Transient hydromagnetic three-dimensional natural convection from an inclined stretching permeable surface

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Abstract

The problem of transient, laminar, three-dimensional natural convection flow over an inclined permeable surface in the presence of a magnetic field and heat generation or absorption effects is considered. The order of the governing differential equations for this investigation is reduced by using transient non-similar transformations. The resulting initial-value problem is solved numerically by an accurate, implicit, finite-difference line by line marching scheme. A parametric study is performed to illustrate the influence of the Prandtl number, Hartmann number, heat generation or absorption coefficient, and the surface suction or injection parameter on the velocity and temperature fields as well as the transient development of the skin-friction coefficients and the Nusselt number. These results are displayed graphically to show special aspects of this flow and heat transfer situation.

Introduction

Recently, Chamkha [10] has investigated the problem of steady-state, laminar, hydromagnetic, three-dimensional free convection flow over a vertical stretching surface in the presence of heat generation or absorption effects. This was done in view of several possible metallurgical applications and processes such as annealing and tinning of copper wires, drawing of continuous filaments through quiescent fluids, extrusion of films and plates, melt spinning, glass blowing, hot rolling, manufacturing of plastic, rubber, metallic and polymer sheets, crystal growing, continuous coating and fibers spinning (see [19], [33]). As mentioned by Vajravelu and Hadjinicolaou [35], the rate of cooling involved in these processes can greatly affect the properties of the end product. This rate of cooling has been shown to be controlled by the use of electrically-conducting working fluids with applied magnetic fields. The use of magnetic fields has also been applied in the process of purification of molten metals from non-metallic inclusions. Some work concerning hydromagnetic flows and heat transfer of electrically-conducting fluids over a stretching surface can be found in the papers by Chakrabarti and Gupta [8], Chiam [15], Chandran et al. [12], and Vajravelu and Hadjinicolaou [35]. In many practical problems, the stretching of the surface can start impulsively in motion from rest with a constant or variable velocity in a stationary fluid. As a result, the velocity and temperature fields will change with time especially at the start of the motion, thus having unsteady flow situation. The objective of this work is to consider transient hydromagnetic three-dimensional natural convection from an inclined linearly stretching porous surface in the presence of magnetic field, heat generation or absorption, and wall suction or injection effects.

Sakiadis [27], [28] was the first to study boundary-layer flow over a stretched surface moving with a constant velocity. He employed a similarity transformation and obtained a numerical solution for the problem. Later, Erickson et al. [18] extended the work of Sakiadis [27], [28] to account for mass transfer at the stretched sheet surface. The numerical results of Sakiadis [27], [28] were confirmed experimentally by Tsuo et al. [34] for continuously moving surface with a constant velocity. In addition, Chen and Strobel [14] and Jacobi [23] have reported results for uniform motion of the stretched surface. Vajravelu and Hadjinicolaou [35] have considered hydromagnetic convective heat transfer from a stretching surface with uniform free stream and in the presence of internal heat generation or absorption effects. Many investigations have concentrated on the problem of a stretched sheet with a linear velocity and different thermal boundary conditions (see, for instance, [6], [13], [16], [17], [21], [22], [30], [37]).

Very recently, many investigations studying the consequent flow and heat transfer characteristics that are brought about by the movement of a stretched permeable and impermeable, isothermal and non-isothermal surface with a power-law velocity variation have been reported. Banks [4] considered the case of an impermeable wall and obtained a similarity solution. Chiam [15] considered steady flow of an electrically-conducting fluid over a surface stretching with a power-law velocity in the presence of a magnetic field. Ali [1], [2], [3] presented various extensions to Banks’ [4] problem in terms of flow and thermal boundary conditions. All of the preceeding references have dealt with two-dimensional flow situations. During the last decade or so, some work have been reported concerning three-dimensional and unsteady flow situations. Wang [38] considered steady-state three-dimensional flow caused by stretching flat surface. Gorla and Sidawi [20] have reported similarity transformations and numerical solutions for the problem of steady, three-dimensional free convection flow on a stretching surface with suction and blowing. Surma Devi et al. [32] have studied unsteady three-dimensional boundary-layer flow due to a stretching surface. Lakshmisha et al. [24] reported numerical solutions for three-dimensional unsteady flow with heat and mass transfer over a continuous stretching surface. Smith [29] has reported an exact solution of the unsteady Navier–Stokes equations resulting from a stretching surface. More recently, Pop and Na [26] analyzed the problem of unsteady flow past a wall which starts to stretch impulsively from rest. Chamkha [11] extended the work of Pop and Na [26] to include the thermal problem and the effects of the presence of a Darcian porous medium, magnetic field, and heat generation or absorption.

Consider the transient, laminar, three-dimensional natural convective boundary-layer flow of an electrically-conducting and heat-generating/absorbing fluid over a semi-infinite inclined permeable surface stretching in the x-direction with a velocity that is linear with the distance along the surface x. Since the velocity would become very large at large x, the latter must be limited at some finite distance x = xa at which point the velocity becomes constant. The y-direction makes an angle α with the horizontal line while the z-direction is normal to the plate surface. A uniform magnetic field is applied in the y-direction. This gives rise to magnetic effects in both the x and z directions. The application of the magnetic field in the y-direction is done so as to allow suppression of convective flow in these directions. This is important in terms of controlling the quality of the product being stretched (see, [35]). In addition, uniform suction or injection is imposed at the plate surface in the z-direction. The coordinate system and flow model are shown in Fig. 1. All fluid properties are assumed constant except the density in the buoyancy terms of the x- and y-momentum equations. Assuming that the edge effects are negligible, all dependent variables will be independent of the y-direction [20]. Under the usual boundary-layer and Boussinesq approximations, the governing equations for this investigation are∂u∂x+∂w∂z=0∂u∂t+u∂u∂x+w∂u∂z2u∂z2+gβ(T−T)cosασρB02u∂v∂t+w∂v∂z2v∂z2+gβ(T−T)sinα∂w∂t+w∂w∂z=−1ρ∂p∂z2w∂z2σB02ρw∂T∂t+w∂T∂z=νPr2T∂z2+Q0ρcp(T−T)where x, y, and z are the coordinates directions. u, v, w, p, and T are the fluid velocity components in the x, y, and z directions, pressure and temperature, respectively. ρ, ν, cp and Pr are the fluid density, kinematic viscosity, specific heat at constant pressure, and the Prandtl number, respectively. g, β, T and α are the gravitational acceleration, coefficient of thermal expansion, ambient temperature, and the inclination angle, respectively. σ, B0, and Q0 are the fluid electrical conductivity, magnetic induction, and dimensional heat generation/absorption coefficient, respectively. It should be noted that in writing , , , , , the magnetic Reynolds number is assumed small so that the induced magnetic field is neglected. Also, the Hall effect of magnetohydrodynamics, Joule heating, and the viscous dissipation are neglected. In many physical situations such as crystal growing, the heat generation or absorption effects in the fluid are greatly dependent on temperature. Sparrow and Cess [31], Moalem [25], Vajravelu and Nayfeh [36], Chamkha [9], and Vajravelu and Hadjinicolaou [35] have considered temperature-dependent heat generation (source) or absorption (sink). Following these authors, the heat generation or absorption term (last term) of Eq. (5) is assumed to vary linearly with the difference of the fluid temperature in the boundary layer and the ambient temperature at z = .

The appropriate boundary conditions for this problem can be written asu(t,x,0)=bx,v(t,x,0)=0,w(t,x,0)=w0,T(t,x,0)=Twu(t,x,∞)=0,v(t,x,∞)=0,∂w∂z(t,x,0)=0,T(t,x,∞)=T where w0 and Tw are the wall suction or injection velocity and temperature, respectively.

In order to minimize the numerical efforts to solve the governing equations, the following transformations are introduced:τ=t,η=zνt,u=bxφ’(τ,η)+ΓcosαM(τ,η),v=TsinαN(τ,η),w=−b2νtφ,p=ρbvG(τ,η),θ(τ,η)=T−TTw−T,Γ=gβ(Tw−T)bSubstituting Eq. (7) into , , , , , reduces the number of independent variables by one and produces the following non-similar transient equations:φ”’+η2φ”+bτ(φφ”−(φ’)2−Ha2φ’)−τφ’τ=0M”+η2M’+bτ(φM’−φ’M−Ha2M+θ)−τMτ=0N”+η2N’+bτ(φN’+θ)−τ∂N∂τ=0G’+φ”+η2φ’−φ2+bτ(φφ’−Ha2φ)−τ∂φ∂τ=0θ”+Prη2θ’+Prbτ(φθ’+γθ)−Prτ∂θ∂τ=0where a prime denotes partial differentiation with respect to η and Ha2B02/(ρb),γ=Qo/(ρcpb) and φw=−w0/ are the square of the magnetic Hartmann number, the dimensionless heat generation/absorption coefficient, and the wall mass transfer coefficient, respectively. It should be noted that positive values of φw indicate fluid suction at the plate surface while negative values of φw indicate fluid blowing or injection at the wall. In addition, it is seen that the advantage of employing the transformations (7) is that they reduce the number of independent variables by one and that similarity equations are obtained at τ = 0 In this way, the initial profiles or conditions for φ, φ′, M, N, G, and θ are obtained by solving these similar equations subject to the boundary conditions.

The transformed boundary conditions becomeφ(τ,0)=φ’(τ,0)=1,φ’(τ,∞)=0M(τ,0)=0,M(τ,∞)=0,N(τ,0)=0,N(τ,∞)=0G(τ,0)=0,φ(τ,0)=1,θ(τ,∞)=0

Important physical parameters for this flow and heat transfer situation are the skin-friction coefficients in the x and y directions and the local Nusselt number. The shear stresses at the stretching surface in both the x and y directions, respectively, are given byτzx∂u∂z(t,x,0)=μνt(bxφ”(τ,0)+ΓcosαM’(τ,0))τzy∂v∂z(t,x,0)=μνtΓsinαN’(τ,0)where μ (=ρν) is the dynamic viscosity of the fluid. The corresponding skin-friction coefficients in the x and y directions are obtained, respectively, by dividing , by the quantity ρ(bx)2/2 which represents the dynamic pressure to yieldCfx=2xRexνtGrxRex2cosαM’(τ,0)+φ”(τ,0)Cfy=2xRexνtGrxRex2sinαN’(τ,0)where Grx = gβ(Tw  T)x3/ν2 and Rex = bx2/ν are the local Grashof and Reynolds numbers, respectively.

The wall heat transfer is given by Fourier’s law of conduction as followsqw=−k∂T∂z(t,x,0)=−kνt(Tw−T)θ’(τ,0)where k is the thermal conductivity of the fluid. The local Nusselt number for this situation can then be defined asNux=hxk=qwxk(Tw−T)=−xνtθ’(τ,0)where h is the local heat transfer coefficient.

, , , , , represent an initial-value problem in which the initial profiles for φ′, M, N, G, and θ are obtained directly by solving the similarity equations obtained by setting τ = 0 in these equations subject to the boundary conditions. Once the initial profiles are obtained, a forward marching technique in τ can be used to obtain the solutions for all of the dependent variables at different times. The implicit finite-difference method discussed by Blottner [5] which is similar to the Keller’s box method (see [7]) have proven to be successful for the solution of similar and non-similar boundary-layer equations. For this reason, it is adopted for the solution of the present investigation.

Non-uniform grid distributions in both the η and τ directions with small initial step sizes were used to accommodate steep changes in the velocity and temperature gradients in the immediate vicinity of the wall and at the start of the flow. The initial step size employed in the η direction was Δη1=0.001 and the growth factor was K1 = 1.03 such that Δηi+1=K1Δηi while an initial step size Δτ1=0.001 and a growth factor K2 = 1.03 such that Δτj+1=K2Δτj were used in the τ direction. A computational domain consisting of 196 grid points in the η direction and 171 grid points in the τ direction was utilized. This gave η  10 and τ  5 The independence of the results from the grid density was ensured and successfully checked by various trial and error numerical experimentations.

All first-order derivatives in τ are replaced by two-point backward-difference formulae. Then, Eq. (8) was converted into a second-order ordinary differential equation by letting F = φ′. Then, the resulting equation in F along with , , were discretized using three-point central difference quotients while the equation φ  F = 0 and Eq. (12) were discretized by the trapezoidal rule. Linearization of the equations was performed by evaluation of the non-linear terms at the previous iteration. At each line of constant τ, linear tri-diagonal algebraic equations resulted which were solved by the Thomas algorithm (see, [5]). The convergence criterion required that the difference between the current and the previous iterations be 10−5. When this condition was satisfied, the solution was assumed converged and the iteration process was terminated.

Comparisons with the works of Pop and Na [26] and Chamkha [11] for unsteady flow over a stretching sheet were conducted and the results were found to be in excellent agreement. It should be mentioned that these comparisons required slight changes in the coefficients of , to make them similar to those reported by Pop and Na [26] and Chamkha [11]. These favorable comparisons lend some confidence to the accuracy of the numerical method.

Section snippets

Results and discussion

Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7 display the effects of all of the Hartmann number Ha, the suction or injection parameter φw the heat generation or absorption coefficient γ and the fluid’s Prandtl number Pr on the profiles of φ′(τ,η) M(τ,η) N(τ,η), φ(τ,η) G(τ,η), and θ(τ,η) at τ = 1, respectively. In these and all subsequent figures, the parametric conditions associated with each of the curves present are given in Table 1. The parametric study of the physical parameters involved in

Conclusion

The problem of transient, laminar, natural convection boundary-layer flow of an electrically-conducting fluid along an inclined porous surface stretching with a linear velocity in the presence of a magnetic field, heat generation or absorption, and fluid wall suction or injection effects was investigated numerically. A new transformation was introduced in which the governing unsteady three-dimensional equations were transformed into non-similar equations. By using this transformation, the

Nomenclature

    b

    a constant having units of inverse time

    B0

    magnetic induction

    cp

    fluid specific heat at constant pressure

    Cfx

    skin-friction coefficient in the x direction

    Cfy

    skin-friction coefficient in the y direction

    g

    acceleration due to gravity

    G

    dimensionless pressure

    Grx

    local Grashof number

    h

    local heat transfer coefficient

    Ha

    magnetic Hartmann number

    k

    fluid thermal conductivity

    M

    function related to dimensionless x-component of velocity

    N

    dimensionless y-component of velocity

    Nux

    local Nusselt number

    p

    pressure

    Pr

    Prandtl number

    q

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