MHD Flow with Hall current and Joule Heating Effects over an Exponentially Stretching Sheet

1 Introduction

The analysis of flow, heat and mass transfer along a stretching surface is of considerable importance in view of its applications to polymer technology and metallurgy. These applications include wire and fibre coating, extrusion process, foodstuff processing, polymer processing, crystal growing, glass blowing, chemical processing equipment, continuous casting and design of various heat exchangers. The study of convective transport with the interaction of the magnetic field has gained attention of many scientists due to its comprehensive applications in engineering. Several processes in metallurgical contain the cooling of continuous strips or filaments such as strengthening and thinning of copper wires, which are sometimes stretched during the process. The proportion of cooling of these strips can be controlled by exposing them to magnetic fields. Similarly, the molten metals can be purified from non-metallic inclusions by utilizing a magnetic field. Several researchers analyzed the MHD flow, heat and mass transfer over a stretching surface in Newtonian and non-Newtonian fluids with various effects. Kameswaran et al. [1] analyzed the effect of radiation on MHD Newtonian fluid over an exponentially stretching sheet. Mandal and Mukhopadhyay [2] studied the flow and heat transfer along a surface stretching exponentially and embedded in porous media with variable surface heat flux. Singh and Agarwal [3] considered the influence of non-uniform heat source/sink and variable thermal conductivity on the MHD flow and heat transfer over an exponentially stretching sheet in a porous medium saturated with Maxwell fluid. Khidir and Sibanda [4] investigated the MHD mixed convective flow, heat and mass transfer along exponentially stretching surface in a porous medium with variable viscosity and Soret and Dufour effects. Gorla et al. [5] discussed the influence of non-uniform heat sink or source and viscous dissipation on MHD flow and heat transfer over an exponentially stretching sheet in a dusty nanofluid saturated porous medium in the presence of suction/injection. Hayat et al. [6] analyzed the unsteady three-dimensional flow by an exponential stretching sheet with effects of thermal and velocity slip conditions. Eid [7] studied the heat generation/absorption and chemical reaction effects on the MHD mixed convective flow of a nanofluid over a surface stretching exponentially in a porous medium.

In most of the MHD flows reported in the literature, the Hall current term in the Ohm’s law was ignored as it has no significant influence for smaller values of the applied magnetic field. However, when the strong magnetic field is applied the effect of Hall current is very important. Hence, the study of Hall current effects on magnetohydrodynamic flows has gained much importance due to its broad range of applications such as in Hall accelerators, refrigeration coils, power generators and pumps, electric transformers, etc. Megahed et al. [8] studied the heat and mass transport along a vertical flat plate by considering the effects of buoyancy force and Hall currents. Abo-Eldahab et al. [9] analyzed the impact of Hall current and internal heat generation/absorption on mixed convective heat transfer over an inclined stretching surface. El-Aziz [10] analyzed the Hall current effect on the flow and heat transfer over an unsteady stretching surface. Motsa and Shateyi [11] considered the MHD boundary layer flow over an unsteady stretching surface in the presence of Hall currents. El-Aziz and Tamer [12] investigated the MHD mixed convection flow past a sheet which is exponentially stretching under the influence of Hall current. Pal [13] investigated the Hall currents effect with radiation over an unsteady stretching surface. Zaib and Shafie [14] studied the effect of Hall currents with soret and dufour effects over an unsteady stretching surface. Zaman et al. [15] reported the influence of Hall current and suction on the MHD flow over an exponentially stretching sheet in a second grade fluid.

Although there are some investigations concerning the heat and mass transfer along an exponentially stretching surface, the combined effects of slip and Hall currents has received no attention so far. Hence, the present article deals with the mixed convection slip flow, heat and mass transfer over porous exponentially stretching sheet subjected to suction or injection with the inclusion of Hall currents.

2 Mathematical formulation

Consider a steady electrically conducting flow of incompressible viscous fluid past a stretching sheet with temperature T_w(x) and concentration C_w(x). The positive x-axis is taken along the sheet in the direction of flow, y-axis is perpendicular to the sheet in the outward direction of the flow and z-axis coincide with the leading edge of the sheet. The sheet velocity varies as an exponential function of the distance x from the slit. The temperature and concentration of ambient medium are assumed to be T_∞ and C_∞, respectively. A strong magnetic field of strength B(x) is applied in y-direction and the influence of Hall current is not neglected. Assuming the magnetic Reynolds number as very small leads to neglect of the induced magnetic field. The flow is three dimensional in view of the cross flow in the z-direction induced by the presence of Hall current. The velocity vector is (u, v, w) and the temperature and concentration of the plate are T and C, respectively.

With the above assumptions together with the Boussinesq and the boundary layer approximations, the equations governing the present flow problem in the presence of Joule heating are given by

where ρ, c_p, g, ν, α, D and β_h are the fluid density, specific heat at the constant pressure, acceleration due to gravity, kinematic viscosity of the fluid, thermal diffusivity, mass diffusivity of the medium and Hall parameter, respectively,

The associated conditions on the surface are

Here B(x)=B0ex2L is the magnetic field term and B₀ is the constant magnetic field, U(x)=U0exL is the stretching velocity and U₀ is the reference velocity, V(x)=V0ex2L is the special velocity at the wall and V₀is the initial strength of suction, N(x)=N0e−x2L is the velocity slip factor. If we take N = 0, no slip case can be achieved.

Introducing the following Similarity variables

into Eqs. (1)–(5), we obtain

where the prime denotes deivative with respect to η,Pr=να is the Prandtl number, Sc=νD is the Schmidt number, S=V02LνU0 is the suction/injection parameter according as S > 0 or S < 0 respectively, λ=N0νU02L is the velocity slip parameter, Re=U0Lν is the Reynolds number, Gr=gβTT0L3ν2 is the Grashof number, ζ=GrRe2 is the mixed convection parameter, Ha=2LσB02ρU0 is the magnetic parameter, B=βCC0βTT0 is the buoyancy ratio and Ec=U02cpT0 is the Eckert number.

The corresponding boundary conditions reduce to

Results of practical interest are the wall shear stress in x and z- directions, heat and mass transfer rates, respectively, acting on the stretching surface and are given by

The local skin friction in x-direction Cfx=2τωxρU2, local skin friction in z-direction Cfz=2τωzρU2, the local Nusselt number Nux=xqwk(Tf−T∞) and local Sherwood number Shx=xqmk(Cw−C∞), are given by

where Rex=xU(x)ν is the local Reynolds number.

3 Method of Solution

The successive linearisation method (SLM) ([16-18]) is implemented to linearize the system of differential equations (8)–(11). The Chebyshev pseudo spectral method is used to solve the resulting linearised system of equations.

In SLM, the functions f (η), g(η), θ(η) and ϕ(η) are assumed to be taken as

where f_n(η), g_n(η), θ_n(η) and ϕ_n(η) (n=1,2,3,...) are functions to be determined and f_i(η), g_i(η), θ_i(η) and ϕ_i(η) (i ≥ 1) are approximations and are obtained by solving recursively the linear part of the system of equations that results from substituting Eq. (15) in the governing equations (8) to (11).

We choose the initial approximation f₀(η), g₀(η), θ₀(η) and ϕ₀(η) so that they satisfy boundary conditions (12).

The succeeding solutions for f_n(η), g_n(η), θ_n(η) and ϕ_n(η)(n ≥ 1) are brought by recursively solving the below linearized system of ordinary differential equations

where the coefficients α_k,n-1 (k = 1, 2,3), β_k,n-1, δ_k,n-1 (k = 1, 2, 3, 4), γ_k,n-1 (k = 1, 2,3,4, 5) and ℰ_k,i-1(k = 1,2,3,4) are the coefficients in terms of known functions f_i(η), g_i (η), θ_i (η) and ϕ_i(η), i = 1, 2, 3,..., n-1 and a=Ha1+βh2.

The associated boundary conditions are

Hence, f(η), g(η), θ(η) and ϕ(η) are obtained as

where the order of SLM approximation is M.

Chebyshev spectral collocation method [19] is used to solve the linearized equations (16) to (19).

The problem is solved for [0,L] instead of [0, ∞), where the parameter L is used to recover the conditions at infinity. To apply this method the domain under consideration [0,L] is transformed to [-1,1] by the transformation ξ=2ηL−1,−1≤ξ≤1.

Approximating the functions f_n, g_n, θ_n and ϕ_n and their derivatives in terms of Chebyshev interpolating polynomials at N + 1 Gauss-Lobatto collocation points ξk=cos⁡πkN,k=0,1,2,...,N and substituting in (16)–(20) leads to the matrix equation

where

Here D=2LD and 𝒟 being the Chebyshev spectral differentiation matrix, I represents (N + 1) × (N + 1) identity matrix and 0 represents (N+ l) × (N+ 1) null matrix.

The boundary condition reduce to

Incorporating boundary conditions (24) in (22) and solving the resulting matrix system, we get the solution.

4 Results and Discussions

To authenticate the correctness of the technique used, results for S = 0, λ = 0, ζ = 0 and H_a = 0 are compared with the results of Magyari and Keller [20]. Table 1 provides the comparative analysis between the results obtained for local skin-friction coefficient -f "(0) and f(∞) by using the present method and the results obtained by Magyari and Keller [20] and found to be in good agreement.

In order to analyze the effect of pertinent parameters, the values of other parameters are fixed as Pr = 0.71, Sc = 0.22, ζ =1.0, 𝔹 = 0.5, S = 0.5, λ = 1.0, β_h = 2.0, H_a = 1.0, Ec = 0.5, N = 150 and L = 30 unless otherwise mentioned.

Figures 1a-1d indicates the effect of suction/injection parameter S on both the velocity components f′(η) and g(η), temperature θ(η) and concentration ϕ(η). From these figures it is noticed that all the physical quantities are decreasing with an enhancement in S. It is known that applying the wall suction have the propensity to lessen both the momentum and thermal boundary layer thickness. This reason decreases in both the velocity and temperature components.

Fig. 1

Effect of Son a) tangential velocity b) cross flow velocity c) temperature and d) concentration profiles.

The impact of mixed convection parameter ζ on f′ (η), g(η), θ(η) and ϕ(η) are depicted in Figs. 2a-2d. From the Fig. 2a, it is seen that the tangential velocity increases with an increase in the values of ζ. This is because, the positive values of ζ induces a favorable pressure gradient which, in turn, increases the fluid flow in the boundary layer. The same effect is observed for the cross flow velocity component as shown in Fig. 2b. From the Figs. 2c-2d, it is noticed that both the temperature and concentration are decreasing with increasing the values of ζ. This is due to the fact that positive values of ζ accelerates the fluid and which results in decreasing both the thermal and concentration boundary layers.

Fig. 2

Effect of ζ on a) tangential velocity b) cross flow velocity c) temperature and d) concentration profiles.

Figures 3a-3d exhibit the influence of magnetic parameter H_a on both the velocities, the temperature and concentration. It is seen from Fig. 3a that the tangential velocity is decreasing with an increase in the value of magnetic parameter H_a. Applying the uniform magnetic field normal to flow direction gives rise to Lorentz force. This force has the tendency to slow down the velocity of the fluid in the boundary layer. Hence, the velocity diminishes with enhancement in H_a. From Fig. 3b, it is seen that there is no cross flow velocity in the absence of the magnetic field (H_a = 0) and it increases gradually with increase in H_a . Hence, for large values of H_a, a cross flow is generated due to the Hall effect. This is clearly depicted in the Fig. 3b. From Figs. 3c-3d, it is evident that the temperature and concentration are increasing with increasing values of H_a.

Fig. 3

Effect of H_a on a) tangential velocity b) cross flow velocity c) temperature and d) concentration profiles.

The influence of Hall parameter on both the velocities, temperature and concentration is depicted in Fig. 4a-4d. From Fig. 4a it is observed that the tangential velocity increases with the increasing value of the Hall parameter β_h. Fig. 4c reveals that the temperature decrease with increasing value of β_h. As the Hall parameter β_h increases, the effective conductivity reduces which in turn reduces the magnetic damping force on the tangential velocity. Hence the tangential velocity increases and temperature decreases with an increase in the Hall parameter. Fig. 4b shows that the cross flow velocity increases with an increase in the value of β_h. Further, it is observed that the cross flow velocity first increases gradually with β_h, attaining an extreme value and then drops to zero. From Fig. 4d, it is noticed that the concentration decrease with increasing value of β_h.

Fig. 4

Effect of β_h, on a) tangential velocity b) cross flow velocity c) temperature and d) concentration profiles.

The variation of heat transfer coefficient (-θ^′(0)) against Pr and mass transfer coefficient (-ϕ′ (0)) against Sc for different values of suction/injection parameter S is presented in the Figs. 5a and 5b. It is seen from these figures that both the heat and mass transfer coefficients are increasing with the increase in the suction/injection parameter. Further, it is understood from the figures that heat and mass transfer rates are increasing with Pr and Sc, respectively. The effect of mixed convection parameter ζ on both the heat and mass transfer coefficients is shown in the Figs. 6a and 6b. These figures indicate that, heat and mass transfer rates are rising for increasing values of the ζ. The influence of slip parameter λ on the heat and mass transfer rates is demonstrated in the Figs. 7a and 7b. It is observed from these figures that, both the heat and mass transfer rates are reduced with increase in the values of slip parameter.

Fig. 5

Effect of S on (a) -θ′(0) and (b) -ϕ′(0).

Fig. 6

Effect of ζ on (a) -θ′(0) and (b) -ϕ′(0).

Fig. 7

Effect of λ on (a) -θ′(0) and (b) -ϕ′(0).

The variation of heat and mass transfer rates with magnetic parameter H_a is presented in the Figs. 8a and 8b. These figures describe that the heat transfer rate is reduced with increasing values of H_a . The same trend is observed in the mass transfer rate also. The decrease in the mass transfer rate is low when compared to that heat transfer rate as shown in the Fig. 8b. Figures 9a and 9b depict the variation of heat and mass transfer rates for various values of β_h. It is noticed from the figures that, the heat and mass transfers are slightly increasing with an increase in β_h. Figures 10a and 10b represents the variation of heat and mass transfer rates for different values of Eckert number Ec. It is evident from the figures that the heat transfer rate is decreased and the mass transfer rate slightly increases with an increase in the values of Eckert number Ec.

Fig. 8

Effect of H_a on (a) -θ′(0) and (b) -ϕ′(0).

Fig. 9

Effect of β_h on (a) -θ′(0) and (b) -ϕ′(0).

Fig. 10

Effect of Ec on (a) -θ′(0) and (b) -ϕ^′(0).

The values of local skin-friction coefficients in x-direction (f″(0)) and z′-direction (g′(0)) are presented in Table 2 for various values of S, λ, β_h, H_a, ζ and Ec. Table 2 depicts that the skin-friction in x-direction is decreasing and in z-direction is increasing with increase in the values of suction/injection parameter S. While, f″(0) is increasing and g′(0) reducing with an increase in the values of the slip parameter λ. It is seen from the table that, both the skin-frictions are increasing with Hall parameter β_h. Further, it is observed that there is no transverse velocity and hence no skin-friction in z-direction in the absence of Hall parameter. The skin-friction in x-direction reduces and in z-direction enhances as H_a increase. In the absence of magnetic field, there is no cross flow velocity and in turn, there is no skin-friction in z-direction. Further, It is noticed that both the skin-frictions are increasing as the values of the mixed convection parameter ζ and Eckert number Ec are increasing.

5 Conclusions

In this paper Hall effect on viscous incompressible fluid flow over an exponentially stretching sheet has been studied. Suction/injection, velocity slip and Joule heating effects are taken into account. Similarity transformations are used to reduce the governing boundary layer equations to ordinary differential equations and then linearized using the successive linearization method. The resulting equations are solved using Chebyshev spectal collocation method. The main findings are as follows.

1. As the magnetic parameter increases, the tangential velocity and heat and mass transfer rate decrease but the cross flow velocity, temperature and concentration increase.

2. The tangential velocity, cross flow velocity, heat and mass transfer rates are increasing, but the temperature and concentration are decreasing with an increase in the values of Hall parameter.

3. Both the tangential and cross flow velocities, temperature and concentration decrease and the rate of heat and mass transfer increase with an increase in suction/injection parameter.