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CHAPTER 5

MHD EFFECTS ON A MOVING SEMI-INFINITE

VERTICAL CYLINDER WITH UNIFORM

HEAT AND MASS FLUX

5.1 INTRODUCTION

The influence of a magnetic field on the viscous incompressible

flow of an electrically conducting fluid has great importance in many

applications in the field of science and technology. One such study is related

to the effects of a free convection MHD flow, which plays a vital role in

agriculture, engineering and petroleum industries. The problem under

consideration has significant applications in the study of geological

formations; in the exploration and thermal recovery of oil; and in the

assessment of aquifers, geothermal reservoirs and underground nuclear waste

storage sites. The intrusive may be taken as a vertical cylinder with power law

heat and mass flux boundary conditions. The time required to set in for the

intrusive magma is very essential. The heat and mass flux problems also rise

in many technological applications, where the devices are cooled by natural

convection, as in the case of electrical heaters and in transformers. The main

concern in such flows is the possibility of the flux boundary conditions. These

types of problems are commonly met in the start-up of a chemical reactor, and

the emergency cooling of nuclear fuel elements. Applications of heat and

mass flux are attractive and significant. The radiative flux is the amount of

energy moving in the form of photons, which are used to determine the

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magnitude and spectral class of a star. The latest research in monopole

magnetism of the radius of the curvature of an unwanted pole is increased

selectively to dilute the magnetic flux density of the pole.

Acharya et al (2000) conducted a study of a steady two-

dimensional free convection and mass transfer flow of a viscous

incompressible electrically conducting fluid through a porous medium over a

vertical infinite surface, with a constant suction velocity and heat flux in the

presence of a magnetic field. Jones et al (1997) developed a new implicit

algorithm for solving the time dependent, non-ideal magnetohydrodynamic

equations. Agrawal et al (1998) discussed thermal and mass diffusion on

hydro magnetic visco-elastic natural convection past an impulsively started

infinite plate in the presence of a transverse magnetic field. Shankar and

Kishan (1997) presented the effect of mass transfer on the MHD flow past an

impulsively started infinite vertical plate. The effects of heat and mass

transfer on the natural convective flow along a vertical cylinder were studied

by Chen and Yuh (1980). Bottemanne (1972) had studied such a problem for

steady heat and mass flux conditions, both experimentally and theoretically.

An analysis was made by Goldstein and Briggs (1964) for the transient free

convection, heat transfer problem from vertical circular cylinders to a

surrounding initially quiescent fluid. The transient is initiated by a change in

the wall temperature of the cylinder.

The solutions for the velocity and penetration distance profiles for

vertical cylinders with a step change in the surface heat flux for an arbitrary

Prandtl number are given. Dring and Gebhart (1966) presented the

experimental results for the transient average temperature of Nichrome wires

in silicone oils and in air. They also compared their experimental results with

a simplified quasistatic theory, which yields simple exponential solutions for

the temperature response. Even for air, the conduction solution was found to

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be better than that predicted by this theory. Chen (1983) investigated the

steady free convection from a vertical needle with a variable wall heat flux,

and found that there was a significant influence of its shape, size and wall

temperature variation upon the flow and heat transfer.

An experimental and analytical study is reported by Even et al

(1968) for transient natural convection in a vertical cylinder. The vertical

cylinder is subjected to a uniform heat flux at the walls for the experimental

study. The temperature of the core fluid was assumed to be uniform, but not

in the horizontal direction. Recently, a combined heat and mass transfer effect

on a moving vertical cylinder of steady and unsteady flow was analyzed by

Takher et al (2000). In the present analysis, consideration is given to a

situation in which the surface of the cylinder is maintained with uniform heat

and mass flux. The unsteady, non-linear and coupled governing equations are

first transformed into a non-dimensional form, and their solutions are

obtained by an efficient Crank-Nicolson implicit finite difference method.

5.2 MATHEMATICAL ANALYSIS

In this analysis, an unsteady laminar incompressible viscous flow

past a moving semi-infinite vertical cylinder of radius r0 with heat and mass

flux is considered. It is assumed that the fluid considered is gray, absorbing-

emitting radiation but non-scattering medium, and the fluid properties

assumed constant except for density variations, which induce only in the body

force. Here, the x-axis is measured along the axis of the cylinder in the

vertically upward direction, and the radial co-ordinate r is taken normal to it.

Initially when ' 0t , it is assumed that the cylinder and the fluid are of the

same temperature T ' , and the species concentration C ' . When ' 0t , it is

assumed that heat is supplied from the cylinder surface to the fluid at a rate of

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heat flux' w

qT

r k, and the diffusing mass is supplied from the cylinder

surface to the fluid at a rate of mass flux*

' wqC

r k and maintained. All the

physical properties are assumed to be constant except for the density

variations, which are given by the usual Boussinesq’s approximation. The

governing boundary layer equations that are based on the balance laws of

mass, linear momentum and energy for this investigation can be written as:

Equation of continuity

( ) ( )0

ru rv

x r (5.1)

Equation of momentum with MHD

2

0

( ' ' ) ( ' ' )'

u u uu v g T T g C C

t x r

Bur u

r r r

(5.2)

Energy equation

' ' ' 1 '

'p

T T T k Tu v r

t x r C r r r (5.3)

Mass diffusion equation

' ' ' '

'

C C C D Cu v r

t x r r r r (5.4)

The appropriate boundary conditions for the velocity, temperature

and concentration of this problem are:

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' 0 : 0, 0, ' ' , ' 't u v T T C C for al 0x and 0r r

*

0

' '' 0 : 0, ,w w

q qT Ct u u v

r k r k at r = r0 (5.5)

0, ' ' , ' 'u T T C C at x = 0 and r r0

' '0, ' , 'u T T C C as r

Introducing the following non-dimensional quantities:

' '

0

2 2 *00 0 0 0 0 0

3 * 3 2 2

0 0 0 0

0 0

' '', , , , , , ,

, , ,Pr ,

w w

ww

w w

vr C C T Tx r u tX R U V t C T

q ru r r u r q r

kk

g q r g q r B rGr Gc Sc M

k u k u D

(5.6)

Equations (5.1), (5.2), (5.3) and (5.4) are reduced to the following

form:

( ) ( )0

RU RV

X R (5.7)

1U U U UU V GrT GcC R MU

t X R R R R (5.8)

1 1

Pr

T T T TU V R

t X R R R R (5.9)

1 1C C C CU V R

t X R Sc R R R (5.10)

The corresponding initial and boundary conditions in non-

dimensional quantities are given by

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0 : 0, 0, 0, 0t U V T C for all X 0 and R 0

0 : 1, 0, 1, 1T C

t U VR R

at R=1 (5.11)

0, 0, 0U T C at X= 0

0, 0, 0U T C as R

5.3 NUMERICAL TECHNIQUE

In order to solve the unsteady, non-linear, coupled and governing

equations from (5.7) to (5.10) under the boundary conditions equation (5.11),

an implicit finite difference scheme of the Crank-Nicolson type is employed.

The region of integration is considered as a rectangle with sides Xmax (=1.0)

and Rmax(=14) where Rmax corresponds to R , which lies well outside the

momentum, and the thermal and concentration boundary layers. The time-

dependent equations are marched in time until a steady solution is obtained.

A convergence criterion based on the relative difference between

the current and previous iterations is employed. When this difference

reaches 510 , the solution is assumed to have converged, and the iterative

process is terminated. After experimenting with a few sets of mesh sizes, the

mesh sizes have been fixed at the level X = 0.02, R = 0.25, with the time

step of t = 0.01. In this case, the spatial mesh sizes are reduced by 50% in

one direction, and later in both the directions, and the results are compared.

It is observed that, when the mesh size is reduced by 50% in the

R-direction, the results differ in the fifth place after the decimal point, while if

the mesh sizes are reduced by 50% in the X-direction or in both directions,

the results are correct to the fourth decimal place. Hence, these mesh sizes are

considered to be appropriate for calculations.

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The finite difference scheme is unconditionally stable. The

truncation error is 2 2( )O t R X and it tends to zero as t, R and X

tend to zero. Therefore, the scheme is compatible. The stability and the

compatibility ensure the convergence.

5.4 RESULTS AND DISCUSSION

The partial differential equations (5.7), (5.8), (5.9) and (5.10) with

the boundary conditions (5.11) have been solved by using the above

mentioned implicit finite difference scheme of the Crank-Nicolson type. The

numerical values of the velocity, temperature, concentration, skin-friction,

Nusselt number and Sherwood number are computed for different values of

the magnetic parameter and Prandtl number.

The transient and steady state velocity profiles for different values

of the magnetic parameter and the Prandtl numbers (for air Pr = 0.71 and

water Pr = 7.0) are shown in Figure 5.1. It is observed that the magnetic field

modifies the velocity gradients near the cylinder. The time taken to reach the

steady state increases as M increases. It is observed that the presence and the

increase of the magnetic fields lead to a decrease in the velocity field; that is,

it retards the flow field. In the absence of the magnetic field, the velocity

overshoots near the cylinder. The lower value of Prandtl number of air

exhibits a greater velocity gradient. Hence, the velocity profile decreases with

an increasing Pr.

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Figure 5.1 Velocity profiles at X = 1.0 for different values of M and Pr

(* - Transient state)

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The transient and steady state temperature profiles for different M and

Pr are shown in Figures 5.2 and 5.3. It is interesting to see that the time

required for reaching the steady state increases with an increasing M and Pr.

Moreover, the thermal boundary layer thickness for air is greater than that for

water. It is also observed that the temperature profiles decrease as Pr

increases, since fluids with a large Pr give rise to less heat transfer. But, the

thermal boundary layer thickness increases with an increasing value of the

magnetic parameter M.

Figure 5.2 Temperature profiles at X = 1.0 for different values of M

(*-Transient state)

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Figure 5.3 Temperature profiles at X = 1.0 for different values of M

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The transient and steady state concentration profiles for different M

and Pr are shown in Figure 5.4. The concentration profile is maxima near the

surface of the cylinder, and it decreases as the radial co-ordinate increases. It

is clear that the concentration profile increases with increasing values of M

and Pr. This is due to the fact that a large Pr corresponds to a thicker

concentration boundary layer, relative to the momentum boundary layer. This

results in a larger concentration gradient on the cylinder.

Figure 5.4 Concentration profiles at X = 1.0 for different values of M

and Pr (*-Transient state)

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The steady state local skin friction values are plotted in Figure 5.5,

against the axial co-ordinate X. It is observed that the local skin friction

decreases as X increases. The wall shear stress increases when M increases.

This behavior is consistent with the velocity profiles shown in Figure 5.1. The

shear stress increases as Pr increases, since the velocity gradient is more for

fluids with a larger Pr than for fluids with a smaller Pr.

Figure 5.5 Local skin-friction

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Figure 5.6 shows the dimensionless steady state local Nusselt

number for different values of the magnetic parameter and Prandtl number. It

is noted that the Nusselt number increases with decreasing M. The local heat

transfer rate increases with an increasing value of Pr; this behavior is

consistent with the temperature profile results, which are shown in Figures 5.2

and 5.3.

Figure 5.6 Local Nusselt number

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The dimensionless steady state local Sherwood number is plotted in

Figure 5.7 for different values of the magnetic parameter and Prandtl number.

It is observed that the Sherwood number increases with decreasing M. It is

also observed that the local mass transfer rate increases with a decreasing

value of Pr. This trend is due to the fact that the concentration profiles

increase with an increasing Pr, which is already observed in Figure 5.4.

Figure 5.7 Local Shrewood number

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The average shear stress, the rate of heat transfer and the rate of

mass transfer are shown in Figures 5.8, 5.9 and 5.10 respectively, as a

function of time for different values of the magnetic parameter and Prandtl

number. It is observed that the rate of shear stress decreases with time and

attains the steady state after a lapse of time. The effects of M and Pr on the

average shear stress are similar to the effect of the local skin friction.

Figure 5.8 Average skin-friction

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Initially, higher values of the average Nusselt number and

Sherwood number are observed, and they decrease with time. It is noted that

there is no change in the average Nusselt number with respect to M. Thus, the

initial heat transfer is due to conduction only. Similarly, there is no change in

the average Sherwood number in the initial period with respect to M and Pr.

This shows that the initial heat and mass transfer are due to conduction only.

In Figures 5.9 and 5.10, it is observed that the average Nusselt number

increases as Pr increases. The average Sherwood number increases as Pr

decreases. It is also observed that both the average Nusselt number and

Sherwood number increase as M decreases.

Figure 5.9 Average Nusselt number

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Figure 5.10 Average Sherwood number

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5.5 CONCLUSION

A numerical study has been carried out for MHD effects on a

moving semi-infinite vertical cylinder with uniform heat and mass flux. The

dimensionless governing equations are solved by the finite difference scheme

of the Crank-Nicolson type. The stability of the finite difference scheme is

analyzed using the Von-Neumann technique. The fluids considered here are

both air and water. The effect of the Prandtl number, Grashof number and

magnetic parameters are studied. The conclusions of the study are as follows:

i) The presence and the increase of the magnetic field lead to a

decrease in the velocity field, whereas in the absence of the

magnetic field, the velocity overshoots near the cylinder.

ii) The thermal boundary layer thickness increases as M

increases, due to the increase in the magnetic field, which

leads to a rise in the temperature distribution.

iii) A higher value of M corresponds to the momentum

boundary layer. This results in a larger concentration

gradient on the cylinder. On the other hand, at small values

of M, the velocity and temperature of the fluid increase

sharply near the cylinder as the time increases.

iv) The local and average skin-friction increases with increasing

values of M and Pr. This trend is just reversed with the

Sherwood number.

v) The local and average Nusselt number increase with an

increasing Pr, and decreasing M.