1

Chapter 1

Introduction

1.1 Objective and Scope

The main objective of this thesis is to investigate the onset of double diffusive

convection in a

i. Darcy-Brinkman reaction-convection in an anisotropic porous layer,

ii. reaction-convection in an anisotropic porous layer with internal heat source,

iii. Maxwell fluid saturated anisotropic porous layer with internal heat source,

iv. Viscoelastic fluid saturated porous layer under the influence of Soret effect with

and without Darcy-Brinkman model,

v. Couple stress fluid saturated anisotropic porous layer with cross-diffusion

effects.

The motivation for the present study is explained below:

Rapidly changing technologies in fluid dynamics play a dominant role and that

trend is sure to flow into the present one. The last century has witnessed the thorough

understanding of the laws and principles of fluid mechanics and the skills of how to

apply them to encounter various challenging problems of the real world. The

applications of fluid mechanics are marked over a broad scale of disciplines of science

and engineering ranging from chemical engineering to geophysics. Aeronautical,

biomedical, civil, marine and mechanical engineers as well as astrophysicists,

geophysicists, space researchers, meteorologists, physical oceanographers, physicists

and mathematicians have used this knowledge to tackle with variety of complex flow

phenomena. The typical complex flows encountered by the researchers often comprise

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of two or more phases in which the interaction between them plays a vital role in

controlling the transport processes such as heat and mass exchange and reaction

kinetics. A quantitative study through a proper theory is essential to understand the

physics of the complex flow behavior and also to obtain invaluable scale-up

information for industrial applications.

Convection is usually one of the important and dominant modes of heat transfer.

The study of Convective instability in a fluid and fluid-saturated porous medium has

attracted considerable interest in the last few decades because of its relevance in a wide

spread applications in many branches of science and technology such as chemical

engineering, geothermal energy utilization, oil reservoir modeling, building of thermal

insulation, nuclear waste disposal and biological processes. It is of practical interest in

the extraction of geothermal energy and more specifically in understanding the

mechanism of transfer of heat from aquifers in the deep interior of earth to shallow

depths. Thermal convection is basically a prime driver in achieving the effective

mixing process in petroleum reservoirs which are regarded as a fixed bed reactor. The

study of convection has also a great impact on many technological applications, for

example, the evaluation of the amount of heat removal from a hypothetical accident in

a nuclear reactor, the prevention of convection and the consequent freezing in roads and

railways, providing effective insulation and so on.

Convection plays a dynamic role in the generation of magnetic fields within the

earth’s core, and it is possibly responsible for the reversal of the geomagnetic field. In

the atmosphere, convection is an important factor in weather prediction at small length

and time scales, and for climate prediction at large time scales. Convective motions are

responsible for much of the mixing of water masses occurring in the oceans.

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Convection is not only confined to natural flows, it also manifests in various forms in

industrial applications.

In recent years there has been a great demand for the polymeric materials due to

their effective reduction in weight in comparison with metals and metal alloys.

Synthetic polymers find numerous applications in many diverse areas varying from

smallest integrated chip to the parts of light combat aircraft. The quality of these

polymeric materials depends on the extent of distortions, which arise during the

solidification process. The distortions make the tensile properties incompatible with

those of metal and its alloys. The main reason for distortions is either due to early or

delayed convection in polymer aqueous solutions and in melts. As a polymer melt at a

given temperature has measurable characteristics, which separates the viscous and

elastic domains, where the viscous and elastic behaviors of polymeric liquid depend on

the applied temperature and its control. Therefore, the onset of thermal convection

plays a predominant role in deciding the quality of the polymeric material.

There are several areas of engineering in which previously puzzling phenomena

have been explained in terms of double diffusive processes. Understanding the physical

behaviour of the ocean is of interest to scientists in a variety of disciplines because

ocean dynamics interacts with atmospheric processes to govern climatic and biological

processes. For example, there is evidence that interactions between the ocean currents

and temperature fields and the atmospheric temperature and wind fields are

mechanisms which influence global weather fluctuations and biological production

over periods of years.

Double diffusive convection has been studied most extensively in

oceanography, where the heat-salt fingering mechanism operating on a scale of

centimeters can affect ocean dynamics and circulation over scales of many kilometers.

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Within porous media and in the context of environmental problems, it is also possible

that this mechanism can lead to surface ground water interactions and influence

subsurface contaminant transport. The subject of double diffusive convection has its

beginning in the year 1956 whose evaluation has been the result of a close interaction

between theoreticians, laboratory experiments and oceanographers. There was little

indication then that in quarter century the phenomenon which is described, and related

processes, would play an important role in oceanography and more recently in

astrophysics, geology, geophysics, biology, chemistry and engineering.

Double diffusive processes can be important in other systems besides aqueous

solutions, and two applications arise in the context of storage and transport of liquid

natural gas. The phenomena of crystal growth have taken on a new practical importance

with the increasing needs of the electronics industry for larger and more chemically

homogeneous crystals. Crystal growth from a solution or melt involves both heat and

mass transfer, and these usually leads to convection in the fluid, often double diffusive

in character. In common practice, convection was regarded as always deleterious and

thus to be avoided. Detailed fluid dynamical studies have begun to contribute to the

understanding of the processes that lead to fluctuations in growth rate and consequent

nonuniformities in crystals.

The substantial part of theoretical and experimental works on convective flow

has usually been concerned with homogeneous isotropic porous structures. In a porous

medium, due to the structure of the solid material in which the fluid flows, there can be

a pronounced anisotropy in such parameters as permeability, thermal diffusivity, or

solute diffusivity. The geological and pedological processes rarely form isotropic

medium as is usually assumed in transport studies. In geothermal system with a ground

structure composed of many strata of different permeabilities, the overall horizontal

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permeability may be up to ten times as large as the vertical component. In geological

processes such as sedimentation, compaction, frost action, and reorientation of the solid

matrix are responsible for the creation of anisotropic natural porous medium.

Anisotropy is generally a consequence of preferential orientation or asymmetric

geometry of porous matrix or fibers and is in fact encountered in numerous systems in

industry and nature. Examples include fibrous materials, sedimentary soils, rock

formations, certain biological materials, columnar dendritic structures formed during

solidification of multicomponent mixtures, and perform of aligned ceramic or graphite

fibers used in casting of metal matrix composites.

There are large number of practical situations in which convection is driven by

internal heat source. Due to internal heating of earth there is a temperature gradient

between the interior and the exterior of the earth’s crust, saturated by multicomponents

fluids, which helps convective flow, thereby transferring the thermal energy toward the

surface of the earth. The internal heat source is the main energy source of celestial

bodies which is generated by radioactive decay and nuclear reaction. The effect of

internal heat generation is important in several applications that include geophysics,

reactor safety analyses, metal waste form development for spent nuclear fuel, fire and

combustion studies, and storage of radioactive materials. There are also situations of

great practical importance where the porous material offers its own source of heat. This

gives a different way in which a convective flow can be set up through the local heat

generation within the porous media. Such a situation can occur through radioactive

decay or through, in the present perspective, a relatively weak exothermic reaction

taking place within the porous material.

Although the problem of Rayleigh-Benard Convection has been extensively

investigated for Newtonian fluids, relatively little attention has been devoted to the

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problem with non-Newtonian fluids. The various non-Newtonian fluids are viscoelastic

fluid, Maxwell fluid, couple stress fluid etc. Recently, non-Newtonian fluids housed in

fluid-based systems, with and without porous matrix, and have been extensively used in

application situations and hence warrant the attention they have been duly getting.

The study of Rayleigh-Benard Convection in viscoelastic fluid may be

important from a rheological point of view. Viscoelastic fluids exhibit unique patterns

of instabilities such as the overstability that is not predicted or observed in Newtonian

flow. There is a great need of investigation to be carried out on the problem of

convective instability in viscoelastic fluids. In the asthenosphere and the deeper mantle,

it is well known that viscoelastic behavior is an important rheological process. The

application areas of viscoelastic fluid saturated porous media are flow through

composites, timber wood, snow systems, and rheology of food transport. The flow of

non-Newtonian fluids in a porous layer is of great interest in different areas of modern

sciences, engineering and technology like material processing, petroleum, chemical and

nuclear industries, geophysics, and bio-mechanics engineering. The performance of a

reservoir depends to a large extent upon the physical nature of crude oil present in the

reservoir. The study of such fluids is based on a generalized Darcy equation which

takes into account the non-Newtonian effects. Such an equation is useful in the study of

mobility control in oil displacement mechanism which improves the efficiency of the

oil recovery. Furthermore some oil sands contain waxy crudes at shallow depth of the

reservoirs which are considered to be viscoelastic fluid. In these situations, a

viscoelastic model of a fluid serves to be more realistic than the Newtonian model.

In recent years the special attention is given to Maxwell and Oldroyd-B fluids,

and these fluids have been applied to a number of disciplines, including biorheology,

geophysics, and chemical and petroleum industries. These investigations have

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demonstrated the usefulness of the Oldroyd-B fluid theory in solving fluid dynamics

problems. Moreover, interest in viscoelastic flows through porous media has also

grown considerably, due to their relevance and importance in diverse areas as like

above disciplines.

The theory of polar fluids has received wider attention in recent years because

the traditional Newtonian fluids cannot precisely describe the characteristics of the

fluid flow with suspended particles. The study of such fluids has applications in a

number of processes that occur in industry, such as the extrusion of polymer fluids,

solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubrication,

and colloidal and suspension solutions. In the category of non-Newtonian fluids couple

stress fluids have distinct features, such as polar effects. The theory of polar fluids and

related theories are models for fluids whose microstructure is mechanically significant.

The exhaustive account of the wide spectrum of applications of buoyancy

driven convection in fluid and fluid-saturated porous layer, with and without additional

constraints, provides the background and motivation for the problems investigated in

the thesis. With motivation directed by applications mentioned above, the main

objective of the thesis is to study double diffusive convection in the presence of

external constraints like chemical reaction, internal heat source, cross diffusions (Soret,

Soret and Dufour effects) in Newtonian and non-Newtonian fluids. The scope of the

present study lies in interpreting and explaining the mechanism of augmenting or

suppressing the convection. The relevant literature survey concerned to the Rayleigh-

Benard convection in single and two component systems with and without external

constraints in both Newtonian and non-Newtonian fluids have been considered in the

next section.

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1.2 Literature Survey

The study of convective instability in a fluid and fluid-saturated porous layer

has been a subject of extensive theoretical and practical importance during last few

decades. Because of the comprehensive importance of thermal convection in various

naturally occurring phenomena as well as in industrial applications, there has been a

great deal of effort by scientists and engineers to study the problem of thermal

convection in fluid and porous medium. The review of the literature pertinent to the

theme of the thesis is as given below.

Double Diffusive Convection

In standard Rayleigh-Benard convection the temperature gradient is the only

driving mechanism for instability. However, both in nature and in physical applications,

there are situations where, in addition to a temperature gradient, there is another,

possibly competing element. For example, in astrophysics this might be a magnetic

field; alternatively, in astrophysics and geophysics the influence of rotation is often

important. Here we shall consider a problem that is very important in oceanography

namely the behaviour of convection with both temperature and salinity (salt) gradients.

This particular problem is known as thermohaline convection or thermosolutal

convection. The key ingredients are two basic state gradients for two quantities (heat

and salt) that both diffuse and do so typically at different rates. For example, heat

diffuses about 80 times faster than salt. In a general context, the problem of convection

with two contributions to the buoyancy, both of which diffuse, is known as Double

diffusive convection.

Double diffusive convection is an important process in oceanography and plays

a role in mantle convection (magma chambers) and some technological applications.

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Double diffusive convection is characterized by well-mixed convecting layers, which

are separated by relatively sharp density steps. These steps may be of the ‘finger’ or

‘diffusive’ kind and both types of interface must enable a net release of potential energy

preferentially transporting the destabilizing property. Salt fingers will occur when

warm salty fluid overlies cooler fresher fluid and diffusive instability will occur when

warm salty fluid underlies the fresh cooler fluid.

1.2.1 Double Diffusive Convection in Fluid Layer

The double diffusive convection is very important in several fields such as high

quality crystal production, liquid gas storage, oceanography, chemical studies, material

sciences, geophysics, atmospheric sciences and so on. Therefore, it has been the subject

of extensive theoretical and experimental interest during the last few years. An

excellent reviews on most of these studies have been reported by Turner (1973, 1974,

1985), Schechter et al. (1974), Huppert and Turner (1981), Platten and Legros (1984),

Schmitt (1994), Whitehead (1995) and Shivakumara (1995).

Veronis (1965) has studied the problem of thermohaline convection in a layer of

fluid heated from below and subjected to a stable salinity gradient. Nield (1967) studied

the problem of thermohaline convection in a horizontal layer of viscous fluid heated

from below and salted from above. Turner and Stommel (1964) demonstrated the

“diffusive-convection” process a few years later. From these beginnings in

oceanography over three decades ago, double diffusion has come to be recognized as an

important convection process in a wide variety of fluid media, including magmas,

metals, and stellar interiors (Schmitt, 1983; Turner, 1985).

The most extensive numerical calculations of the extended Rayleigh-Benard

convection problem have been undertaken by Huppert (1976), and Huppert and Moore

(1976). These authors have studied nonlinear double diffusive convection between two

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long horizontal planes, heated and salted from below, by a combination of perturbation

analysis and direct numerical solution of the governing equations, the possible forms of

large amplitude motion are traced out as a function of the four non-dimensional

parameters which specify the problem: the thermal Rayleigh number, the saline

Rayleigh number, the Prandtl number and the diffusivity ratio.

Rajagopal et al. (2009) have investigated thermal-convection in a fluid with a

viscosity that depends on both the temperature and pressure, within the context of a

generalization of the Oberbeck-Boussinesq approximation. Assuming that the viscosity

is an analytic function of the temperature and pressure. They reported that the linear as

well as the non-linear stability of the problem of Rayleigh-Benard convection and also

shows that the principle of exchange of stability holds and the Rayleigh numbers for the

linear and non-linear stability coincide. In the next section we shall review works on

double diffusive convection in porous layer.

1.2.2 Double Diffusive Convection in Porous Layer

The problem of double diffusive convection in porous media has received much

attention during the last few decades and has practical importance in many fields such

as geophysics, oceanography, ecology, chemistry, and metallurgy. Specific areas of

application range from the flow of groundwater to oil recovery, underground storage of

waste products, food processing, and building insulation. The problem of double-

diffusive convection in a porous medium has been extensively investigated and the

growing volume of work devoted to this area is well documented by Ingham and Pop

(2000, 2005), Nield and Bejan (2006), Vafai (2000, 2005) and Vadasz (2008).

The study of double diffusive convection in porous medium is first under taken

by Nield (1968) on the basis of linear stability theory for various thermal and solutal

boundary conditions. He observed that the stabilizing solute gradients inhibit

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convection and can also introduce the possibility of oscillatory motion called

‘overstability’, when the slower diffusing species has a stabilizing gradient and a faster

diffusing species has a destabilizing gradient (e.g., hot salty water lying below cool

fresh water). He has also predicted that salt fingers within a porous medium are

possible. Then the analysis is extended by Taunton et al. (1972).

The onset of double diffusive convection in a horizontal porous layer has been

investigated by Rudraiah et al. (1982) using a weak nonlinear theory. Brand and

Steinberg (1983a) have studied the instabilities which can occur when a layer of a

mixture of two miscible fluids in a porous medium is heated from below or from above.

Brand and Steinberg (1983b) have investigated the nonlinear effects in the convective

instability of a binary mixture in a porous medium near threshold. The linear stability

analysis of the thermosolutal convection in a sparsely packed porous layer was made by

Poulikakos (1986) using the Darcy–Brinkman model. Rudraiah et al. (1986) have

applied linear and nonlinear stability analysis and showed that sub critical instabilities

are possible in the case of two component fluids. Trevisan and Bejan (1986) studied

heat and mass transfer by natural convection in a vertical slot filled with porous

medium.

Murray and Chen (1989) have carried out an experimental study to examine

double diffusive convection in a porous medium. The onset of convection was detected

by a heat flux sensor and by the temperature distribution in the porous medium. Sheela

(1990) has investigated the onset of double diffusive convection in a composite layer

by considering both the boundaries either rigid / free (without deformation), using the

Darcy-Brinkman equation for the porous layer with the continuity of velocity, shear

stress, normal stress, heat, heat flux, mass and mass flux at the interface. Malashetty

(1993) has investigated the linear stability of the thermodiffusive equilibrium of a

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binary mixture of two miscible fluids in a horizontal plane porous layer. Kaloni and

Guo (1996) have considered a nonlinear double diffusive convection system, which is

the model for incompressible flows through a porous medium in which typically one

substance diffused more rapidly than the other and has a significant influence on

convection adopting a Brinkman-Forchheimer model. Shivakumara and Sumithra

(1999) have investigated the non-Darcian effects on double diffusive convection in an

isotropic porous medium and conditions for both simple and Hopf bifurcation have

been obtained.

Double diffusive convection within a horizontal porous layer is studied both

analytically and numerically by Kalla et al. (2001). The enclosure is heated and cooled

along vertical walls by uniform heat fluxes and a solutal gradient is imposed vertically.

In the formulation of the problem the Darcy model is used and the density variation is

taken into account by the Boussinesq approximation. Now we shall review some of the

important works related to anisotropic porous layer.

1.2.3 Double Diffusive Convection in an Anisotropic Porous Layer

Early studies on convection in a porous medium have usually been concerned

with homogeneous isotropic porous structures. However, in most of the practical

situations the porous layer will be rarely homogeneous and isotropic. In spite of the

practical importance, in contexts varying from fibrous insulating material to

sedimentary rocks, only few studies have been reported on convection in an anisotropic

porous medium uniformly heated from below. The review of research on convective

flow through anisotropic porous media has been excellently documented by McKibbin

(1985, 1992), Storesletten (1998, 2004), and Vasseur and Robillard (1993).

Castinel and Combarnous (1974) have conducted an experimental and

theoretical investigation on the Rayleigh–Benard convection in an anisotropic porous

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medium. They derived the stability criterion and performed experiments to find the

supercritical heat transport and temperature field. Epherre (1977) extended the stability

analysis to a porous medium with anisotropic thermal diffusivity.

The theoretical investigation of convection in an anisotropic porous media is

made by Kvernvold and Tyvand (1979). The effect of anisotropy in mechanical and

thermal properties on thermohaline convection in a porous layer has been analyzed by

Tyvand (1980). Nilsen and Storesletten (1990) presented an analytical study of two

dimensional natural convection in horizontal rectangular channels filled with an

anisotropic porous medium with a linear temperature distribution in the vertical

direction. They derived the critical Rayleigh numbers for the onset of convection and

examined the steady flow patterns at moderately supercritical Rayleigh numbers.

Tyvand and Storesletten (1991) have studied the problem concerning the onset

of convection in an anisotropic porous layer in which the principal axes obliquely

oriented to the gravity vector. As a result, new flow patterns with tilted plane of motion

or tilted lateral cell walls were obtained. It was also found that the critical Rayleigh

number was always reduced when compared with a perpendicular or parallel

orientation of fibers versus boundaries. Chen and Hsu (1991) have analyzed the onset

of thermal convection in a single-component fluid layer bounded above by a rigid wall

and saturating an underlying porous medium whose permeability and thermal

diffusivity are anisotropic and inhomogeneous.

The onset of thermal convection due to heating from below in a system

consisting of a fluid layer overlying a porous layer with anisotropic permeability and

thermal diffusivity is examined by Chen et al. (1991). The effect of anisotropy of

thermal instability in a fluid saturated porous medium subjected to an inclined

temperature gradient of finite magnitude is studied by Parthiban and Patil (1993). Qin

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and Kaloni (1994) have investigated convective instabilities in anisotropic porous

media considering anisotropy in permeability. The effect of anisotropy on the onset of

convection in a horizontal porous layer with constant heat flux is studied by Mamou et

al. (1998). Alex and Patil (2000) carried out an analysis to investigate convective

instability due to centrifugal acceleration in an anisotropic porous medium. Kim et al.

(2001) carried out a numerical study to investigate the thermal characteristics of an

aluminum foam heat sink horizontally placed in a porous channel modeled as a

hydraulically and thermally anisotropic.

Bennacer et al. (2001), and Bennacer and Beji (2002) have investigated a

thermosolutal convection in a two dimensional rectangular cavity filled with saturated

homogeneous porous medium that is thermally anisotropic. Khalili and Huettel (2002)

have carried out convective instability caused by a non-uniform temperature gradient

due to vertical thoroughflow and internal heat generation in an anisotropic porous layer,

using a Darcy-Forchheimer model.

Bennacer et al. (2005) have studied the analytical and numerical investigation of

double diffusive convection in a multilayer anisotropic porous medium. Darcy model

with classical Boussinesq approximation is used in formulating the mathematical

model. An analytical study of natural convection in an anisotropic porous layer

subjected to centrifugal body forces are considered by Govender (2006).

Recently, Malashetty and Swamy (2010) have studied the linear and nonlinear

stability analyses of double diffusive convection in a horizontal anisotropic porous

layer saturated with a Boussinesq fluid heated and salted from below. So far, we have

surveyed the studies concerned to plane Rayleigh-Benard convection in the absence of

external constraints. Now, we shall look into some of the works in the presence of

external constraints in both fluid and porous layer.

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1.2.4 Convective Instability with Chemical Reaction

Thermal convection is considered to be an important and in many practical

cases a major mechanism for the transport and deposition of salts and other chemicals

in sedimentary basins. A variety of chemical reactions can occur as fluid, carrying

various dissolved species, moves through a permeable matrix. The nature of the

resulting dissolution or precipitation depends on the reaction kinetics and the influence

of temperature, pressure, and other factors on them has been studied by Phillips (2009).

The effect of chemical reactions on convective motion is not fully known and has

received relatively little attention. Influence of chemical reaction on the onset of

convection in a porous medium was first introduced by Steinberg and Brand (1983,

1984). Their analysis is restricted to the regime where the reaction rate was sufficiently

fast that the solutal diffusion could be neglected.

Gitterman and Steinberg (1983) studied the onset of convective instability in

binary mixtures with fast chemical reactions. They have presented a linear stability

analysis for chemically driven instabilities in a binary mixture with a chemical reaction

that is fast compared with the diffusive rate. Analytic expressions are found for the

criteria for the onset of stationary and oscillatory convection and oscillatory frequency,

which depend on the rate and the heat of the reaction. Kordylewski and Krajewski

(1984) have paid attention to the interaction of chemical reaction and free convection in

a porous medium. They formulated the problem based on Darcy’s law with the

Boussinesq approximation, assuming that a zero-order exothermic reaction occurs in a

fluid phase and that local thermal equilibrium exists between the fluid and solid phases.

The effect of the Rayleigh number on the critical conditions of thermal ignition was

investigated.

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Gatica et al. (1987) have performed stability analysis of an isothermal first order

and non-isothermal zero order reaction in the presence of free convection. Critical

Rayleigh numbers for both cases were calculated analytically. They found that the

critical Rayleigh number values compared favorably with the numerical simulation of

the full governing equations. Gatica et al. (1989) and Viljoen et al. (1990) have

examined the effect of exothermic-reaction on the stability of the porous system. Their

study is limited to the case where the thermal and solutal diffusivities are equal so that

overdamped oscillations are not possible. Vafai et al. (1993) have found numerical

solution for chemically driven convection in a porous cavity with isothermal wall at the

top and bottom surfaces and thermally insulated sidewalls. Both the inertia and the

viscous forces have been taken into consideration in the momentum equation.

Malashetty et al. (1994) made a linear stability analysis to study the onset of

convective instability in a horizontal inert porous layer saturated with a fluid

undergoing zero-order exothermic chemical reaction. Assuming two different thermal

boundary conditions at the lower boundary, i.e. an isothermal wall and adiabatic wall,

they found that, with chemical reactions, the fluid in the porous medium is more prone

to instability as compared to the case in which chemical reactions are absent.

Linear stability analysis for chemically driven instabilities in binary liquid

mixtures with fast chemical reaction was studied by Malashetty and Gaikwad (2003).

They obtained analytical expressions for the onset of stationary and oscillatory

convection and concluded that both stationary instability and an oscillatory instability

can occur as the first bifurcation, depending on sign and the value of the heat of

reaction. Pritchard and Richardson (2007) have been considered the effect of

temperature dependent solubility on the onset of thermosolutal convection in an

isotropic porous medium. A linear stability analysis was used to investigate how the

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dissolution or precipitation of concentration affects the onset of convection and

selection of an unstable wavenumber. Using a Galerkin method the analysis was further

extended to predict the structure of the initial bifurcation, and they compared analytical

results with numerical integration of the nonlinear equations.

On the basis of Brinkman model, the onset of double-diffusive (thermo-solutal)

convection with a reaction term in a horizontal sparsely packed porous media has been

investigated by Wang and Tan (2009). They have reported that the Darcy number

destabilizes the flow in stationary as well as oscillatory modes, however, effects of the

Lewis number and reaction term depend on the values of the solutal Rayleigh number.

Malashetty and Biradar (2011a) have studied the onset of double diffusive reaction-

convection in an anisotropic porous layer. They have reported that the effect of

chemical reaction as well as anisotropy of the medium may be stabilizing or

destabilizing.

1.2.5 Convective Instability with Internal Heat Source

The effect of internal heat generation is very important in several applications

that include reactor safety analyses, metal waste form development for spent nuclear

fuel, fire and combustion studies, and storage of radioactive materials. Therefore, we

now review some of the literature pertaining to the effect of internal heating on

thermoconvective flow.

Sparrow et al. (1964) have studied analytically the problem of thermal

instability of an internally heated fluid as well as heated from below, with various

boundary conditions and showed that with increasing heat generation rate the fluid is

prone to instability. Thermal convection in a horizontal porous layer with internal heat

sources is studied by Tveitereid (1977). Bejan (1978) studied the natural convection in

an infinite porous medium with a concentrated heat source. Bhattacharya and Jena

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(1984) investigated the effect of a uniform distribution of heat source, which gives rise

to a nonlinear temperature profile in the quiescent state, on the stability of a horizontal

layer of micropolar fluid heated from below. A uniform heat-generation term across an

enclosure with isothermal vertical walls and adiabatic horizontal walls was studied by

Haajizadeh et al. (1984), and Rao and Wang (1991). Khalili and Shivakumara (1998)

have studied the onset of convection in a horizontal porous layer including the effects

of throughflow and a uniformly distributed internal heat generation for different types

of hydrodynamic boundary conditions.

Rionero and Straughan (1990) have analyzed convection in a porous medium

with internal heat source and variable gravity effects. Onset of convection in a fluid

saturating a horizontal layer of an anisotropic porous medium with internal heat source

subjected to inclined temperature gradient was studied by Parthiban and Patil (1997).

Shivakumara and Suma (2000) have investigated the effect of throughflow and constant

internal heat generation on the onset of convection using rigid and perfectly conducting

boundaries. Herron Isom (2001) has studied the onset of convection in a porous

medium with internal heat source and variable gravity. Effects of throughflow and

internal heat generation on convective instabilities in an anisotropic porous layer were

performed by Khalili and Huettel (2002). The experimental investigation of natural

convection induced by internal heat generation has been analyzed by Tasaka et al.

(2005). This paper attempted the calibration by a horizontal temperature gradient in a

horizontal fluid layer with heat conduction, to quantitatively investigate the temperature

field of the internally heated conduction. Hill (2005) studied double-diffusive

convection in a fluid saturated porous layer with a concentration based internal heat

source. An analytic solution for small Rayleigh number in a finite container with

isothermal walls and uniform heat generation within the porous medium has been given

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by Joshi et al. (2006). Magyari et al. (2007a, 2007b) have examined the effect of the

source term on steady free convection boundary layer (large Rayleigh) flows over a

vertical plate in a porous medium.

The instability parameter is either a Darcy-external or Darcy-internal Rayleigh

number which has been determined numerically using Galerkin technique. Influence of

Darcy number on the onset of convection in a porous layer with a uniform heat source

was studied by Borujerdi et al. (2008) and obtained a smooth monotonic variation in

the critical Rayleigh number. Double-diffusive penetrative convection simulated via

internal heating in an anisotropic porous layer with throughflow was studied by Capone

et al. (2011). Bhadauria et al. (2011) have studied the natural convection in a rotating

anisotropic porous layer with internal heat generation using a weak nonlinear analysis.

Bhadauria (2012) analyzed double diffusive natural convection in an anisotropic porous

layer in the presence of an internal heat source using a linear and weak nonlinear

analysis.

1.2.6 Double Diffusive Convection with Cross-Diffusion Effects in

Porous Layer

When heat and mass transfer occur simultaneously in a moving fluid, the

relation between the fluxes and the driving potentials are of more intricate in nature. It

has been found that an energy flux can be generated not only by temperature gradient

but also by composition gradients as well. The energy flux caused by a composition

gradient is called the Dufour or diffusion-thermo effect. On the other hand, mass flux

can also be created by temperature gradient is called the Soret or thermal-diffusion

effect. In gas mixtures the Dufour effect is of a similar magnitude to the Soret

correction. However, in liquids the Dufour effect is comparatively negligible. In many

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studies the Dufour and Soret effects are neglected on the basis that these are second

order effects and are much smaller in magnitude than the effects described by the

Fourier’s and Ficks’ law. However, some recent studies have shown that the Soret and

Dufour effects in fluid flow through porous medium cannot always be neglected.

Double diffusive convection in a fluid-saturated porous media with thermo-

diffusive effect is of practical interest in many engineering applications such as

petrology, hydrology, solidification of binary alloys as well as many other applications.

According to fact that the fluid density depends on solute concentration, it leads to a

competition between thermal and compositional gradients. If the cross-diffusion terms

are included in the species transport equations, then the situation will be quite different.

Owing to the cross-diffusion effects, each property gradient has a significant influence

on the flux of the other property.

An experimental and theoretical study of the Soret driven thermosolutal

convection in a binary fluid mixture has been made by Hurle and Jakeman (1971).

They found that when the water-methanol mixture is heated from below, initially the

oscillatory flow was observed and later it was bifurcated towards the finite amplitude

motion. Oscillatory motions in Benard cell due to the Soret effect has been studied by

Platten and Chavepeyer (1973). They reported that as predicted by the linear theory

oscillatory motions are observed in the two-component system with negative Soret

coefficients and the order of magnitude for the period of oscillations is confirmed by

experiments.

Thermal convection in a binary fluid driven by the Soret and Dufour effects has

been investigated by Knobloch (1980). He has shown that equations are identical to the

thermosolutal problem except for a relation between the thermal and solute Rayleigh

numbers. The double-diffusive convection in a porous medium in the presence of the

21

Soret and Dufour coefficients has been studied by Rudraiah and Malashetty (1986) for

a Darcy porous medium using linear analysis, which was extended to include weak

nonlinear analysis by Rudraiah and Siddheshwar (1998). The effect of temperature

dependent viscosity on double diffusive convection in an anisotropic porous medium in

the presence of the Soret coefficient has been studied by Patil and Subramanian (1992).

Straughan and Hutter (1999) have analyzed the double diffusive convection

with Soret effect in a porous layer using Darcy-Brinkman model. Malashetty and

Gaikwad (2001) have investigated the effect of cross diffusion terms on double

diffusive convection in a porous medium in the presence of horizontal gradients.

Bahloul et al. (2003) have carried out an analytical and numerical study of the double-

diffusive convection in a shallow horizontal porous layer under the influence of the

Soret effect. Natural convection with the Soret effect in a binary fluid saturating a

shallow horizontal porous layer is studied both numerically and analytically by

Bennacer et al. (2003). Bourich et al. (2005) have studied fluid flow, and heat and mass

transfer, induced by Soret-driven thermosolutal convection in a horizontal porous layer

using Brinkman-extended Darcy model.

Mansour et al. (2006) have investigated the multiplicity of solutions induced by

thermosolutal convection in a square porous cavity heated from below and subject to

horizontal solute gradient in the presence of the Soret effect. Charrier-Mojtabi et al.

(2007) present an analytical and numerical stability analysis of the Soret-driven

convection in a porous cavity saturated by a binary fluid. The role of the separation

ratio, characterizing the Soret effect and the normalized porosity are investigated

theoretically and numerically. Mansour et al. (2008) have investigated the Soret effect

on thermosolutal convection developed in a horizontal shallow porous layer salted from

below and subject to cross fluxes of heat. Narayana et al. (2008) have considered the

22

Soret-driven thermosolutal convection induced by inclined thermal and solutal

gradients in a shallow horizontal layer of a porous medium. Mosta (2008) has

addressed the problem of double diffusive convection in a porous layer filled with a

fluid in the presence of temperature gradient (Soret effect) and concentration gradient

(Dufour effect).

Gaikwad et al. (2009a, 2009b) investigated the linear and nonlinear double

diffusive convection in a fluid saturated anisotropic porous layer with the Soret effect

and cross-diffusion effects. Khadiri et al. (2010) have considered numerically the Soret

effect on double diffusive convection in a square porous cavity. Awad et al. (2011)

have studied convection from a semi-finite plate in a fluid saturated porous layer with

cross-diffusion and radiative heat transfer. Malashetty and Biradar (2012) have carried

out an analytical study of linear and nonlinear double diffusive convection in a fluid-

saturated porous layer with the Soret and Dufour effects. So far, we have reviewed

works on the convective instability in Newtonian fluids. We shall review in the next

section works on the convective instability in non-Newtonian fluids.

1.2.7 Convective Instability of Non-Newtonian Fluids and Saturated

Porous Layer

The flow of non-Newtonian fluids is of great interest in different areas of

modern sciences, engineering and technology like material processing, petroleum,

chemical and nuclear industries, carbon dioxide geologic sequestration and bio-

mechanics engineering. Although the problem of Rayleigh-Benard convection has been

extensively investigated for Newtonian fluids, relatively little attention has been

devoted to the thermal convection in viscoelastic fluids (see e.g., Li and Khayat, 2005a

and references there in). The study of Rayleigh-Benard convection in viscoelastic fluid

23

may be important from a rheological point of view because the onset of convection

provides potentially useful techniques to investigate the suitability of a constitutive

model adopted for certain viscoelastic fluids. Some oil sands contain waxy crudes at

shallow depth of the reservoirs which are considered to be viscoelastic fluid. In these

situations, a viscoelastic model of a fluid serves to be more realistic than the Newtonian

model. Viscoelastic fluids exhibit unique patterns of instabilities such as the

overstability.

Herbert (1963) and Green (1968) have analyzed the problem of oscillatory

convection in an ordinary viscoelastic fluid of the Oldroyd type under the condition of

infinitesimal disturbances. The nature of convective motions in a thin horizontal layer

of viscoelastic fluid which is heated from below, in the classical Rayleigh-Benard

convection geometry, has been the subject of discussion in the literature for nearly four

decades (see e.g., Vest and Arpaci (1969), Sokolov and Tanner (1972), Rosenblat

(1986), Martinez-Mardones and Perez-Garcia (1990, 1992), Larson (1992), Khayat

(1994, 1995a, 1995b, 1995c, 1996, 1997, 1999), Martinez-Mardones et al. (1996)).

Rudraiah et al. (1989, 1990) have studied the stability of Viscoelastic fluid

saturated porous layer using Darcy and Brinkman models. The combined effect of non-

uniform temperature gradient and Coriolis acceleration on the thermal convection in

densely packed porous layer saturated with a viscoelastic fluid has been investigated by

Siddheshwar and Srikrishna (2000). Using a single term Galerkin technique,

Siddheshwar and Srikrishna (2001) studied the effect of non-uniform temperature

gradient on the onset of Rayleigh-Benard convection in a viscoelastic fluid saturated

porous layer. Kaloni and Lou (2002) have considered the problem of convection in a

viscoelastic Maxwell fluid. They have investigated the stability characteristics of the

Hadley circulations which occur in the fluid layer.

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A theoretical analysis of thermal instability in a horizontal porous layer

saturated with viscoelastic liquid is carried out by Kim et al. (2003). It is found that the

overstability is a preferred mode for a certain parameter range. They further reported on

the basis of nonlinear theory that the convection has the form of a supercritical and

stable bifurcation independent of the values of the elastic parameters. Yoon et al.

(2004) have examined the onset of thermal convection in a horizontal porous layer

saturated with viscoelastic liquid using linear theory. A simple constitutive model was

employed to examine the effects of relaxation times. It is shown that the oscillatory

instabilities can set in before stationary mode. Kaloni and Lou (2005) used the energy

method to study the viscoelastic fluid convection problem in a thin horizontal layer,

subjected to an applied inclined temperature gradient. Li and Khayat (2005b) analyzed

the influence of inertia and elasticity on the onset and stability of Rayleigh-Benard

thermal convection for highly elastic polymeric solutions with constant viscosity.

Shivakumara et al. (2006) have studied viscoelastic fluid convection in a

sparsely packed horizontal porous layer heated from below using linear stability

analysis. The viscoelastic fluid flow was modeled by using a modified Brinkman-

Lapwood-extended Darcy model with the fluid viscosity different from the effective

viscosity accounting viscoelastic properties and friction due to macroscopic shear.

Malashetty et al. (2006) studied the linear stability of viscoelastic fluid saturated porous

layer using a thermal non-equilibrium model by considering the Oldroyd-B type fluid.

They reported that due to the competition between the processes of viscous relaxation

and thermal diffusion, the first convective instability to be oscillatory rather than

stationary.

Tan and Masuoka (2007) have studied the stability of a horizontal layer of

Maxwell fluid in a porous medium heated from below using the modified-Darcy-

25

Brinkman-Maxwell model and the critical value of the Rayleigh number, wave number

and frequency for overstability at the onset were determined. It was established that the

relaxation time had destabilizing and the porosity parameter had stabilizing effects on

the system. Sheu et al. (2008) investigated the chaotic convection of viscoelastic fluids

in porous media and deduced that the flow behavior may be stationary, periodic, or

chaotic. Zhang et al. (2008) made an excellent analysis of linear and nonlinear thermal

convection in a porous medium saturated with Oldroyd-B fluid using a modified Darcy-

Brinkman-Oldroyd model.

Martinez-Mardones et al. (2000, 2003), Laroze et al. (2005, 2006, 2007a,

2007b) have studied the Rayleigh-Benard convection in binary viscoelastic fluids.

Stability analysis of double diffusive convection of Maxwell fluid in a porous medium

heated from below has been investigated by Wang and Tan (2008). Malashetty et al.

(2009a, 2009b) studied the double diffusive convection in a viscoelastic fluid saturated

isotropic and anisotropic porous layer, respectively. They found that the competition

between the processes of thermal, solute diffusions, and viscoelasticity that causes the

convection to set in through oscillatory rather than stationary mode. Awad et al. (2010)

used the Darcy-Brinkman-Maxwell model to study linear stability analysis of a

Maxwell fluid with cross-diffusion and double-diffusive convection. They found that

the effect of relaxation time is to decrease the critical Darcy-Rayleigh number. Stability

analysis of Soret-driven double diffusive convection of a Maxwell fluid in a porous

medium has been investigated by Wang and Tan (2011). Malashetty and Biradar

(2011b) have studied the onset of double diffusive convection in a binary Maxwell

fluid saturated porous layer with cross diffusion effects. Kumar and Bhadauria (2011)

have studied the effect of rotation on the onset of double diffusive convection in a

horizontal saturated porous layer of a viscoelastic fluid using linear and non-linear

26

analyses. Swamy et al. (2012) have analyzed the onset of Darcy-Brinkman convection

in a binary viscoelastic fluid saturated porous layer. Narayana et al. (2012) have

performed linear and nonlinear stability analysis of binary Maxwell fluid convection in

a porous medium with the Soret and Dufour effects. Now we shall review some of the

works related to couple stress fluids.

The theory of polar fluids has received wider attention in recent years because

the traditional Newtonian fluids cannot precisely describe the characteristics of the

fluid flow encountered in many practical problems such as the extrusion of polymer

fluids, solidification of liquid crystals, cooling of metallic plates in a bath, exotic

lubricants and colloidal fluids and suspension solutions. In the category of non-

Newtonian fluids couple stress fluid has distinct features, such as polar effects and

whose microstructure is mechanically significant. The constitutive equations for couple

stress fluids were given by Stokes (1966). The theory proposed by Stokes is the

simplest one for micro-fluids, which allows polar effects such as the presence of couple

stress, body couple, and non-symmetric tensors.

Sharma and Thakur (2000) investigated the thermal stability of an electrically

conducting couple stress fluid saturated porous layer in the presence of a magnetic

field. They reported that the couple stress postpone the onset of stationary convection.

Sunil et al. (2002) have studied the stability of superposed couple stress fluids in a

porous medium with magnetic effect. They derived a sufficient condition for the non-

existence of overstability. Sharma and Sharma (2004) have studied the onset of

convection in a couple stress fluid saturated porous layer in the presence of rotation and

a magnetic field. Siddheshwar and Pranesh (2004) analytically studied linear and

nonlinear convection in a couple stress fluid layer. The linear and nonlinear double

diffusive convection with Soret effect in couple stress liquids have been considered by

27

Malashetty et al. (2006). Gaikwad et al. (2007) have investigated linear and nonlinear

double diffusive convection in a couple stress liquid by considering both Soret and

Dufour effects.

Shivakumara (2010) has studied onset of convection in a couple stress fluid

saturated porous medium with nonuniform temperature gradients. Malashetty et al.

(2010) examined the double diffusive convection in a couple stress fluid saturated

porous layer using linear and weakly nonlinear theories. Malashetty and Kollur (2011)

investigated the onset of double diffusive convection in a couple-stress fluid saturated

anisotropic porous layer. Malashetty et al. (2012) have analyzed the Soret effect on

double diffusive convection in a Darcy porous medium saturated with couple stress

fluid using linear and nonlinear analyses.

1.3 Plan of Work

The literature survey made above has helped me to take up the problems

investigated in this thesis. The outline of the thesis is as follows.

Chapter 1 is introductory and presents the objective and scope of the thesis. A

detailed survey of the literature relevant to title of the thesis is included.

Chapter 2 deals with the classification of fluids, basic equations, boundary

conditions and dimensionless parameters relevant to the problems are considered.

Chapter 3 explains the linear and nonlinear stability analysis of Darcy-

Brinkman reaction convection in an anisotropic porous layer subjected to chemical

equilibrium on the boundaries is investigated. The Darcy-Brinkman model is employed

for the momentum equation. The results obtained are discussed.

In Chapter 4, the onset of double diffusive reaction convection in an

anisotropic porous layer with internal heat source is studied using both linear and

28

weakly nonlinear stability analyses. The Darcy model for the momentum equation has

been used and results obtained are discussed.

In Chapter 5, onset of double diffusive convection in a Maxwell fluid saturated

anisotropic porous layer with internal heat source is investigated using a linear and

weak nonlinear stability analyses. The modified Darcy-Maxwell model is employed for

the momentum equation. The results obtained are reported.

Chapter 6 presents the onset of double diffusive convection in a binary

viscoelastic fluid saturated porous layer, heated and salted from below in the presence

of the Soret effect is studied using both linear and nonlinear stability analyses. The

modified Darcy law for the viscoelastic fluid of the Oldroyd type is used to model the

momentum equation and results obtained are discussed.

In Chapter 7, Soret effect on Darcy-Brinkman convection in a binary

viscoelastic fluid saturated porous layer is analyzed using both linear and weakly

nonlinear stability analyses. The modified Darcy-Brinkman-Oldroyd model including

the time derivative term is employed for the momentum equation and results obtained

are reported.

The Chapter 8 explains the onset of double diffusive convection in a couple

stress fluid saturated anisotropic porous layer with cross-diffusion effects is studied

using linear stability analysis. The Darcy model that includes the time derivative term is

employed for the momentum equation. The results obtained are discussed.

The general conclusions drawn on the results obtained for the investigation

carried out in the thesis are presented in the Chapter 9.