MHD natural convection from a heated vertical wavy surface with variable viscosity and thermal conductivity.

Choudhury, M. ; Hazarika, G.C. ; Sibanda, P. 等

Introduction

Solutions for the flow over a specified wavy surface are desirable for understanding the process of wave growth under the action of wind and the effects of surface configuration on drag (Caponi et. al [3]). Solutions for flow over small amplitude wavy surfaces were obtained by Miles [17] and by Benjamin [1]. Lyne [15] used the method of conformal transformation to investigate the steady streaming generated by an oscillatory viscous flow over a wavy wall. One of the reasons why a roughened surface is more efficient in heat transfer is its capability to promote fluid motion near the surface. Among others, Moulic and Yao [18] studied natural convection along a wavy surface with uniform heat flux. Yao [27] concluded that total heat transfer rate for a wavy surface of any kind, in general, is greater than that of a flat surface.

Since the work of Schmidt and Beckman [23], the study of natural convection has been one of the most important research topics in heat transfer problems. The problem of natural or mixed convection along a sinusoidal wavy surface has received considerable attention because such a surface can be viewed as an approximation to certain geometries of practical relevance in heat transfer [2, 26, 27].

Fluid viscosity and thermal conductivity (hence thermal diffusivity) play an important role in the flow characteristic of laminar boundary layer problems. Fluid properties are significantly affected by the variation of temperature. The increase in temperature leads to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so the heat transfer rate at the wall is affected. In the cooling of electronic equipments, it is relatively frequent to find circumstances in which variable property effects are significant and cannot be neglected.

Sparrow and Gregg [24] was the first to study the variable fluid property in natural convection. Later, Zhong et. al [29], Kafousias and Williams [13], Zamora and Hernandez [28], Hernandez and Zamora [9] and Maleque and Sattar [16] considered variable property effects on natural convection flow. Hazarika and Lahkar [8] observed that a significant variation takes place in velocity and temperature distribution with the variation of the viscosity and thermal conductivity parameters. Hossain et. al [10] studied natural convection of fluid with variable viscosity from a heated vertical wavy surface.

Magnetoconvection plays an important role in many industrial applications. Such flows also occur naturally in geophysical and astrophysical problems during the cooling or heating of liquid metals (Nagata [19]). The presence of a magnetic field may, in some instances, have the effect of limiting the effectiveness of cooling systems by increasing wall temperatures, Sharma and Singh [22], Uda et. al.[25]. The application of a magnetic field suitably oriented to the flow in a shock layer may modify the flow pattern and this in turn may cause a change in the heat transfer characteristics of the body, Gupta et. al [7]. A magnetic field applied transverse to the plate causes a reduction in heat transfer at the plate (Gupta [6]). Hossain et.al. [11] studied magnetohydrodynamic free convection along a vertical wavy surface. Hossain and Pop [12] investigated the magnetohydrodynamic boundary layer flow and heat transfer from a continuous moving wavy surface.

The aim of this paper is to study the effects of temperature dependent viscosity and thermal conductivity on natural convection flow of a viscous incompressible electrically conducting fluid from a vertical wavy surface. The flow is subject to a uniform transverse magnetic field. Both the fluid viscosity and thermal conductivity vary as inverse linear functions of temperature. Recent studies of a similar nature with temperature dependent viscosity and thermal conductivity have been carried out by, among others, Sharma and Singh [22] for flow along a vertical non-conducting plate with internal heat generation and by Rahman et. al. [21] for MHD natural convection of an electrically conducting fluid, also along a vertical flat plate. The recent study by Prasad et. al. [20] considered the MHD flow of a viscoelastic fluid and heat transfer over a stretching sheet. The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface had earlier been considered by Elbashbeshy and Bazid [5].

Formulation of the problem

Consider the steady laminar free convective boundary layer flow of a viscous incompressible electrically conducting fluid from a semi-infinite vertical wavy surface. The geometric model considered is a wavy surface similar to that in Hossain et al. [10] and shown schematically in Fig. 1.

The [bar.x]--axis is taken along the vertical surface in the flow direction and the [bar.y] axis normal to the surface. The flow is subject to a uniform magnetic field of strength [B.sub.0] applied transverse to the direction of the flow. The plate is electrically non-conducting. The surface temperature is held uniform at [T.sub.w] warmer than the ambient temperature [T.sub.[infinity]] .The surface modulation is described by [bar.y] = [bar.[sigma]]([bar.x])

where [bar.[sigma]] is a surface geometry function. The mathematical formulation proposed in Hossain et. al. [10] allows [bar.[sigma]] to be of arbitrary shape but a sinusoidal surface was used as an example in the computations. The sinusoidal profile of the wavy surface is given by [[bar.[sigma]].sub.x] = [alpha] sin x where [alpha] is an amplitude function.

[FIGURE 1 OMITTED]

Following Lai and Kulacki [14], the fluid viscosity is assumed to be an inverse linear function of the temperature T of the form;

[mu] = [[mu].sub.[infinity]]/[1 + [gamma](T-[T.sub.[infinity]])] or 1/[mu] = a (T-[T.sub.r]) (1)

where a = [gamma]/[[mu].sub.[infinity]] and [T.sub.r] = [T.sub.[infinity]] -1/[gamma], is the fluid temperature, [mu] is the coefficient of dynamic viscosity, [[mu].sub.[infinity]] is the coefficient of viscosity at the free stream, a, [T.sub.r] and [gamma] are constants whose values depend on the reference state and the thermal property of the fluid. In general, for liquids a > 0 and for gases a < 0. For [gamma] = 0, the fluid viscosity is constant throughout the flow field.

The variation of thermal conductivity with temperature is considered to be as follows, see also Hazarika and Lahkar [8];

1/k = [1 + [epsilon](T - [T.sub.[infinity]])]/[k.sub.[infinity]] or 1/k = c (T - [T.sub.k]) (2)

where c = [epsilon]/[k.sub.[infinity]] and [T.sub.k] = [T.sub.[infinity]] -1/[epsilon], k is the thermal conductivity of the fluid, [k.sub.[infinity]] is the thermal conductivity of the ambient fluid, c, [T.sub.k] and [epsilon] are constants whose values depend on the reference state and the thermal property of the fluid. For liquids c > 0 while for gases c < 0.

Under the usual Boussinesq approximation (see Rahman et al. [21], Sharma and Singh [22]), the flow is governed by the following boundary layer equations

[partial derivation][bar.u]/ [partial derivation][bar.x] + [partial derivation][bar.v]/[partial derivation][bar.y] = 0 (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[bar.u] [partial derivation]T/[partial derivation][bar.x] + [bar.v] [partial derivation]T/[partial derivation][bar.y] = 1/[rho][c.sub.p] [DELTA] x (k[DELTA]T) (6)

where [bar.u] and [bar.v] are the [bar.x], [bar.y] components of the velocity field with the bar denoting dimensional quantities, [[DELTA].sup.2] = [[partial derivation].sup.2]/[partial derivation][[bar.x].sup.2] + [[partial derivation].sup.2]/[partial derivation][[bar.y].sup.2], g is the gravitational acceleration, [rho] is the density of the fluid, [bar.p] is the fluid pressure, [c.sub.p] is the specific heat at constant pressure, [beta] is the coefficient of thermal expansion and [B.sub.0] is the applied magnetic field strength.

The boundary conditions are,

[bar.u] = 0, [bar.v] = 0 T = [T.sub.w] at [bar.y] = [y.sub.w] = [bar.[sigma]]([bar.x]) (7a)

[bar.u] [right arrow] 0, T [right arrow] [T.sub.w] as [bar.y] [right arrow] [infinity]. (7b)

To non-dimensionalize equations. (3)--(6) we introduce the following scales;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8c)

where Gr is the thermal Grashof number, L is the fundamental wavelength, p is the dimensionless pressure and [p.sub.[infinity]] is the ambient pressure, [theta] is the non-dimensional temperature, [[theta].sub.r] and [[theta].sub.k] are a viscosity measuring parameter and a transformed dimensionless reference temperature respectively. The parameter A is a thermal influence parameter (see Cramer and Pai [4]) and M is the Hartmann number. The parameter [[theta].sub.r] is positive for gases and negative for liquids if [T.sub.w] > [T.sub.[infinity]].

Using the transformations (8a)-(8c) in equations (3)--(6) and ignoring terms of small orders in the Grashof number Gr we have,

[partial derivation]u/[partial derivation]x + [partial derivation]v/[partial derivation]y = 0 (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where Pr = [[mu].sub.[infinity]][c.sub.p]/[k.sub.[infinity]] is the Prandtl number. The associated boundary conditions are

u = 0, v = 0, [theta] = 1 on y = 0 (13a) u [right arrow] 0, [theta] [right arrow] 0 as y [right arrow] [infinity] . (13b)

Equation (11) indicates that the pressure gradient along the y--direction is O([Gr.sup.1/4]) which implies that the lowest order pressure gradient along the x-direction can be determined from the inviscid flow solution.

Eliminating the pressure terms between equations (10) and (11) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

From a technological point of view it is important to know the local skin-friction [C.sub.f] and the local Nusselt number Nu . These are given in non-dimensional form by

[C.sub.f] = (1 + [[sigma].sup.2.sub.x][[theta].sub.r]/[[theta].sub.r] - 1 [Gr.sup.3/4][([partial derivation]u/[partial derivation]y).sub.y = 0] (15)

[N.sub.u] = -(1 + [[sigma].sub.x])[[theta].sub.k]/[[theta].sub.k] - 1 [Gr.sup.1/4] [([partial derivation][theta]/[partial derivation]y).sub.y = 0]. (16)

Results and Discussions

The system of differential equations (9), (12) and (14) subject to the boundary conditions (13) was solved numerically. A finite difference solution is straightforward since the computational grids are fitted to the shape of the wavy wall. Central differences were used for the diffusion terms and the forward difference scheme for the convection terms. After experimenting with a few set of mesh sizes, the mesh sizes were fixed at [DELTA]x = 0.1and [DELTA]y = 0.01which gave sufficient accuracy for Pr = 0.7, Grashof number Gr = 0.2 and wave amplitude [alpha] = 0.2.

Table 1 displays the values of the skin-friction coefficient and Nusselt number at the surface for different values of [[theta].sub.r]. Increasing the viscosity parameter leads to increases in the values of both the skin friction coefficient and the heat transfer coefficient. This finding is similar to the results obtained by Prasad et al. [20] in their study of the effects of variable viscosity on MHD viscoelastic flow and heat transfer over a stretching sheet and arises because the fluid is able to move more easily close to the heated surface since its viscosity is lower relative to the constant viscosity case.

Table 2 shows the effect of increasing the thermal conductivity parameter on the skin friction and the heat transfer coefficients when M = 0.5, Pr = 0.7. The related study by Rahman et al. [21] for flow along a vertical plate with heat generation used M = 0.10 and Pr = 0.733.The results are qualitatively similar and show that increasing the thermal conductivity of the fluid leads to an increase in the skin-friction coefficient but a decrease in the Nusselt number. This may partly be explained by the fact that that increasing thermal conductivity has the effect of accelerating and increasing the temperature of the fluid.

Table 3 shows the effect of increasing the magnetic field intensity on the skin friction coefficient and the Nusselt number for constant values of viscosity and thermal conductivity. Simulations show that the increasing the magnetic field intensity leads to a decrease in the skin-friction coefficient as well as in the heat transfer coefficient. This is broadly in line with the recent findings by Sharma and Singh [22] who also observed that the heat transfer coefficient however decreases with Prandtl numbers. The study by Rahman et al. [21] however found that an increase in the magnetic field intensity leads to an increase in the surface temperature.

The tangential and normal velocity components and the temperature distributions are displayed in Figs.1-5 for various viscosity, thermal conductivity and magnetic parameter values. Figs. 1 and 2 depict the effects of the viscosity variation parameter [[theta].sub.r] on the tangential and normal velocity components for fixed values of thermal conductivity parameter [[theta].sub.k] = -15, Hartman number M = 0.5 and Prandtl number Pr = 0.7. Increases in the values of 0r lead to an increase in the velocity field away from the surface. This results are however different to the case of a viscoelastic fluid where Prasad et al. [20] showed that the velocity decreases with increasing values of the viscosity parameter. The effect of an increase in the value of this parameter on the temperature distribution is not significant in the entire boundary layer region. The velocity and the temperature profiles for values of the thermal conductivity parameter [[theta].sub.k] = -15, -12, -9, -5 and viscosity parameter [[theta].sub.r] = -15 are depicted in Figs. 3 and 4. The results are in line with the findings in Prasad et al. [20] and show that the velocity and temperature profiles increase with increases in [[theta].sub.k]. This effect is however not very significant in the case of the normal velocity component.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Fig. 5 shows the velocity profiles for Hartman numbers M = 0,5, 10, uniform

viscosity [[theta].sub.k] = -10 and thermal conductivity and [[theta].sub.r] = -10. The velocity curves show that the rate of transport is reduced with increases in M . This clearly confirms that the transverse magnetic field opposes the transport phenomena confirming the earlier findings in, for example, Prasad et al. [20], Rahman et al. [21] and Sharma and Singh [22]. This is because the variation of the Hartmann number leads to the variation of the Lorentz force due to the magnetic field and the Lorentz force produces resistance to transport phenomena.

[FIGURE 5 OMITTED]

Conclusion

We have investigated the effect of temperature-dependent viscosity and thermal conductivity on the MHD flow of an incompressible electrically conducting fluid along a semi-infinite vertical plate with a wavy surface. The study shows that increasing the viscosity variation leads to increases in both the skin friction and the heat transfer coefficient. The effect of increasing the thermal conductivity is also to increase the skin friction while reducing the Nusselt number. Both the skin friction and the heat transfer coefficients decrease with increases in the magnetic field parameter. The numerical simulations thus show that fluid viscosity and thermal conductivity (hence thermal diffusivity) play an important role in the flow characteristics of laminar boundary layer problems. Fluid properties are significantly affected by the variation of the viscosity and thermal conductivity due to temperature changes. The effects of the Lorentz force due to applied magnetic field has the effect of retarding the transport phenomena.

Nomenclature

A thermal influence parameter

[B.sub.0] = Magnetic field induction

[C.sub.f] = local skin friction

[C.sub.p] = Specific heat at constant pressure

Gr = Grashof number

g = Acceleration due to gravity

k = Thermal conductivity

L = wavelength

M = Hartmann number

Nu = local Nusselt number

p = fluid pressure

Pr = Prandtl number

T = fluid temperature

[bar.u], [bar.v] = dimensional stream wise and normal velocity components

[bar.x], [bar.y] = dimensional tangential and normal coordinate axis

Greek Symbols

[alpha] = surface wave amplitude

[beta] = coefficient of thermal expansion

[gamma], [epsilon] = constants based on the thermal property of the fluid

[mu] = coefficient of dynamic viscosity

[rho] = fluid density

[bar.[sigma]] = surface geometry function

[upsilon] = coefficient of kinematic viscosity

[theta] = dimensionless temperature term

[[theta].sub.k] = thermal conductivity variation parameter

[[theta].sub.r] = viscosity variation parameter

Subscripts

k, r = reference state values

[infinity] = ambient free-stream values

w = wall surface conditions

References

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M. Choudhury (1) *, G.C. Hazarika (2) and P. Sibanda (3)

(1) Department of Mathematics, N.N.S. College, Titabar, Assam-785630, India

* Corresponding author E-mail: mirabpgc@gmail.com

(2) Department of Mathematics, Dibrugarh University, Dibrugarh, Assam-786004, India.

(3) School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa E-mail: sibandap@ukzn.ac.za

Table 1: Effect of variable viscosity [[theta].sub.r] on the
skin-friction coefficient and Nusselt number when M = 0.5, Pr = 0.7,
[[theta].sub.k] = -15.

[[theta].sub.r] [C.sub.f] Nu

-15 0.027088 -1.90584
-13 0.028313 -1.90318
-11 0.029985 -1.90317
-9 0.032397 -1.90317
-7 0.036184 -1.90311
-5 0.042990 -1.90306
-3 0.058832 -1.90291
-1 0.137251 -1.90223

Table 2: Values of skin-friction coefficient and Nusselt number for
different [[theta].sub.k] and M = 0.5, Pr = 0.7,
[[theta].sub.r] = -15.

[[theta].sub.k] [C.sub.f] Nu

-15 0.027088 -1.071795
-13 0.028155 -1.08392
-11 0.029799 -1.10102
-9 0.032675 -1.127015
-7 0.039155 -1.171557
-5 0.073148 -1.267624

Table 3: Values of skin-friction coefficient and Nusselt number for
different M and [[theta].sub.k] = -10, Pr = 0.7,
[[theta].sub.r] = -10.

M [C.sub.f] Nu

0 0.036579 -3.04150
2 0.036402 -3.04165
4 0.036090 -3.04169
6 0.035575 -3.04176
8 0.034866 -3.04184
10 0.033974 -3.04196