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Chapter one

Introduction :

Thermal instability theory has attracted considerable interest and has been

recognized as a problem of fundamental importance in many fields of fluid

dynamics. The earliest experiments to demonstrate the onset of thermal

instability in fluids in a definitive manner are those of Benard [1,2]. He worked

with very thin layers, standing on a levelled metallic plate which was

maintained at a uniform temperature. In his experiments Benard deduced that a

can set in. The correct interpretation of Benard's experiments was given by

Rayleigh [3] who showed that what decides the stability or otherwise of fluid

heated from below is the numerical value of a non-dimensional parameter, (now

called the Rayleigh number)

g
R=         d4                                                    (1.1 )


where g is the acceleration due to gravity, d the depth of the layer,  the adverse

temperature gradient and ,  and  are respectively the coefficients of volume

expansion, thermal conductivity, and kinematic viscosity. Jeffreys [4,5], Low

[6] and Pellew  Southwell [7] discussed the Benard problem using different

boundary conditions (rigid and mixed) and they solved the problem using a
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finite difference method. Their results were in qualitative agreement with the

experimental work of Benard.

Thompson [8] examined the modifications produced in the Rayleigh-

Jeffreys theory of slow thermal convection by the addition of magnetic fields.

The presence of such fields in an electrically conducting fluid usually has the

effect of inhibiting the development of instabilities. Chandrasekhar [9] extended

the work of Thompson [8] to include different boundary conditions and tables

of the critical Rayleigh numbers were presented for the cases of stationary

convection and overstability. He showed that overstability is possible provide

Pm > Pr and Q exceeds a critical value where Pm and Pr are respectively the

magnetic and viscous Prandtl numbers and Q ( known as Chandrasekhar

number ) has the form

 2H 2 2
Q=         d                                                      (1.2)


where  is the magnetic permeability,  is the fluid density,  is the resistivity

and H is the magnitude of the magnetic field. The case when the impressed

magnetic field acts in a direction different from gravity is discussed by

Chandrasekhar [10]. The problems discussed by Chandrasekhar [9], [10] have

been investigated experimentally by Nakagawa [11,12] for different layers of

mercury. The results obtained coincide with the theoretical values given by

Chandrasekhar. The instability of a horizontal layer of fluid heated from below

and subjected to Coriolis forces has been studied by Chandrasekhar [13] and
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Chandrasekhar  Elbert [14] for stationary convection and overstability

respectively. They showed that the effect of these forces is to inhibit the onset

of convection. The extent of the inhibition depends on the value of the non-

dimensional parameter (known as Taylor number )

4 2
T=           d4                                                   (1.3)
   2

where  is the magnitude of the angular velocity. The simultaneous effect of

magnetic field and Coriolis forces on the thermal instability of a layer of fluid

heated from below has been investigated by Chandrasekhar [15,16] for

stationary convection and overstability respectively, where the boundaries are

free. The results obtained reveal some very unexpected features and Nakagawa

[17] proved experimentally the results obtained.

In the studies mentioned above the magnetic Benard problem have been

discussed for magnetic fluids which have a linear constitutive relationship

between the magnetic field H and the magnetic induction B. However Roberts

[18] suggested that a non-linear relationship between the magnetic field and the

magnetic induction may be appropriate for certain classes of materials. This

non-linear relationship has been used by Muzikar & Pethick [19] for materials

relevant to neutron stars. Abdullah & Lindsay [20,21] used the non-linear

relationship suggested by Roberts [18] to discuss the instability of the magnetic

Benard problem for a vertical and non-vertical magnetic field respectively.

They showed that this non-linear relationship has no effect on the development
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of instabilities through the mechanism of stationary convection which is the

preferred process in terrestrial applications. However in non-terrestial

applications, the nonlinearity influences the onset of overstable convection and

overstability is the preferred mechanism. Roberts model has been applied by

Abdullah [22,23] to examine the instability of a horizontal layer of a

magnetohydrodynamic fluid in the presence of both magnetic field and Coriolis

forces.

Horton & Rogers [24], Lapwood [25], Wooding [26] and Elder [27] have

used the analysis of Rayleigh to examine theoretically and numerically the

convection of a fluid through a porous medium using Darcy's law, which states

that the fluid flowing with macroscopic velocity through a porous medium

experiences a resistance. This resistance is proportional to the fluid velocity and

to its viscosity and is inversely proportional to the permeability. In the equation

of motion, the resistance term, calculated from Darcy's law, will replace the

usual viscosity term. The flow through a porous mass is described by a

modification of Darcy's law. Such a modification was necessary in order to

obtain consistent boundary conditions. A modification of Darcy's law has been

suggested by Brinkman [28,29]. His modification assumes that the viscosity

term in the Navier-Stokes equations should be included.

There has been considerable interests in the study of the stability of

different problems in the presence of porous medium by several workers.

Yamamoto & Iwamura [30 ] examined the flow with convective acceleration
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through a porous medium using Brinkman model, Sharma and Thakur [31]

studied the frictional effect of the collisions of ionized with neutral atoms on the

Rayleigh-Taylor instability of a composite mixture through porous medium in

hydromagnetics, Rudraiah, et al. [32] investigated the effect of non-uniform

thermal gradient, caused by either sudden heating or cooling at the boundaries

or by distributed heat sources on convective instability in a fluid saturated

porous medium using the Brinkman model. Georgiadis and Catton [33] studied

numerically the Prandtl number effect on Benard convection in porous media.

Kladias & Prasad [34] conducted a series of studies of Benard convection in

porous media by employing the Darcy-Brinkman-Forchheimer (DBF) model.

Jan & Abdullah [35] examined the convective instability of a horizontal porous

layer permeated by a conducting fluid in the presence of a uniform vertical

magnetic field using Brinkman's model. Abdullah [36] extended the work of

Jan & Abdullah [35] to include a non-linear magnetic fluid based on Robert's

model [18].

The effect of the earth's magnetic field on the stability of a layer of porous

medium is of interest in geophysics particularly in the study of the earth's core

where the earth's mantle, which consists of conducting fluid, behaves like a

porous medium which can become convectively unstable as a result of

differential diffusion. Another application of the result of flow through a porous

medium in the presence of a magnetic field lies in the study of the stability of a

convective flow in the geothermal region.
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The problem of stability of thermohaline conducting fluid layer in the

presence of magnetic field through a porous medium is of importance in

geophysics, soil sciences and ground water hydrology. In such problems

buoyancy forces can arise not only from density differences due to variations in

temperature but also from those due to variations in solute concentrations.

The analysis of such problems has many applications such as the fluid

movement in soil, particularly salt water movement in oil reservoirs and waste

and fertilizer migration in saturated soil.

Previous studies showed that the presence of salinity in convection

problems usually has the effect of inhibiting the development of instabilities

when the solute concentration decrease upwards. However it has an opposite

effect when the solute concentration increases upwards. Moreover if the fluid is

heated from above and soluted from below, then the temperature at the bottom

of the fluid will be relatively high comparing to other parts of the fluid. Solar

ponds are a good application of this phenomena and it has been considered in

detail by Shirtcliff [37] and Hoare [38]. The thermohaline convection problem

is considered by Stern [39] when the top surface is maintained at a higher

temperature and salinity than the bottom surface. This situation exists over most

of oceans, i.e. salty warm water usually lies above fresher cold water.

The stability of water stratified by both heat and salt has been studied by

Walin [40] in the case when the salt concentration increases upwards and the

layer is heated from below. Veronis [41] studied a horizontal layer of fluid that
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is stable by salt but unstable by temperature. It was shown that, with a very

stable salt stratification, overstable motions can arise for a Rayleigh number (or

destabilizing temperature gradient) having a value that is approximately one per

cent of the value necessary for instability to occur as exchange of stabilities. A

linear stability analysis is used by Nield [42] to study the onset of convection

induced by thermal and solute concentration gradients, in a horizontal layer of

viscous fluid. Veronis [43] studied the effect of a stabilizing gradient of solute

on thermal convection and found that the presence of a stabilizing gradient of

solute inhibits convection and also introduces the possibility of oscillatory

motions. AL-Aidrous & Abdullah [44] studied the thermosolutal convection in

a layer of fluid in the presence of vertical magnetic field when the layer is

heated and soluted from above, heated and soluted from below, heated from

above and soluted from below and heated from below and soluted from above.

This problem has been extended by Abdullah [45] for a non-linear magnetic

fluid.

The onset of salt-finger convection in a porous layer has been considered

by Nield [46] for a general set of boundary conditions when the fluid is heated

and soluted from below. Taunton & Lightfoot [47] extended Nield's [46]

analysis of the onset of convection in a porous medium and determined the

conditions at which ''salt fingers '' develop in the presence of both temperature

and concentration gradients. Sharma & Sharma [48] studied the thermohaline

convection in a layer of conducting fluid heated and salted from below in
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porous medium in the presence of a uniform vertical magnetic field. They

showed that the salinity gradient and magnetic field have stabilizing effect on

the system, but the medium permeability has a destabilizing effect. Chakrabarti

& Gupta [49] examined the effect of rotation on thermohaline convection in a

horizontal layer of a saturated porous medium. The effect of permeability of the

porous medium on the stability characteristics was also studied. The results of

their work have bearing on the onset of convection in geothermal areas where

the ground water flows through a porous medium and is subjected to the earth's

rotation.

The hydrodynamic problem of the onset of finger convection in a porous

layer of constant porosity underlying a fluid layer has been studied by Chen &

Chen[50]. Patil et al. [51] investigated the onset of thermohaline convection in

anisotropic rotating porous layer of infinite horizontal extent. Sharma & Misra

[52] studied the thermosolutal instability in porous medium of a layer of

compressible fluid heated from below and subjected to a stable solute gradient.

The thermosolutal instability of a layer of rotating compressible fluid in porous

medium was also considered and the roles of medium permeability.

Thermosolutal convection in a porous medium in the presence of uniform

rotation and uniform magnetic field, separately, have been considered by

Sharma & Kumari [53]. Khare & Sahai [54] have studied the thermosolutal

instability of a heterogenous fluid in porous medium by using Brinkman [28,29]

equation which is a general equation of motion of fluid through porous medium
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accounting for both the viscous force and the frictional force offered by solid

particles on the fluid which is usually called Darcy resistance. This problem has

been extended by Khare & Sahai [55] to include the effect of magnetic field. A

numerical method based on finite elements was used by Murthy et. al. [56] to

study the stability of double diffusive flow in a rectangular box containing a

porous medium. Sherin & Patil [57] investigated the effect of variable gravity

fields on soret driven thermohaline convection in a porous medium. Mharzi et

al. [58] investigated numerically the two-dimensional cooperating and opposing

steady state thermosolutal natural convection phenomenon occurring inside a

square cavity that is separated in two fluid filled regions by a fluid saturated

porous medium. The Brinkman model for thermosolutal convection in a vertical

annular porous layer has been studied by Bennacer et al. [59]. An analytical and

numerical investigation of the Sort effect on natural convection in a horizontal

porous layer subject to uniform fluxes of heat has been studied by Bourich et al.

[60]. Hatim [61] studied the Benard convection in a horizontal porous layer of

conducting fluid in the presence of a vertical magnetic field using Brinkman

model. Numerical results for stationary convection and overstability cases were

presented for different boundary conditions.

In this thesis we shall study, thermosolutal convection instability in a

horizontal porous layer permeated by an incompressible, thermally and

electrically conducting fluid using Brinkman model in the presence of a non-

linear relationship between magnetic field and the magnetic induction for both
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stationary and overstability cases. It is an extension of Hatim's work [61] which

was done for a linear relation between magnetic field and magnetic induction.

Analytical solutions were obtained when both boundaries are free and

numerical results were presented for the cases of free and rigid boundaries.

The numerical computations were performed using expansions of

Chebyshev polynomials. This method will be described in detail in chapter two.

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