Sage Journals: Discover world-class research

Abstract

MHD free convection and mass transfer flow over a continuously moving vertical porous plate under the action of strong magnetic field is investigated. In this analysis the hall and ion-slip current in the momentum equation are considered for high speed fluid flows and the level of concentration of foreign mass have been taken very high. The governing equations of the problem contain a system of partial differential equations. The coupled partial differential equations are solved numerically by explicit finite difference method. The results of this investigation are discussed for the different values of the well-known parameters with different time steps. Finally, the obtained finite difference solutions are compared with analytical solution by the graphical representation.

1. Introduction

The use of magnetic field that influences mass generation process in electrically conducting fluid flows has important engineering applications. Hall currents effects are likely to be important in many astrophysical and geophysical situations as well as in engineering problems such as Hall accelerators, constructions of turbines, and centrifugal machines. The effects of Hall and ion-slip currents in steady MHD free convective flow past a semi-infinite vertical porous plate in a rotating frame of reference when the heat flux maintained as constant at the plate was studied by Ram. Takhar and Ram [1] investigated the effects of hall current on hydromagnetic free convective flow through a porous medium. Later, they dealt with MHD free convection from an impulsively moving infinite vertical plate in a rotating fluid with Hall and ion-slip currents. Pop and Watanable studied the effects of Hall current on MHD free convection flow past a semi-infinite vertical flat plate. The effects of Hall and ion-slip currents on free convective heat generating flow in a rotating fluid were analysed by Ram. Abo-Eldahab and El Aziz [2] studied the hall and ion-slip effects on MHD free convective heat generating flow past a semi-infinite vertical flat plate. Seddeek and Abdelmeguid [3] studied hall and ion-slip effects on magnetomicropolar fluid with combined forced and free convection in boundary layer flow over a horizontal plate. Abo-Eldahab and El Aziz [4] investigated viscous dissipation and Joule heating effect on MHD free convection from a vertical plate with power law variation in surface temperature in the presence of Hall and ion slip currents. In the present study, the transient flow and mass transfer of a viscous incompressible electrically conducting fluid within a vertical semi-infinite porous plate is studied with the consideration of both the Hall current and ion slip currents. Finally, nondimensional coupled similar and nonsimilar equations are solved by explicit finite difference method. Numerical results have presented for the range of Prandtl number, Schmidt Number, and other well-known parameters which are taken arbitrarily for the fluid.

2. Mathematical Formulation

The plate as well as the fluid is considered at the same temperature and the concentration level everywhere in the fluid is the same.

Also it is assumed that the fluid and the plate are at rest; after that the plate is moving with a constant velocity U in its own plane and instantaneously at time $t > 0$ , the temperature of the plate and the species concentration are raised to $T_{w} (> T_{\infty})$ and $C_{w} (> C_{\infty})$ , respectively, as illustrated in Figure 1, where T_w, C_w are the temperature and species concentration at the wall and T_∞, C_∞ are the temperature and concentration of the species far away from the plate, respectively. In addition the analysis is based on the following assumptions.

Figure 1:

Geometrical configuration of thermal boundary layer.

(I) All the physical properties of the fluid are considered to be constant excepttheinfluenceofvariationsof density with temperature which considered only on the body force term, in accordance with the Boussinesq's approximation.

(II) Since the plate is of semi infinite extent and the fluid motion is unsteady so all the flow variables will depend only upon y and time $(t)$ .

(III) The hall and ion slip terms in the momentum equation are considered for observing the pattern of velocity variation.

(IV) The equation of conservation of electric charge, $\nabla \cdot J = 0$ gives $J_{y} = constant$ , where the current density $J = (J_{x}, J_{y}, J_{z})$ , because the direction of propagation is considered only along the y-axis and $J$ does not have any variation along the y-axis. The constant is assumed to be zero, since J_y = 0 everywhere in the flow. The generalized Ohm's law (G. W. Sutton and A. Sharman) within the presence of electric field

\begin{matrix} J_{x} = \frac{σ_{e}}{α_{e}^{2} + β_{e}^{2}} [α_{e} (E_{x} - w B) + β_{e} (E_{z} + u B)], \\ J_{z} = \frac{σ_{e}}{α_{e}^{2} + β_{e}^{2}} [α_{e} (E_{z} + u B) - β_{e} (E_{x} - w B)], \end{matrix}

(1)

where $α_{e} = 1 + β_{i} β_{e}$ , β_e is the hall parameter, β_i is the ion-slip parameter, and σ_e is the electric conductivity. When $E_{x} = E_{z} = 0$ , then (1) becomes

\begin{matrix} J_{x} = \frac{B σ_{e}}{α_{e}^{2} + β_{e}^{2}} (β_{e} u - α_{e} w), \\ J_{z} = \frac{B σ_{e}}{α_{e}^{2} + β_{e}^{2}} (α_{e} u + β_{e} w) . \end{matrix}

(2)

For the strong magnetic field Lorentz force becomes

\begin{matrix} J \land B = [\frac{- B^{2} σ_{e}}{α_{e}^{2} + β_{e}^{2}} (α_{e} u + β_{e} w)] i + [\frac{B^{2} σ_{e}}{α_{e}^{2} + β_{e}^{2}} (β_{e} u - α_{e} w)] k . \end{matrix}

(3)

Within the framework of the above-stated assumptions the generalized equations relevant to the one-dimensional unsteady free convective mass transfer problem are governed by the following system of coupled partial differential equations.

Momentum Equation in x-Direction

Consider that

\begin{matrix} \frac{\partial u}{\partial t} - v_{0} \frac{\partial u}{\partial y} = g β (T - T_{\infty}) + g β^{*} (C - C_{\infty}) + υ \frac{\partial^{2} u}{\partial y^{2}} + 2 Ω w - \frac{υ}{K} u - \frac{B^{2} σ_{e} (α_{e} u + β_{e} w)}{ρ (α_{e}^{2} + β_{e}^{2})} . \end{matrix}

(4)

Momentum Equation in z-Direction

Consider that

\begin{matrix} \frac{\partial w}{\partial t} - v_{0} \frac{\partial w}{\partial y} = υ \frac{\partial^{2} w}{\partial y^{2}} - 2 Ω u - \frac{υ}{K} w + \frac{B^{2} σ_{e} (β_{e} u - α_{e} w)}{(α_{e}^{2} + β_{e}^{2}) ρ} . \end{matrix}

(5)

Energy Equation

Consider that

\begin{matrix} \frac{\partial T}{\partial t} - v_{0} \frac{\partial T}{\partial y} = \frac{σ}{ρ c_{p}} \frac{\partial^{2} T}{\partial y^{2}} + Q (T_{\infty} - T) . \end{matrix}

(6)

Concentration Equation

Consider that

\begin{matrix} \frac{\partial C}{\partial t} - v_{0} \frac{\partial C}{\partial y} = D_{m} \frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{m} k_{T}}{T_{m}} \frac{\partial^{2} T}{\partial y^{2}} \end{matrix}

(7)

with the corresponding initial and boundary conditions are

\begin{matrix} t = 0, u = 0, w = 0, T = T_{\infty}, \\ C = C_{\infty} \forall values of y \end{matrix}

(8)

\begin{matrix} u (y, t) = U, w (y, t) = 0, T (y, t) = T_{w}, \\ C (y, t) = C_{w} at y = 0, t > 0 \\ u (y, t) = 0, w (y, t) = 0, T (y, t) \to T_{\infty}, \\ C (y, t) \to C_{\infty} as y \to \infty, t > 0 . \end{matrix}

(9)

Since the solutions of the governing (4) to (7) under the initial conditions (8) and boundary conditions (9) will be based on the finite difference method it is required to make the equations dimensionless. For this purpose we now introduce the following dimensionless quantities:

\begin{matrix} Y = \frac{y U}{υ}, \bar{U} = \frac{u}{U}, \bar{W} = \frac{w}{U}, τ = \frac{t U^{2}}{υ}, \\ \bar{T} = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, \bar{C} = \frac{C - C_{\infty}}{C_{w} - C_{\infty}} . \end{matrix}

(10)

Applying all those nondimensional variables into the equations (4) to (7), the basic equations relevant to the problem are in dimensionless form

\begin{matrix} \frac{\partial \bar{U}}{\partial τ} - λ \frac{\partial \bar{U}}{\partial Y} = G_{r} \bar{T} + G_{m} \bar{C} + \frac{\partial^{2} \bar{U}}{\partial Y^{2}} + 2 R' \bar{W} - γ \bar{U} - \frac{M}{(α_{e}^{2} + β_{e}^{2})} (α_{e} \bar{U} + β_{e} \bar{W}), \\ \frac{\partial \bar{W}}{\partial τ} - λ \frac{\partial \bar{W}}{\partial Y} = \frac{\partial^{2} \bar{W}}{\partial Y^{2}} - 2 R' \bar{U} - γ \bar{W} + \frac{M}{(α_{e}^{2} + β_{e}^{2})} (β_{e} \bar{U} - α_{e} \bar{W}), \\ \frac{\partial \bar{T}}{\partial τ} - λ \frac{\partial \bar{T}}{\partial Y} = \frac{1}{P_{r}} \frac{\partial^{2} \bar{T}}{\partial Y^{2}} - \frac{α \bar{T}}{P_{r}}, \\ \frac{\partial \bar{C}}{\partial τ} - λ \frac{\partial \bar{C}}{\partial Y} = \frac{1}{S_{c}} \frac{\partial^{2} \bar{C}}{\partial Y^{2}} + S_{o} \frac{\partial^{2} \bar{T}}{\partial Y^{2}} \end{matrix}

(11)

with the corresponding initial and boundary conditions are

\begin{matrix} τ = 0, \bar{U} = 0, \bar{W} = 0, \bar{T} = 0, \\ \bar{C} = 0 \forall values of Y \end{matrix}

(12)

\begin{matrix} \bar{U} = 1, \bar{W} = 0, \bar{T} = 1, \bar{C} = 1 at Y = 0, τ > 0 \\ \bar{U} = 0, \bar{W} = 0, \bar{T} \to 0, \bar{C} \to 0 as Y \to \infty, τ > 0 . \end{matrix}

(13)

The nondimensional quantities introduced in (11) are defined as

\begin{matrix} λ = \frac{v_{o}}{U}, γ = \frac{υ^{2}}{K U^{2}}, α = \frac{Q υ^{2} ρ c_{p}}{U^{2} σ}, \\ R^{'} = \frac{Ω υ}{U}, M = \frac{σ B^{2} υ}{ρ U^{2}}, P_{r} = \frac{ρ c_{p} υ}{σ}, \\ G_{r} = \frac{g β (T_{w} - T_{\infty}) υ}{U^{3}}, G_{m} = \frac{g β^{*} (C_{w} - C_{\infty}) υ}{U^{3}}, \\ S_{c} = \frac{υ}{D_{m}}, S_{o} = \frac{D_{m} κ_{T} (T_{w} - T_{\infty})}{υ T_{m} (C_{w} - C_{\infty})} . \end{matrix}

(14)

3. Shear Stress, Nusselt Number, and Sherwood Number

From the velocity field, the effects of various parameters on the plate shear stress have been calculated. The following equations represent the shear stress at the plate.

Shear stress $τ_{x} = μ {(\partial \bar{U} / \partial Y)}_{Y = 0}$ , and $τ_{z} = μ {(\partial \bar{W} / \partial Y)}_{Y = 0}$ which are proportional to ${(\partial \bar{U} / \partial Y)}_{Y = 0}$ and ${(\partial \bar{W} / \partial Y)}_{Y = 0}$ , respectively.

From the temperature field, the effects of various parameters on the heat transfer coefficients have been analysed. The following equations represent the heat transfer rate that is well-known Nusselt number.

Nusselt number $N_{u} = - μ {(\partial \bar{T} / \partial Y)}_{Y = 0}$ which is proportional to $- {(\partial \bar{T} / \partial Y)}_{Y = 0}$ .

And from the concentration field, we analyze the effects of various parameters on the mass transfer coefficients. The following equation represents the mass transfer rate that is well-known Sherwood number.

Sherwood number $S_{h} = - μ {(\partial \bar{C} / \partial Y)}_{Y = 0}$ which is proportional to $- {(\partial \bar{C} / \partial Y)}_{Y = 0}$ .

4. Numerical Solutions

In this section, we attempt to solve the governing second-order coupled dimensionless partial differential equations with the associated initial and boundary conditions. For simplicity the explicit finite difference method has been used to solve (11) subject to the initial and boundary conditions (12) and (13). The present problem requires a set of finite difference equation. In this case the region within the boundary layer is divided by some perpendicular lines of $Y -axis$ , where $Y -axis$ is normal to the medium as shown in Figure 2. It is assumed that the maximum length of boundary layer is $Y_{\max} (= 20)$ as corresponds to $Y \to \infty$ . That is, Y varies from 0 to 20 and the numberofgridspacing in Y directionsis $m (= 100)$ , hence the constant mesh size along $Y axis$ becomes $Δ Y = 0.20 (0 \leq Y \leq 20)$ with a smaller time step $Δ t = 0.005$ . Let ${\bar{U}}^{n}$ , ${\bar{W}}^{n}$ , ${\bar{T}}^{n}$ , and ${\bar{C}}^{n}$ denote the values of $\bar{U}$ , $\bar{W}$ , $\bar{T}$ , and $\bar{C}$ at the end of a time step, respectively. Using the explicit finite difference approximation we have

\begin{matrix} {(\frac{\partial \bar{U}}{\partial τ})}_{i} = \frac{{\bar{U}}_{i}^{n + 1} - {\bar{U}}_{i}^{n}}{Δ τ}, {(\frac{\partial \bar{U}}{\partial Y})}_{i} = \frac{{\bar{U}}_{i + 1}^{n} - {\bar{U}}_{i}^{n}}{Δ Y}, \\ {(\frac{\partial^{2} \bar{U}}{\partial Y^{2}})}_{i} = \frac{{\bar{U}}_{i + 1}^{n} - 2 {\bar{U}}_{i}^{n} + {\bar{U}}_{i - 1}^{n}}{{(Δ Y)}^{2}}, \\ {(\frac{\partial \bar{W}}{\partial τ})}_{i} = \frac{{\bar{W}}_{i}^{n + 1} - {\bar{W}}_{i}^{n}}{Δ τ}, {(\frac{\partial \bar{W}}{\partial Y})}_{i} = \frac{{\bar{W}}_{i + 1}^{n} - {\bar{W}}_{i}^{n}}{Δ Y}, \\ {(\frac{\partial^{2} \bar{W}}{\partial Y^{2}})}_{i} = \frac{{\bar{W}}_{i + 1}^{n} - 2 {\bar{W}}_{i}^{n} + {\bar{W}}_{i - 1}^{n}}{{(Δ Y)}^{2}}, \\ {(\frac{\partial \bar{T}}{\partial τ})}_{i} = \frac{{\bar{T}}_{i}^{n + 1} - {\bar{T}}_{i}^{n}}{Δ τ}, {(\frac{\partial \bar{T}}{\partial Y})}_{i} = \frac{{\bar{T}}_{i + 1}^{n} - {\bar{T}}_{i}^{n}}{Δ Y}, \\ {(\frac{\partial^{2} \bar{T}}{\partial Y^{2}})}_{i} = \frac{{\bar{T}}_{i + 1}^{n} - 2 {\bar{T}}_{i}^{n} + {\bar{T}}_{i - 1}^{n}}{{(Δ Y)}^{2}}, \\ {(\frac{\partial \bar{C}}{\partial τ})}_{i} = \frac{{\bar{C}}_{i}^{n + 1} - {\bar{C}}_{i}^{n}}{Δ τ}, {(\frac{\partial \bar{C}}{\partial Y})}_{i} = \frac{{\bar{C}}_{i + 1}^{n} - {\bar{C}}_{i}^{n}}{Δ Y}, \\ {(\frac{\partial^{2} \bar{C}}{\partial Y^{2}})}_{i} = \frac{{\bar{C}}_{i + 1}^{n} - 2 {\bar{C}}_{i}^{n} + {\bar{C}}_{i - 1}^{n}}{{(Δ Y)}^{2}} . \end{matrix}

(15)

From the system of partial differential equation (11) with substituting the above relations into the corresponding differential equation we obtain an appropriate set of finite difference equations,

\begin{matrix} \frac{{\bar{U}}_{i}^{n + 1} - {\bar{U}}_{i}^{n}}{Δ τ} - λ \frac{{\bar{U}}_{i + 1}^{n} - {\bar{U}}_{i}^{n}}{Δ Y} = \frac{{\bar{U}}_{i + 1}^{n} - 2 {\bar{U}}_{i}^{n} + {\bar{U}}_{i - 1}^{n}}{{(Δ Y)}^{2}} + G_{r} {\bar{T}}_{i}^{n} + G_{m} {\bar{C}}_{i}^{n} + 2 \bar{R} {\bar{W}}_{i}^{n} - γ {\bar{U}}_{i}^{n} - \frac{M}{(α_{e}^{2} + β_{e}^{2})} (α_{e} {\bar{U}}_{i}^{n} + β_{e} {\bar{W}}_{i}^{n}) \end{matrix}

(16)

\begin{matrix} \frac{{\bar{W}}_{i}^{n + 1} - {\bar{W}}_{i}^{n}}{Δ τ} - λ \frac{{\bar{W}}_{i + 1}^{n} - {\bar{W}}_{i}^{n}}{Δ Y} = \frac{{\bar{W}}_{i + 1}^{n} - 2 {\bar{W}}_{i}^{n} + {\bar{W}}_{i - 1}^{n}}{{(Δ Y)}^{2}} - 2 \bar{R} {\bar{U}}_{i}^{n} - γ {\bar{W}}_{i}^{n} + \frac{M}{(α_{e}^{2} + β_{e}^{2})} (β_{e} {\bar{U}}_{i}^{n} - α_{e} {\bar{W}}_{i}^{n}), \end{matrix}

(17)

\begin{matrix} \frac{{\bar{T}}_{i}^{n + 1} - {\bar{T}}_{i}^{n}}{Δ τ} - λ \frac{{\bar{T}}_{i + 1}^{n} - {\bar{T}}_{i}^{n}}{Δ Y} = \frac{1}{P_{r}} \frac{{\bar{T}}_{i + 1}^{n} - 2 {\bar{T}}_{i}^{n} + {\bar{T}}_{i - 1}^{n}}{{(Δ Y)}^{2}} - \frac{α}{P_{r}} {\bar{T}}_{i}^{n}, \end{matrix}

(18)

\begin{matrix} \frac{{\bar{C}}_{i}^{n + 1} - {\bar{C}}_{i}^{n}}{Δ τ} - λ \frac{{\bar{C}}_{i + 1}^{n} - {\bar{C}}_{i}^{n}}{Δ Y} = \frac{1}{S_{c}} \frac{{\bar{C}}_{i + 1}^{n} - 2 {\bar{C}}_{i}^{n} + {\bar{C}}_{i - 1}^{n}}{{(Δ Y)}^{2}} + S_{o} \frac{{\bar{T}}_{i + 1}^{n} - 2 {\bar{T}}_{i}^{n} + {\bar{T}}_{i - 1}^{n}}{{(Δ Y)}^{2}} \end{matrix}

(19)

and the initial and boundary conditions with the finite difference scheme are

\begin{matrix} τ = 0, {\bar{U}}_{i}^{0} = 0, {\bar{W}}_{i}^{0} = 0, \\ {\bar{T}}_{i}^{0} = 0, {\bar{C}}_{i}^{0} = 0, \\ {\bar{U}}_{0}^{n} = 1, {\bar{W}}_{0}^{n} = 0, {\bar{T}}_{0}^{n} = 1, {\bar{C}}_{0}^{n} = 1, τ > 0, \end{matrix}

(20)

\begin{matrix} (21) & {\bar{U}}_{L}^{n} = 0, {\bar{W}}_{L}^{n} = 0, {\bar{T}}_{L}^{n} = 0, \\ {\bar{C}}_{L}^{n} = 0, where L \to \infty, τ > 0 . \end{matrix}

(21)

Here the subscripts i designate the grid points with Y coordinates and the superscript n represents a value of time, $τ = n Δ τ$ where $n = 0, 1, 2, 3, 4, \dots$ . From the initial condition (20), the values of ${\bar{U}}^{n}$ , ${\bar{W}}^{n}$ , ${\bar{T}}^{n}$ , and ${\bar{C}}^{n}$ are known at τ = 0. At the end of any time-step $Δ τ$ , the new temperature ${\bar{T}}^{n + 1}$ , the new concentration ${\bar{C}}^{n + 1}$ , the new primary velocity ${\bar{U}}^{n + 1}$ , and the new secondary velocity ${\bar{W}}^{n + 1}$ , at all interior nodal points, may be obtained by successive applications of (18), (19), (16), and (17), respectively. This process is repeated in time and provided the time step is sufficiently small, ${\bar{U}}^{n}$ , ${\bar{W}}^{n}$ , ${\bar{T}}^{n}$ , and ${\bar{C}}^{n}$ should eventually converge to values which approximate the solution of (11). Also the numerical values of the local Shear Stress, Nusselt number, and Sherwood number are evaluated by five-point approximate formula for the derivatives. These converged solutions are shown graphically in Figure 3 to Figure 38.

Figure 2:

Finite difference space grid.

Figure 3:

Primary velocity profile for different value of hall parameterβ_ewith τ = 5, τ = 10, andτ = 50.

5. Stability and Convergence Analysis

Since an explicit procedure is being used, the analysis will remain incomplete unless we discuss the stability and convergence of the finite difference scheme. For the constant mesh sizes the stability criteria of the scheme may be established as follows.

The general terms of the Fourier expansion for ${\bar{U}}^{n}$ , ${\bar{W}}^{n}$ , ${\bar{T}}^{n}$ and ${\bar{C}}^{n}$ at a time arbitrarily called t = 0 are all $e^{i α Y}$ , apart from a constant, where $i = \sqrt{- 1}$ . At a time $t = τ$ , these terms become

\begin{matrix} \bar{U} : ψ (τ) e^{i α Y}, \\ \bar{W} : ξ (τ) e^{i α Y}, \\ \bar{T} : θ (τ) e^{i α Y}, \\ \bar{C} : φ (τ) e^{i α Y}, \end{matrix}

(22)

and after the time step these terms will become

\begin{matrix} \bar{U} : ψ' (τ) e^{i α Y}, \\ \bar{W} : ξ' (τ) e^{i α Y}, \\ \bar{T} : θ' (τ) e^{i α Y}, \\ \bar{C} : φ' (τ) e^{i α Y} . \end{matrix}

(23)

Substituting (22) and (23) into (22) to (25), we obtain the following equations upon simplification:

\begin{matrix} \frac{ψ' (τ) - ψ (τ)}{Δ τ} - λ \frac{ψ (τ) (e^{i α Δ Y} - 1)}{Δ Y} = \frac{2 ψ (τ) (\cos α Δ Y - 1)}{{(Δ Y)}^{2}} - \frac{M}{(α_{e}^{2} + β_{e}^{2})} [α_{e} ψ (τ) + β_{e} ξ (τ)] + G_{r} θ (τ) + G_{m} φ (τ) + 2 \bar{R} ξ (τ) - γ ψ (τ), \end{matrix}

(24)

\begin{matrix} \frac{ξ' (τ) - ξ (τ)}{Δ τ} - λ \frac{ξ (τ) (e^{i α Δ Y} - 1)}{Δ Y} = \frac{2 ξ (τ) (\cos α Δ Y - 1)}{{(Δ Y)}^{2}} + \frac{M}{(α_{e}^{2} + β_{e}^{2})} [β_{e} ψ (τ) - α_{e} ξ (τ)] - 2 \bar{R} ξ (τ) - γ ψ (τ), \end{matrix}

(25)

\begin{matrix} \frac{θ' (τ) - θ (τ)}{Δ τ} - λ \frac{θ (τ) (e^{i α Δ Y} - 1)}{Δ Y} = \frac{1}{P_{r}} \frac{2 θ (τ) (\cos α Δ Y - 1)}{{(Δ Y)}^{2}} - \frac{α}{P_{r}} θ (τ), \end{matrix}

(26)

\begin{matrix} \frac{φ' (τ) - φ (τ)}{Δ τ} - λ \frac{φ (τ) (e^{i α Δ Y} - 1)}{Δ Y} = \frac{1}{S_{c}} \frac{2 φ (τ) (\cos α Δ Y - 1)}{{(Δ Y)}^{2}} + S_{o} \frac{2 θ (τ) (\cos α Δ Y - 1)}{{(Δ Y)}^{2}} . \end{matrix}

(27)

Equations (24), (25), (26), and (27) can be written in the following form:

\begin{matrix} ψ^{'} = D ψ + E ξ + F θ + G φ, \\ ξ^{'} = H ψ + I ξ, \\ θ^{'} = J θ, \\ φ^{'} = K θ + L φ, \end{matrix}

(28)

where

\begin{matrix} D = 1 - γ Δ τ - \frac{2 Δ τ}{{(Δ Y)}^{2}} (1 - \cos α Δ Y) - λ \frac{Δ τ}{Δ Y} (1 - e^{i α Δ Y}) - \frac{M α_{e} Δ τ}{(α_{e}^{2} + β_{e}^{2})}, \\ E = [2 \bar{R} - \frac{M β_{e}}{(α_{e}^{2} + β_{e}^{2})}] Δ τ, \\ F = G_{r} Δ τ, \\ G = G_{m} Δ τ, \\ H = [\frac{M β_{e}}{(α_{e}^{2} + β_{e}^{2})} - 2 \bar{R}] Δ τ, \\ I = 1 - γ Δ τ - \frac{2 Δ τ}{{(Δ Y)}^{2}} (1 - \cos α Δ Y) - λ \frac{Δ τ}{Δ Y} (1 - e^{i α Δ Y}) - \frac{M α_{e} Δ τ}{(α_{e}^{2} + β_{e}^{2})}, \\ J = 1 - \frac{α}{P_{r}} Δ τ - \frac{1}{P_{r}} \frac{2 Δ τ}{{(Δ Y)}^{2}} (1 - \cos α Δ Y) - λ \frac{Δ τ}{Δ Y} (1 - e^{i α Δ Y}), \\ K = S_{o} \frac{2 Δ τ}{{(Δ Y)}^{2}} (\cos α Δ Y - 1), \\ L = 1 - \frac{1}{S_{c}} \frac{2 Δ τ}{{(Δ Y)}^{2}} (1 - \cos α Δ Y) - λ \frac{Δ τ}{Δ Y} (1 - e^{i α Δ Y}) . \end{matrix}

(29)

Equations (28) may be expressed in matrix form as follows:

\begin{matrix} [\begin{bmatrix} ψ' \\ ξ' \\ θ' \\ φ' \end{bmatrix}] = [\begin{bmatrix} D & E & F & G \\ H & I & 0 & 0 \\ 0 & 0 & J & 0 \\ 0 & 0 & K & L \end{bmatrix}] [\begin{bmatrix} ψ \\ ξ \\ θ \\ φ \end{bmatrix}], \end{matrix}

(30)

that is, $η' = T η$ , where

\begin{matrix} η^{'} = [\begin{bmatrix} ψ^{'} \\ ξ^{'} \\ θ^{'} \\ φ^{'} \end{bmatrix}], T = [\begin{bmatrix} D & E & F & G \\ H & I & 0 & 0 \\ 0 & 0 & J & 0 \\ 0 & 0 & K & L \end{bmatrix}], \\ η = [\begin{bmatrix} ψ \\ ξ \\ θ \\ φ \end{bmatrix}] . \end{matrix}

(31)

From the above matrix T we obtain the following eigenvalues:

\begin{matrix} λ_{1} = J, λ_{2} = L, \\ λ_{3} = \frac{D + I + \sqrt{{(D - I)}^{2} + E H}}{2}, \\ λ_{4} = \frac{D + I - \sqrt{{(D - I)}^{2} + E H}}{2} . \end{matrix}

(32)

For stability, each eigenvalue $λ_{1}$ , $λ_{2}$ , $λ_{3}$ , and $λ_{4}$ must not exceed unity in modulus. Hence, the stability condition is

\begin{matrix} | J | \leq 1, | L | \leq 1, | \frac{D + I + \sqrt{{(D - I)}^{2} + E H}}{2} | \leq 1, \\ | \frac{D + I - \sqrt{{(D - I)}^{2} + E H}}{2} | \leq 1, \forall α . \end{matrix}

(33)

We have

\begin{matrix} J = 1 - \frac{α}{P_{r}} Δ τ - \frac{1}{P_{r}} \frac{2 Δ τ}{{(Δ Y)}^{2}} (1 - \cos α Δ Y) - λ \frac{Δ τ}{Δ Y} (1 - e^{i α Δ Y}), \end{matrix}

(34)

or,

\begin{matrix} J = (1 - \frac{α}{P_{r}} a - λ b - \frac{2 c}{P_{r}}) + λ \frac{Δ τ}{Δ Y} e^{i α Δ \bar{Y}} + \frac{1}{P_{r}} \frac{2 Δ τ}{{(Δ Y)}^{2}} \cos α Δ Y, \end{matrix}

(35)

where $a = Δ τ$ , $b = Δ τ / Δ Y$ , and $c = 2 Δ τ / {(Δ Y)}^{2}$ .

The coefficients a, b, and c are all real and nonnegative. We can demonstrate that the maximum modulus of Joccurs when $α Δ Y = m π$ , where m is integer and hence J is real. The value of $| J |$ is greater when m is odd integers, in which case

\begin{matrix} J = (1 - \frac{α a}{P_{r}} - λ b - \frac{2 c}{P_{r}}) - λ b - \frac{2 c}{P_{r}} \end{matrix}

(36)

\begin{matrix} J = 1 - [\frac{α a}{P_{r}} + 2 (λ b + \frac{2 c}{P_{r}})] . \end{matrix}

(37)

To satisfy the first condition $| J | \leq 1$ , the most negative allowable value is $J = - 1$ . Therefore the first stability condition is

\begin{matrix} \frac{α a}{P_{r}} + 2 (λ b + \frac{2 c}{P_{r}}) \leq 2, \end{matrix}

(38)

or,

\begin{matrix} \frac{α}{P_{r}} Δ τ + 2 (λ \frac{Δ τ}{Δ Y} + \frac{2}{P_{r}} \frac{Δ τ}{{(Δ Y)}^{2}}) \leq 2, \end{matrix}

(39)

Same calculation applied for the second condition $| L | \leq 1$ as

\begin{matrix} 1 - 2 (λ \frac{Δ τ}{Δ Y} + \frac{2}{S_{c}} \frac{Δ τ}{{(Δ Y)}^{2}}) \leq 1, \end{matrix}

(40)

as well as

\begin{matrix} 2 (λ \frac{Δ τ}{Δ Y} + \frac{2}{S_{c}} \frac{Δ τ}{{(Δ Y)}^{2}}) \leq 2 . \end{matrix}

(41)

Hence, the stability conditions of the problem are as furnished below

\begin{matrix} \frac{α}{P_{r}} Δ τ + 2 (λ \frac{Δ τ}{Δ Y} + \frac{2}{P_{r}} \frac{Δ τ}{{(Δ Y)}^{2}}) \leq 2, \\ λ \frac{Δ τ}{Δ Y} + \frac{2}{S_{c}} \frac{Δ τ}{{(Δ Y)}^{2}} \leq 1 . \end{matrix}

(42)

Form the above equations (42) the convergence limit for the model of flow are $P_{r} \geq 0.26$ and $S_{c} \geq 0.26$ .

6. Results and Discussion

In case of observing the physical significance of the model, the numerical values of primary velocity $(\bar{U})$ , secondary velocity $(\bar{W})$ , temperature $(\bar{T})$ , and concentration $(\bar{C})$ within the boundary layer for different values of hall parameter $(β_{e})$ , ion-slip parameter $(β_{i})$ , magnetic parameter $(M)$ , rotational parameter $(R')$ , Prandtl number $(P_{r})$ , suction parameter $(λ)$ , Schmidt number $(S_{c})$ , Grashof number $(G_{r})$ , modified Grashof number $(G_{m})$ , Soret number $(S_{o})$ , and heat source parameter $(α)$ have been analyzed with the help of a computer programming language Compaq Visual Fortran 6.6 a. The finite difference explicit method has been used as the solution tool. To obtain the steady-state solution of the computation have been carried out up to τ = 5 to τ = 50. We observe that the numerical values of primary velocity $\bar{U}$ , secondary velocity $\bar{W}$ , temperature $\bar{T}$ , and concentration $\bar{C}$ , however show little change after τ = 20; the solutions for τ = 50 is steady-state solution.

The transient primary and secondary velocity profiles have been shown in Figure 3 to Figure 18 for different values of β_e, β_i, M, S_o, R′,P_r, λ, S_c in case of cooling plate. The effect of hall parameter β_e as well as for the different values of τ is represented for the primary velocity distribution in Figure 3. In this figure we observe that the primary velocity decreases with the decrease of hall parameter β_e but as compare to τ the primary velocity increases with the increase of τ. In Figure 4, for τ = 5, τ = 10, and τ = 50, the secondary velocity increases for the decrease of hall parameter β_e. From Figure 5, we analyze that the primary velocity has a minor effect with the decrease of ion-slip parameter β_i. But the secondary velocity increases gradually with the decrease of ion-slip parameter β_i in Figure 6. The effect of the magnetic parameter on the velocity profile is represented in Figure 7. It is observed that $\bar{U}$ decreases near the plate in case of strong magnetic field but far away from the plate the decrease of magnetic parameter M leads to an increase in velocity. In Figure 8, the secondary velocity has decreased with the decrease of magnetic parameter M. From Figures 9 and 10 the primary velocity and the secondary velocity both increase with the decreases of rotational parameter R′. In Figures 11 and 12, Prandtl number $P_{r}$ has minor effect on primary and secondary velocity profile. The primary velocity increases gradually for the increases of τ, which is illustrated in Figure 11. On the other hand Figure 12 shows the opposite flow pattern for secondary velocity profile. Figures 13 and 14, for the primary and the secondary velocity, respectively, increases and decreases with the decrease of suction parameter λ. Figures 15 and 16, respectively, represent the primary and the secondary velocity for different values of Schmidt number S_cas 0.97 for methanol at $2 5^{°}$ C temperature and 1 atmospheric pressure, 0.78 corresponds to ammonia and 0.3 corresponds to helium. In that case, the primary velocity increases with the decrease of Schmidt number. In Figures 4, 6, and 14 the secondary velocity shows the opposite characteristic as compared to the primary velocity profile. Figures 17 and 18 illustrate the primary velocity profile and secondary velocity profile, respectively, for different values of Soret number S_o, where primary velocity decreases with the decrease of Soret number S_o and the secondary velocity profile increases with the decrease of Soret number S_o for all the values of τ.

Figure 4:

Secondary velocity profile for different value of hall parameter β_ewith τ = 5, τ = 10, and τ = 50.

Figure 5:

Primary velocity profile for different value of ion-slip parameter β_iwith τ = 5, τ = 10, and τ = 50.

Figure 6:

Secondary velocity profile for different value of ion-slip parameterβ_iwithτ = 5, τ = 10, and τ = 50.

Figure 7:

Primary velocity profile for different value of magnetic parameterMwithτ = 5, τ = 10, and τ = 50.

Figure 8:

Secondary velocity profile for different value of magnetic parameterMwithτ = 5, τ = 10, and τ = 50.

Figure 9:

Primary velocity profile for different value of rotational parameterR′with τ = 5, τ = 10, and τ = 50.

Figure 10:

Secondary velocity profile for different value of rotational parameterR′ with τ = 5, τ = 10, and τ = 50.

Figure 11:

Primary velocity profile for different value of Prandtl parameter P_r with τ = 5, τ = 10, and τ = 50.

Figure 12:

Secondary velocity profile for different value of Prandtl parameter P_r with τ = 5, τ = 10, and τ = 50.

Figure 13:

Primary velocity profile for different value of suction parameter λwith τ = 5, τ = 10, and τ = 50.

Figure 14:

Secondary velocity profile for different value of suction parameter λ with τ = 5, τ = 10, and τ = 50.

Figure 15:

Primary velocity profile for different value of Schmidt number S_c with τ = 5, τ = 10, and τ = 50.

Figure 16:

Secondary velocity profile for different value of Schmidt number S_c with τ = 5, τ = 10, and τ = 50.

Figure 17:

Primary velocity profile for different value of Soret number S_o with τ = 5, τ = 10, and τ = 50.

Figure 18:

Secondary velocity profile for different value of Soret number S_o with τ = 5, τ = 10, and τ = 50.

Now we analyze shear stress $(τ_{x})$ and $(τ_{z})$ for x direction and z direction, respectively, Nusselt number $(N_{u})$ and Sherwood number $(S_{h})$ for different values of hall parameter $(β_{e})$ , ion-slip parameter $(β_{i})$ , magnetic parameter $(M)$ , Rotational parameter $(R')$ , Prandtl number $(P_{r})$ , Suction parameter $(λ)$ , Soret number $(S_{o})$ , and heat Source parameter $(α)$ . Figure 19 illustrates the shear stress τ_x for different values of hall parameter β_e, where we obtain that shear stress decreases with the decrease of hall parameter β_e. Figure 20 illustrates the shear stress τ_z exactly opposite characteristic as compared to τ_x. In Figures 21 and 22; shear stress has same effect for ion slip parameter β_i as we have observed for hall parameter β_e in Figures 19 and 20. In Figures 23 and 24, shear stress τ_x increases with the decrease of magnetic parameter M. But shear stress τ_z gradually decreases with the decrease of magnetic parameter M. In Figures 25 and 26, a decreasing and increasing effects for shear stress τ_x and shear stress τ_z have shown the decrease of Soret number S_o, respectively. Shear stress τ_x has increasing effect on decreases of heat source parameter α and Prandtl number P_r as for the opposite case shear stress τ_z has the decreasing effect with the decrease of heat source parameter α and Prandtl number P_r, which is illustrated from Figures 27, 28, 29, and 30, which is true in the real case because the increase of P_r is due to an increase in the viscosity of the fluid. A minor effect is observed for the suction parameter λ in Figures 31 and 32 where τ_x and τ_z increase and decreases, respectively, with the decrease of suction parameter λ. For the Figures 33 and 34 Schmidt number S_c taken in particular, 0.30 corresponds to helium, 0.78 corresponds to ammonia, 1.0 for carbon dioxide, and 0.97 corresponds to methanol at $2 5^{°}$ C temperature and 1 atmospheric pressure which represent the specific condition of the flow, where τ_x and τ_z have exactly the opposite effect for the corresponding Schmidt number S_c.

Figure 19:

Shear stress τ_x for different values of hall parameter β_e.

Figure 20:

Shear stress τ_z for different values of hall parameter β_e.

Figure 21:

Shears tress τ_x for different values of ion-slip parameterβ_i.

Figure 22:

Shear stress τ_z for different values of ion-slip parameterβ_i.

Figure 23:

Shear stress τ_x for different values of magnetic parameter M.

Figure 24:

Shear stress τ_z for different values of magnetic parameterM.

Figure 25:

Shear stress τ_x for different values of Soret numberS_o.

Figure 26:

Shear stress τ_z for different values of Soret numberS_o.

Figure 27:

Shear stress τ_x for different values of heat source parameterα.

Figure 28:

Shear stress τ_z for different values of heat source parameterα.

Figure 29:

Shear stress τ_x for different values of Prandtl numberP_r.

Figure 30:

Shear stress τ_z for different values of Prandtl numberP_r.

Figure 31:

Shear stressτ_xfor different values of suction parameter λ.

Figure 32:

Shear stress $τ_{z}$ for different values of suction parameter λ.

Figure 33:

Shear stressτ_xfor different values of Schmidt numberS_c.

Figure 34:

Shear stressτ_zfor different values of Schmidt numberS_c.

In Figures 35 and 36, Nusselt number N_u has shown for the effect of suction parameter λ and Prandtl number P_r respectively. For both the case Nusselt number N_u decreases with the decrease of suction parameter λ and prandtl number P_r. The profiles for Sherwood number S_h for the different values of prandtl number P_r and Schmidt number S_c are represented in Figures 37 and 38, where Sherwood number S_h has increasing effect on decrease of prandtl number P_r and Schmidt number S_c.

Figure 35:

Nusselt number N_u for different values ofsuction parameterλ.

Figure 36:

Nusselt number N_u for different values of Prandtl number P_r.

Figure 37:

Sherwood number S_h for different values of Prandtl numberP_r.

Figure 38:

Sherwood number S_hfor different values of Schmidt number S_c.

7. General Discussions

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions. We can solve differential equations on both analytical and numerical way (Figures 39, 40, 41, 42, 43, and 44). In some of the cases all the problems are impossible to solve in analytical way. But numerical process is more acceptable in that case. In this chapter we discussed about the comparison between analytical and finite difference solution with graphical representation within the perspective of our problem.

Figure 39:

(a) Analytical solution of primary velocity profile for different value of hall parameter β_e. (b) Numerical solution of primary velocity profile for different value of hall parameter β_e.

Figure 40:

(a) Analytical solution of secondary velocity profile for different value of hall parameter β_e. (b) Numerical solution of secondary velocity profile for different value of hall parameter β_e.

Figure 41:

(a) Analytical solution of primary velocity profile for different value of magnetic parameter M. (b) Numerical solution of primary velocity profile for different value of magnetic parameter M.

Figure 42:

(a) Analytical solution of secondary velocity profile for different value of magnetic parameter M. (b) Numerical solution of secondary velocity profile for different value of magnetic parameter M.

Figure 43:

(a) Analytical solution of temperature profile for different value of heat source parameter α. (b) Numerical solution of temperature profile for different value of heat source parameter α.

Figure 44:

(a) Analytical solution of concentration profilefor different value of Schmidt number S_c. (b) Numerical solution of concentration profile for different value of Schmidt number S_c.

8. Comparison

To obtain a considerable comparison the same parametric values with the steady-state condition for both analytical and numerical solution are taken. In Comparison Figure 29(a) and Comparison Figure 29(b) primary velocity profile for different values of hall parameter β_e shown by applying analytic and finite difference method, respectively, for both cooling and heating of the porous plate, where we observe that the primary velocity is decreased with the decrease of hall parameter β_e for cooling of the plate in both of the two Comparison Figures 29(a) and 29(b); for heating of the plate the primary velocity is increased with the decrease of hall parameter β_e. Comparison Figures 30(a) and 30(b) represents the secondary velocity profile for analytic and finite difference method, respectively, within both the case of cooling and heating plate, where secondary velocity profile shows the same characteristics for the solution of analytic and finite difference method. Comparison Figures 31(a) and 31(b) represent the primary velocity profile and Comparison Figures 32(a) and 32(b) represent the secondary velocity profile respectively for different values of magnetic parameter M, where for the analytic and numerical solution both the primary and secondary velocity acted towards the same flow pattern. Comparison Figures 33(a) and 33(b) represent that temperature profile is increases with the decrease of heat source parameter α for both the case of analytic and numerical solutions. Comparison Figures 34(a) and 34(b) illustrate that for the decreases of Schmidt number S_c concentration profile reacts exactly the same for analytic and finite difference solution.

From the above discussion, the analytical and the numerical solutions of primary velocity profile, secondary velocity profile, transiesnt temperature profile, and transient concentration profile have comparatively similar flow pattern for different parameters.

9. Conclusions

MHD mass transfer problem by free convection flow of an electrically conducting incompressible viscous fluid past an electrically conducting continuously moving semi-infinite vertical porous plate under the action of strong magnetic field is taken into account. The resulting governing system of dimensionless coupled nonlinear partial differential equations are numerically solved by an explicit finite difference method. The results are discussed for different values of important parameters as hall parameter, ion slip parameter, magnetic parameter, rotational parameter, heat source parameter, Soret number, Grashof number, modified Grashof number, Prandtl number, and Schmidt number. The important findings of this investigation with graphical representation are listed below.

The primary velocity increases with the decrease of M, R′, P_r, λwhile the transient primary velocity decreases with the decrease of β_e, β_i, G_r, G_m.

The secondary velocity increases with the decrease of β_e, β_i, G_r, G_m while the transient secondary velocity decreases with the decrease of M, P_r, λ.

The transient temperature increases with the decrease of α and λ. Particularly, the fluid temperature is more for air than water and it is less for lighter than heavier particles.

The transient concentration profiles are increasingly affected by P_r, λ and decreasingly affected by S_o, also α has a minor effect for concentration profile. Particularly, the species concentration is higher for water than air as well as it is more for lighter than heavier particles.

The shear stress τ_x increases with the decrease of M, α, P_r, λ while it decreases with the decrease of β_e, β_i, G_r, S_o, S_c, G_m.

The shear stress τ_z decrease with the decrease of M, α, P_r, λ, while it increases with the decrease of β_e, β_i, G_r, S_o, S_c, G_m. The shear stress is greater for air than water.

The rate of heat transfer increase with the increases of Prandtl number P_r and suction parameter λ. The rate of heat transfer is higher for water than air.

The rate of mass transfer increases with the decrease of P_r, S_o and S_c. The coefficient of mass transfer is greater for heavier than lighter particles. On the opposite case mass transfer has a decreasing effect with the decrease of λ.

In this analysis the effect of different parameters on the velocity, temperature, and concentration flow pattern is discussed. In all those parameters hall and ion slip terms are the main objective for our work, which has attracted the interest of any investigators in view of its important applications in many engineering problems such as power generators, MHD accelerators, refrigeration coils, transmission lines, electric transformers, and heating elements. Also the comparison between analytical and numerical solution gives clear information about the work.

Footnotes

Nomenclature

References

Takhar

H. S.

and Ram

P. C.

, “Effects of hall current on hydromagnetic free-convective flow through a porous medium,” Astrophysics and Space Science, vol. 192, no. 1, pp. 45–51, 1992.

Abo-Eldahab

E. M.

and El Aziz

M. A.

, “Hall and ion-slip effects on MHD free convective heat generating flow past a semi-infinite vertical flat plate,” Physica Scripta, vol. 61, no. 3, pp. 344–348, 2000.

Seddeek

M. A.

and Abdelmeguid

M. S.

, “Hall and ion-slip effects on magneto-micropolar fluid with combined flow and free convection in boundary layer over a horizontal plate,” The KSIAM Journal, vol. 8, no. 2, pp. 51–73, 2004.

Abo-Eldahab

E. M.

and El Aziz

M. A.

, “Viscous dissipation and Joule heating effects on MHD-free convection from a vertical plate with power-law variation in surface temperature in the presence of hall and ion-slip currents,” Applied Mathematical Modelling, vol. 29, no. 6, pp. 579–595, 2005.

MHD Free Convection and Mass Transfer Flow from a Vertical Plate in the Presence of Hall and Ion-Slip Current

Abstract

1. Introduction

2. Mathematical Formulation

3. Shear Stress, Nusselt Number, and Sherwood Number

4. Numerical Solutions

5. Stability and Convergence Analysis

6. Results and Discussion

7. General Discussions

8. Comparison

9. Conclusions

Footnotes

Nomenclature

References