the series solutions converge for τ = 1 and the particular solution is where X = f , g, s, and θ ; X 0 = f 0 , g 0 , s 0 , 0 and θ 0 . Therefore, we get the general approximate analytical solutions f , g , s , , θ in terms of special solutions f * , g * , s * , * , θ * as s (η) = s * (η) + C 7 e η + C 8 e −η , (η) = * (η) + C 9 e η + C 10 e −η , θ (η) = θ * (η) + C 11 e η + C 12 e −η (44) We solve Eqs. (43, 44) one after the other in the order = 1, 2, 3, . . . by means of the symbolic computation software Mathematica. It is shown that the solution for the velocity profile can be expressed as an infinite series of any desired order. It is customary in HAM analysis to draw c X -curves to identify the interval of optimal convergence control parameter (i.e., c f and c g , c s , c and c θ ) inside which any suitable values can be chosen. In Fig. 2

Convergence control parameters
In order to choose the optimal values of the convergence control parameters c X , where X = f , g, s, and θ , we compute the averaged residual errors suggested by Liao [15].

Results and discussion
We have studied the effect of various physical parameters such as magnetic parameter M, ratio of the velocity gradient c, unsteady parameter λ on velocity and temperature profiles as well as on skin friction coefficients and heat transfer rates. The approximate analytical solutions for the velocity and temperature are obtained in the form of a power series solution using computer Algebra software Mathematica. The expressions for f , g, s, and θ are obtained up to 10th degree polynomial. It is to be mentioned here that, though computations could be done easily up to 16th order of deformations equations, to retain the results up to sixth decimal places accuracy we have stopped at the 10th order. Computations have been carried out for several combinations of parameters λ and M, c, Pr, only some representative results are presented here in the form of Tables 1, 2, 3, 4  and Figs. 2, 3 , 4, 5, 6, 7, 8, 9. For the special case of M = λ = 0 (in the absence of magnetic and unsteady parameter), Table 2 presents the values of skin friction results for steady case ( f (0), g (0)) with that of [9] for nodal and saddle point regions. It was found to be in very good agreement from the 10th order approximation itself. The comparison of heat transfer results for steady case −s (0) is presented in Table 3   has the tendency to slow down the flow near the surface, at the same time increasing fluid temperature. As the magnetic field increases, the velocity profile decreases while the temperature profile increases, but the temperature profiles are thinner for higher values of M. The temperature in the absence of viscous dissipation s(η) gradually reaches the free stream values, but the behaviour of temperature profiles in the presence of viscous dissipation and Ohmic heating are significantly different in the sense that it increases sharply close to the boundary layer, attains a maximum value and starts decreasing rapidly to attain its free stream values. The variation of surface shear stress rates in x-and y-directions (i.e., f (0) and g (0)) and the heat transfer rate (−s (0), − (0) and −θ (0)) with the unsteady parameter λ for different magnetic parameter M and the different viscous dissipation parameters α, β with ratio of velocity gradient c on the heat transfer rates profile for a fixed value of Pr = 0.7 are presented in Figs. 5, 6, 7, 8, 9. It observed that the decrease in unsteady parameter λ decreases the surface shear stress rates. It is to be noted that the surface shear stress rates ( f (0)   )) in x-and y-directions for every M curve becomes zero for a certain value of unsteady parameter λ. For example, as seen from Fig. 5, g (0) vanishes at λ = −0.438, −2.189 for M = 0.5, 4.0, respectively. Also, reverse flow occurs for negative values of λ. This means that the decelerating flow behaves like an adverse pressure gradient. Similar trend has been observed by Kumari et al. [9]. The boundary layer thickness increases with decreasing unsteady parameter, which means that the velocity in z-direction increases which in turn increases the heat transfer rate. Hence, as the unsteady parameter λ decreases the heat transfer rate (−s (0), − (0), −θ (0)) increases as observed from Fig. 6.   (0), g (0)), and the heat transfer rates −s (0) at M = 0.0, Pr = 0.7 with those of [9] The effect of the viscous dissipation parameters α, β on the surface heat transfer rate (− (0), −θ (0)), for various values of M, c are presented in Figs. 7 and 8. From Fig. 7, it is observed that, as the unsteady parameter increases, the surface heat transfer (− (0) and −θ (0)) decreases with respect to increase in the magnetic parameter and viscous dissipation parameters. For a given α, β and M, −θ (0)) values are always higher than − (0) values. This is due to the presence of Ohmic heating. Similar to Fig. 7, in Fig. 8, the effect with respect to stretching ratio c is presented. Here surface heat transfers (− (0) and −θ (0)) are found to be higher for saddle point region (c = −0.5) compared to its value in the nodal point region (c = 0.5).
A comparison result of the shear stress rates (f (0) and g (0)) and the surface heat transfer rate (−s (0)) for two values of c with those of Kumari [9] is shown in Fig. 9. As the unsteadiness in the flow increases the shear stress rates increase where as heat transfer rates decrease.

Conclusions
The flow and heat transfer analysis is presented for a three-dimensional viscous fluid flow in the neighborhood of a stagnation point. An approximate analytical expression in the form of a series solution is derived for velocity and temperature using Homotopy Analysis Method. The coefficients of the series solution contain all the study parameters such as c, M and Pr, λ, α, β. The Computer Algebra software Mathematica is used to perform these semi-analytical calculations. The effect of various parameters such as magnetic parameter M, ratio of the velocity gradient parameter c and unsteady parameter λ are analysed on velocity and temperature as well as on surface shear stress rates and heat transfer rates. The accelerating free stream velocity, i.e., λ > 0 or magnetic parameter M tend to delay or prevent flow reversal. The heat transfer and surface shear stress are found to increase with the increase in the magnetic parameter. Hence it is suggested that by controlling the magnetic field on suitable strength, one can control the heat transfer rates as well as delay the occurrence of flow reversal near stagnation point region.