# Finite Difference Analysis of MHD Laminar Mixed Different Analysis of MHD Laminar Mixed Convection...

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Serials PublicationsISSN: 0973-6794

* E-mail: reddybrb@gmail.com

Finite Difference Analysis of MHD Laminar MixedConvection With Vertical Channel in Downflow

D. Venkata Rami Reddy1, M. Rama Chandra Reddy2 and B. Rama Bhupal Reddy3

1Lecturer in Mathematics, Government Polytechnic College, Ananthapur2Reader in Mathematics, S. K. S. C. Degree College, Proddutut3Associate Professor in Mathematics, K. S. R. M. College of Engineering, Kadapa-516 003, A.P. - India

ABSTRACT

In this paper, finite difference analysis of MHD laminar mixed convection in vertical channel is studied. The distance betweenthe plates is b and g is the gravitational force. The buoyancy effect is taken into account according to the Boussinesqapproximation. The flow problem is described by means of parabolic equations and the solutions are obtained by the use of animplicit finite difference technique coupled with a marching procedure. The velocity, the temperature and the pressure profilesare shown in graphs and their behaviour is discussed for different values of magnetic parameter M, buoyancy parameter Gr/Reand Prandtl number Pr.

Keywords: Mixed Convection, Vertical Channel, MHD

1. INTRODUCTION

Different orientation are heating conditions of the channelcan induce different kinds of heated buoyant flows whichenhance the heat transfer in different manners. Forasymmetric heated channel, when the buoyancyparameter is not large, this small amount of the buoyantflow induced along the heated wall can either assist oroppose the main flow and cause either enhancement orreduction in the heat transfer. When the buoyancyparameter becomes large, the buoyant flow along the sidewall become substantial. Depending upon the flowdirection of the main stream, the buoyancy flow can causedifferent kinds of flow reversals which will alter the entireflow characteristic and enhance the heat transfer indifferent manners.

The laminar mixed convection in vertical or inclinedducts has been widely studied in the literature. Indeed,this research area has several technical applications, suchas heat exchangers, cooling systems for electronicdevices, solar collectors. Interesting results of theresearches on this topic are collected in Refs. [1, 17].Early exact solutions for the differential equationsdescribing mixed convection in vertical channels andpipes were given by Ostrach [16] and Morton [14].Ostrach [16] used a linearly changing wall temperatureprofile for vertical flow. He found instability and flowinversion for heating from below. Morton [14] predicted

parallel flow for small Ra numbers in downward heatedflow. He predicted that the flow would become unsteadyand turbulent when the Raleigh number Ra reached 33.Quientiere [17] presented approximate analyticalsolutions for mixed convection between parallel plates.He related the heat transfer results to the inlet pressuredefect and flowrate.

Yao [22] studied mixed convection by the use ofstability analyses. Extensive work has been done to solvethe flow field equation for mixed convection flows bynumerical methods. The level of details of the results ofthese studies is highly dependent on the computingresources used. In the 1960s and 1970s general resultswere found for heat transfer effects for mixed convectionflows by Lawrence [13], and Zeldin and Schmidt [23].Beginning in the 1980s, numerical studies started beingable to better resolve the velocity profiles of vertical pipeflows with mixed convection as shown in studies byIngham [11], Aung [2], Cebeci [5], and Hebchi andAcharya [10]. Recent numerical studies by Evans [8] andChang [7] have focused on the oscillatory flow reversalbehavior that can occur in mixed convection flows.

Barletta [2] studied on the existence of parallel flowfor mixed convection in an inclined duct. Barletta andZanchini [3] studied mixed convection with variableviscosity in an inclined channel with prescribed walltemperatures. Finite difference analysis of Laminar mixedconvection in vertical channels is studied by Reddy[18].

I J C I R A6(1) 2012, pp. 19-26

mailto:reddybrb@gmail.comFinite

20 IJCIRA D. Venkata Rami Reddy, M. Rama Chandra Reddy and B. Rama Bhupal Reddy

Sparrow and Cess [20] considered the effective ofmagnetic field on the free convection heat transfer froma surface. Garandet et al. [9] discussed buoyancy drivenconvection in a rectangular enclosure with a transversemagnetic. Chamkha[6] studied on a laminar hydro-magnetic mixed convection flow in a vertical channelwith symmetric and Asymmetric wall heating conditions.Mazimber et al. [15] studied Hall effects all combinedfree and forced convective hydro-magnetic flow througha channel.

2. NOMENCLATURE

b Distance between the parallel plates

Cp

Specific heat of the fluid

g Gravitational body force per unit mass (acceleration)

Gr Grashoff number, gb (T1 T

0) b3/2

Ho

Applied Magnetic Field

J Current Density vector

k Thermal conductivity of fluid

M Magnetic Parameter

p Local pressure at any cross section of the verticalchannel

p0

Hydrostatic pressure

P Dimensionless pressure inside the channel at anycross section, (p p

0) /r u

02

Pr Prandtl Number, m Cp / k

q Velocity Vector

q Constant heat fluxRe Reynolds number, (b u

0)/

T Dimensional temperature at any point in the channel

T0

Ambient of fluid inlet temperature

Tw

Isothermal temperatures of circular heated wall

T1, T

2Isothermal temperatures of plate 1 and plate 2 of

parallel plates

u Axial velocity component

u Average axial velocity

u0

Uniform entrance axial velocity

U Dimensionless axial velocity at any point, u/u0

v Transverse velocity component

V Dimensionless transverse velocity, v/u0

x Axial coordinate (measured from the channelentrance)

X Dimensionless axial coordinate in Cartesian, x / ( bRe)

y Transverse coordinate of the vertical channel betweenparallel plates

Y Dimensionless transverse coordinate, y/b

Dimensionless temperature Density of the fluid

0Density of the fluid at the channel entrance

Dynamic viscosity of the fluid

e

Magnetic Permeability

Electrical Conductivity Kinematic viscosity of the fluid, / Volumetric coefficient of thermal expansion

3. FORMULATION OF THE PROBLEM

We consider a steady developed laminar mixedconvection flow between two vertical channels. Thevertical plates are separated by a distance b. TheCartesian coordinate system is chosen such that the x-axis is in the vertical direction that is parallel to the flowdirection and the gravitational force g always actingdownwards independent of flow direction. The y-axis isorthogonal to the channel walls, and the origin of theaxes is such that the positions of the channel walls are y= 0 and y = b. One wall is maintained at constant heatflux and the other is at isothermal condition. The fluidproperties are assumed to be constant except for thebuoyancy terms of the momentum equation. A uniformtransfers magnetic field of strength H

o is applied

perpendicular to the walls. The geometry is illustrated inFigure 1.

The governing equations for the steady viscous flowof an electrically conducting fluid in the presence ofexternal magnetic field with the following assumptionsare made:

(i) The flow is steady, viscous, incompressible anddeveloped.

(ii) The flow is assumed to be two-dimensionalsteady, and the fluid properties are constantexcept for the variation of density in thebuoyancy term of the momentum equation.

(iii) The electric field E and induced magnetic fieldare neglected [19, 21]

(iv) Energy dissipation is neglected.

After applying the above assumptions the boundarylayer equations appropriate for this problem are

Continuity Equation

0 + = U V

X Y(1)

Finite Different Analysis of MHD Laminar Mixed Convection with Vertical Channel in Downflow 21

X Momentum Equation

22

2Re

+ = + U V dP Gr U

U V M UX Y dX Y

(2)

Energy Equation

2

2

1

Pr

+ =

U VX Y Y

(3)

Where M2 = 2e

Ho

2 b2/

The form of continuity equation can be written inintegral form as

1

0

1=U dY (4)

The boundary conditions are

Entrance conditions

At X = 0, 0< Y< 1 : U = 1, V = 0, = 0, P = 0 (5a)

No slip conditions

At X > 0, Y = 0 : U = 0, V = 0

At X > 0, Y = 1 : U = 0, V = 0 (5b)

Thermal boundary conditions

At X > 0, Y = 0 :0

1=

= Yy

At X > 0, Y = 1 : =0 (5c)

In the above, dimensionless parameters have beendefined as:

U = u / u0, V = vb/u, X = x / (b Re), Y = y / b

P = (p p0 ) / u

02, Pr = C

p / k, Re = (b u

0)/u (6)

Gr = g (T1 T

0) b3/2, = (T T

0) / (T

1 T

0)

The systems of non-linear equations (1) to (3) aresolved by a numerical method based on finite differenceapproximations. An implicit difference technique isemployed whereby the differential equations aretransformed into a set of simultaneous linear algebraicequations.

4. NUMERICAL SOLUTION

The solution of the governing equations for developingflow is discussed in this section. Considering the finitedifference grid net work of figure 2, equations (2) and(3) are replaced by the following difference equationswhich were also used in [4]

Figure 1: Schematic View of the System and Coordinate AxesCorresponding to Down-flow

Figure 2: Mesh Network for Difference Representations

( ) ( ) ( ) ( ) ( ) ( )1, , 1, 1 1, 1, ,2

+ + + + +

U i j U i j U i j U i j

U i j V i jX Y

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