Chapter 1 Introduction -...

Chapter 1

Introduction

The advancement in the subject of fluid dynamics was started in 1755 when Euler gave

his famous equations of fluid flow for ideal (inviscid) fluids in his paper entitled ”General

principles of the motion of fluids”. Fluid dynamics is a subset of that science that looks

at the materials which are in motion. Hydrodynamics looks specifically at liquids in

motion. Fluid dynamics refers to a subcategory of the science of fluid mechanics, with

the other subcategory being fluid statics, which deals with fluids that are at rest while

fluid dynamics is concerned with fluids that are in motion. Any matter in a gas or liquid

state can be considered as a fluid. Fluid dynamics is governed by the laws of conservation

which states that the total amount of energy, mass, and linear momentum in a closed

system remain constant, and that energy and mass can neither be created nor destroyed.

They may change forms but cannot disappear. This law constitute some of the most basic

assumptions in science. Another governing principle of fluid dynamics is the continuum

assumption, also called the continuum hypothesis. Fluids are known to be composed of

microscopic, discret particles, thus this hypothesis states that they are continuous, and

that their properties vary evenly throughout. The history of fluid dynamics can be found

in Rouse and Ince (1957) and Tokaty (1971). Anderson (1997) presented the history of

both fluid dynamics and aerodynamics.

1.1 Scope and Motivation

The stretching surface in a quiescent or moving fluid is important in number of industrial

manufacturing processes that includes both metal and polymer sheets. An interesting

fluid mechanical application is found in polymer extrusion processes, where the object

8

CHAPTER 1. INTRODUCTION 9

on passing between two closely placed vertical solid blocks is stretched in a region of

fluid-saturated porous medium. The stretching imparts a unidirectional orientation to

the extrudate, thereby improving its mechanical properties. The liquid is meant to cool

the stretching sheet whose property depends greatly on the rate at which it is cooled and

stretched in porous medium. The fluid mechanical properties desired for the outcome

of such a process depends mainly on the rate of cooling and the stretching rate. It is

important that proper cooing fluid is chosen and flow of the cooling liquid caused due

to the stretching sheet can be controlled so as to arrive to the desired properties for the

outcome. As a result, one has to pay considerable amount of attention for both flow and

heat transfer characteristic of the cooling fluid. The quality of the final product depends

on the rate of heat transfer at the stretching surface. The temperature distribution, thick-

ness and width reduction are function of draw ratio and stretching distance. It is worth

mentioning that there are several practical applications in which significant temperature

differences between the body surface and the ambient fluid exist. The temperature dif-

ferences cause density gradients in the fluid medium and free convection effects become

more important in the presence of gravitational force. There arise some situations where

the stretching sheet moves vertically in the cooling liquid. In this situation, the fluid

flow and the heat transfer characteristic are determined by two mechanisms namely, the

motion of stretching sheet and the buoyancy force.

Convection heat transfer and fluid flow through porous medium is a phenomenon of

great interest from both theoretical and practical point of view because of its applications

in many engineering and geophysical fields such as geothermal and petroleum resources,

solid matrix heat exchanges, thermal insulation drying of porous solids, enhanced oil

recovery, cooling of nuclear reactors and other practical interesting designs. The possible

use of porous media adjacent to surfaces to enhance heat transfer characteristics have

lead to extensive research in heat transfer and flows over bodies embedded in a porous

media. Physically, the problem of mixed convection flow past a stretching sheet embedded

in a porous medium arise in some metallurgical processes which involve the cooling of

continuous strips or filaments by drawing them through quiescent fluid and the rate of

cooling can be better controlled and final product of desired characteristics can be achieved

if the strips are drawn through porous media.

A new dimension is added to the study of mixed convection flow past a stretching

sheet embedded in a porous medium by considering the effect of thermal radiation. Ra-

diative heat transfer flow is very important in manufacturing industries for the design


of reliable equipments, nuclear plants, gas turbines and various propulsion devices for

aircraft, missiles, satellites and space vehicles. Also, the effect of thermal radiation on

the forced and free convection flows are important in the content of space technology and

processes involving high temperature. Thermal radiation effect plays a significant role in

controlling heat transfer process in polymer processing industry. The quality of the final

product depends to a certain extent on heat controlling factors. Also, the effect of thermal

radiation on flow and heat transfer processes is of major important in the design of many

advanced energy convection systems which operate at high temperature. Thermal radia-

tion occurring within these systems is usually the result of emission by the hot walls and

the working fluid. Thermal radiation effects become more important when the difference

between the surface and the ambient temperature is large. Thus thermal radiation is one

of the vital factors controlling the heat and mass transfer. Another important effect of

considering thermal radiation is to enhance the thermal diffusivity of the cooling liquid

in the stretching sheet problem. Thus the knowledge of radiation heat transfer in the

system can perhaps lead to a desired product with sought characteristics.

The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid

in the boundary layer flow due to the stretching of the sheet is of considerable interest

in modern metallurgical and metal-working process. Many metallic materials are man-

ufactured after they have been refined sufficiently in the molten state. Therefore, it is

a central problem in metallurgical chemistry to study the heat transfer on liquid metal

which is perfect electric conductor. Thus a careful examination of the needs in the sys-

tem suggests that it is advantageous to have a controlled cooling system. An electrically

conducting liquid can be regulated by external means through a variable magnetic field.

The problem is a prototype for many other practical problems also, similar to those of

polymer extrusion processes.

The thermal-diffusion and diffusion-thermo effects are interesting macroscopically phys-

ical phenomenon in fluid mechanics. The heat transfer caused by concentration gradient

is called the diffusion-thermo or Dufour effect. On the other hand, mass transfer caused

by temperature gradients is called Soret or thermal diffusion effect. Thus Soret effect is

referred to species differentiation developing in an initial homogeneous mixture submitted

to a thermal gradient and the Dufour effect referred to the heat flux produced by a con-

centration gradient. Usually, in heat and mass transfer problems the variation of density

with temperature and concentration give rise to a combined buoyancy force under natu-

ral convection. The heat and mass transfer simultaneously affect each other that create


cross-diffusion effect. Soret and Dufour effects have been found to appreciably influence

the flow field in mixed convection boundary layer over a vertical surface embedded in a

porous medium.

In many practical applications mass transfer takes place by diffusive operations which

involve the molecular diffusion of species in the presence of two types of chemical reac-

tions namely, homogeneous and heterogeneous. The diffusive species can be generated

or absorbed due to different types of chemical reaction with the ambient fluid which can

greatly affect the properties and quality of finished products. Thus the study of heat

and mass transfer in the presence of chemical reaction is of great practical importance to

engineers and scientists in many branches of science and engineering.

In view of the above mentioned applications, it is important to study the boundary

layer viscous flow over moving surface in porous media with heat and mass transfer con-

sidering various aspects of the physical properties of the fluid in the Ph.D. thesis work.

Numerical methods is employed to solve the momentum, energy and mass-diffusion equa-

tions by considering effects of thermal radiation, variable viscosity and thermal conduc-

tivity, buoyancy force, magnetic and electric field, viscous dissipation and Ohmic heating,

non-uniform heat source/sink, non-Darcy (or second order quadratic drag) effects in the

porous medium. Also, two different types of boundary conditions for heat transfer anal-

ysis, namely the prescribed surface temperature (PST) and the prescribed surface heat

flux (PHF) conditions are considered for the problems investigated in the thesis, more

emphasis is given on the effects of thermal radiation on heat and mass transfer problems

in porous medium of constant porosity and on related work with Soret and Dufour effects

with first-order chemical reaction.

Motivated by the above studies, in the present study emphasis is given on MHD

convective heat and mass transfer from a vertical stretching sheet embedded in a porous

medium considering Darcy and Darcy-Forchheimer-Brinkman flow models. The flow is

subjected to a transverse magnetic field normal to the plate. The problem addressed here

is a fundamental one that arise in many practical situations such as polymer extrusion

process. Highly non-linear momentum and heat transfer equations are solved numerically

using fifth-order Runge-Kutta Fehlberg method with shooting technique. The effects of

various parameters on the velocity and temperature profiles as well as on local skin-friction

co-efficient and local Nusselt number are presented graphically and in tabulated form. The

effect of thermal radiation, variable viscosity, viscous dissipation and Ohmic heating on

MHD non-Darcy mass diffusion of species over a continuous stretching sheet with electric


and magnetic fields subject to a transverse magnetic field normal to the plate would

also be taken up. Two different types of boundary conditions for heat transfer analysis,

namely the prescribed surface temperature (PST) and the prescribed surface heat flux

(PHF) conditions are considered.

It is also important to study the unsteady two-dimensional MHD non-Darcian mixed

convection heat and mass transfer past a semi-infinite vertical permeable plate embedded

in a porous medium in the presence of Soret and Dufour effects with suction or injection,

thermal radiation and first-order chemical reaction. The problems are important in many

practical situations such as polymer extrusion process and would also be useful in magnetic

material processing and chemical engineering systems.

1.2 Classification of Fluids

Fluids are in general classified in the following categories:

1.2.1 Ideal and Real Fluids

The ideal fluids are those which are incapable of sustaining any tangential force (shearing

stresses) or action in the form of shear but the normal force (pressure) acts between

the adjoining layers of fluid. This means that an ideal fluid offers no internal resistance

to change its shape. Ideal fluids are also known as inviscid fluids or perfect fluids or

frictionless fluids. Those fluids which have low viscosity such as air, water etc. may be

treated as ideal fluids.

Real fluids are known as viscous fluids. A fluid is said to be viscous when normal as

well as shearing stresses exist. Due to shearing stress a viscous fluid offers resistance to

the body moving through it as well as between the particles of the fluid itself. Heavy oils

and syrup may be treated as viscous fluids. Water and air flow much easier than syrup

and heavy oil which demonstrate the existence of a property of the fluid which controls

the rate of fluid flow. This property of fluids is known as viscosity or internal friction.

Viscous fluids when compared with ideal fluids may be characterized by the following two

properties:

(i) When a viscous fluid flows along a well it adheres to the wall i.e., the layer of fluids

is in immediate contact with the wall and has no velocity relative to it,

(ii) shearing stresses arises whenever the fluid properties are distorted. The viscosity,


which is also known as an internal friction, of a fluid is that characteristic of the real fluid

which is capable to offer resistance to shearing stress.

1.2.2 Newtonian and Non-Newtonian Fluids

A fluid in which the components of the stress tensor are linear functions of the first spatial

derivatives of the velocity components. These functions involve two material parameters

taken as constants throughout the fluid, although depending on ambient temperature and

pressure. The constant ratio of the shearing stress τ to the rate of shear is the viscosity

of the liquid i.e., τ = µ∂u∂y, when µ is the viscosity of the fluid. In common terms, this

means the fluid continues to flow, regardless of the forces acting on it. For example, water

is Newtonian, because it continues to exemplify fluid properties no matter how fast it is

stirred or mixed.

If the fluid viscosity varies with the rate of deformation, then it is said to Non-

Newtonian fluid. Non-Newtonian fluids are characterized by different features, such as

viscosity and elasticity for example, the viscosity of polymeric liquids changes with the

shear rate, so it is known as non-Newtonian fluids. Thus Non-Newtonian fluids are those

in which there is no shear stress and there exists a non-linear relation between τ and ∂u∂y.

1.3 Types of flows

1.3.1 Steady and Unsteady Flows

If a flow is such that the properties at every point in the flow do not depend upon time then

it is called a steady flow. Mathematically, for steady flows ∂P∂t

= 0, where P = P (x, y, z)

is any property like pressure, velocity or density. Unsteady or non-steady flow is one in

which the properties do depend on time.

1.3.2 Laminar and Turbulent Flows

Laminar flow is referred to as streamline or viscous flow. In laminar flow, (i) layers of

fluid flowing over one another at different speeds with virtually no mixing between layers,

(ii) fluid particles move in a definite path or streamlines, and (iii) viscosity of the fluid

plays a significant role.


Turbulent flow is characterized by the irregular movement of particles of the fluid.

The particles travel in irregular paths with no observable pattern and no definite layers.

1.3.3 Compressible and Incompressible Flows

A compressible fluid is one in which the fluid density changes when it is subjected to

high pressure-gradients. For gasses, changes in density are accompanied by changes in

temperature which complicates the analysis of the compressible flow. In a compressible

fluid, the imposition of a force at one end of a system does not result in an immediate flow

throughout the system. Instead, the fluid compresses near where the force was applied,

i.e., its density increases locally in response to the force. The compressed fluid expands

against neighbouring fluid particles causing the neighbouring fluid itself to compress and

setting in motion a wave pulse that travels throughout the system.

An incompressible fluid is one in which the fluid density does not change with pres-

sure. Liquid and gas may be modeled as incompressible fluids in both microscopic and

macroscopic calculations.

1.3.4 Viscosity of Fluids

A real fluid flowing in a pipe experiences frictional forces due to friction with the walls of

the pipe which results in friction within the fluid itself and there by converting some of

its kinetic energy into thermal energy. The frictional forces that try to prevent different

layers of fluid from sliding past each other are called viscous forces. Viscosity is a measure

of a fluid resistance to relative motion within the fluid. We can measure the viscosity of

a fluid by measuring the viscous drag between two plates. The viscosity of fluids depend

strongly on temperature. The viscosity of a liquid decreases with increasing temperature

and viscosity of liquid increases with increasing temperature. In liquids viscosity is due

to the cohesive forces between the molecules and in gases the viscosity is due to collisions

between the molecules. If the viscosity is a constant, independent of flow speed, then the

fluid is termed a Newtonian fluid.

1.4 Flow Through Porous Media

Studies on flow through porous media has attracted considerable research attention in

recent years because of its several important applications notably in the flow through


packed beds, extraction of energy from the geothermal regions, filtration of solids from

liquids, flow of liquids through ion-exchange beds, the evaluation of the capability of heat

removal from particulate nuclear fuel debris that may result from accident in a nuclear

reactor and in chemical reactors for economical separation or purification of mixtures.

A porous medium of volume is a fixed solid matrix with a connected void space

through which a fluid can flow or consists of solid particles (which are deformable or non-

deformable) so that fluid can flow through voids and passages. Let Vv be the volume of

voids. When fluid flows through the interconnected voids and passages of a porous medium

V , the walls of these voids and passages from small tunnels through which fluid can flow.

The study of motion of fluid in a porous medium on pore scale is called microscopic scale.

The study of fluid flow at microscopic scale is complicated and unrealistic because of the

complexity of the micro-geometry of porous media. A more realistic approach to study

dynamics of flow through porous media is under the assumption of continuum macroscopic

phenomena. Usually the spacial averages are used to transfer properties of porous media

from microscopic scale to macroscopic scale. Therefore, the definition of porosity and

permeability is essential.

1.4.1 Porosity

Most important geometrical property of the porous media is to porosity. The rheological

properties of fluids often change with the geometry, of the porous medium, thus it is

important to measure the porosity. Porosity is defined as the percentage of a volume

of medium that is empty space that contributes to the fluid flow. Mathematically it is

the ratio between the unit volume of void space Vv to the total volume containing both

fluid and solid which may be either sphere or cube. Then the porosity, ϵ, of such porous

medium is defined as

ϵ =void volume

total volume=VvV

(1.1)

where 0 < ϵ < 1. If Vv = V then it is the case for free fluid.

1.4.2 Permeability

Flow through a porous medium in the macroscopic continuum approach is described by

the Darcy’s law. For an anisotropic porous medium Darcy’s law can be expressed as

qi = −Kij∂h

∂xi(1.2)


where qi (i = 1, 2, 3) is the Darcy velocity, Kij, a tensor, the hydraulic conductivity of

porous media and h is the water head at a point xi which depends on the pressure p and

density ρ and is a macroscopic quantity. for an isotropic porous medium, Kij reduces to

a scalar K and then the Darcy law, given by Eq. (1.2), becomes

qi = −K ∂h

∂xi, (i = 1, 2, 3) (1.3)

The hydraulic conductivity κ of the porous medium depends on the properties of both

solid and fluid aspect of porous media and given by

κ =kρg

µ(1.4)

where k is the permeability having dimension of (Length)2, g is the gravity and ρ is the

density and µ is the viscosity. Thus permeability measures quantitatively the ability of

the porous medium to permit fluid flow.

1.4.3 Darcy’s Law

In fluid dynamics and hydrology, the observation of Henry Darcy (1856) on the public

water supply at Dijon and experiments on steady flow suggested Darcy’s law which is

analogous to Fourier’s law in the field of heat conduction, or Fick’s law in diffusion theory.

−→q =κ

µ∇P (1.5)

where −→q is the filtration velocity or Darcy flux and ∇P is the pressure gradient vector.

This value of the filtration velocity (Darcy flux), is not the velocity which the water

travelling through the pores is experiencing. The porosity ϵ is very small in a densely

packed porous medium. The usual Darcy equation is valid in a densely packed porous

medium saturated with laminar flow, which is written as

−→q = −kµ(∇p+ ρ−→g ). (1.6)

Under the following two approximations, the basic equations of motion in porous media

are valid:

(i)The saturated porous medium is homogeneous and isotropic so that the porosity

and permeability are constant. The porous medium is assumed to consist of sparsely

distributed particles so that viscous shear and inertial effects play an important role in

addition to Darcy resistance.


(ii) The usual MHD approximations are valid even in flow through porous media.

Limitations: Darcy model takes into account of the frictional force which is offered

due to the presence of solid particles to the fluid rather than the boundary and internal

effects.

1.4.4 Brinkman Model

Henry P. G. Darcy (1803–1858), Director of public works in Dijon has worked on the

design and execution of a municipal water supply system. He discovered a law known as

’Darcy law’ which states that as the rate of flow is proportional to pressure drop through

a bed of fine particles. It is Mathematically expressed as

−→Q = −κ

µ

dP

dx(1.7)

where κ represents the permeability of the material, Q is a volumetric flow rate per unit

cross-sectional area. The total effect, as the fluid slowly percolates through the porous of

the medium, must be represented by a macroscopic law which is applicable to masses of

fluid large compared with the dimensions of the porous structure of a medium (Lapwood,

1948), which is the basic of Darcy law. One of the approximate boundary layer type

of equations in a porous medium is the Brinkman model. Brinkman model consists of

viscous term ν∇2−→q in addition to the Darcy resistance term (µ/κ)−→q in the momentum

equation.

1.4.5 Darcy-Forchheimer Model

Darcy equation (1.6) is linear in the seepage velocity −→q . It holds when −→q is sufficiently

small which means that the Reynolds number of the flow is based on a typical pore or

particle diameter is of order unity or smaller. As −→q increases, the transition as Reynolds

number is increased in the range 1 to 10 so that the flow in the pores is still laminar.

Thus the breakdown is linearity is due to the fact that the drag is formed due to solid

obstacles which is comparable with the surface drag due to friction of the form ρCb√b|−→q |−→q

in addition to linear drag µκ−→q . Thus according to Joseph et al. (1982) the appropriate

modification to Darcy’s equation is to replace (1.6) by

∂−→q∂t

= −∇p− µ

κ−→q − cbk

−1/2ρ|−→q |−→q + ρ−→q (1.8)


where cb is the drag coefficient and other quantities have the same meaning as defined

earlier Eq. (1.8) is known as Darcy-Forchheimer equation and the quadratic drag term in

Eq. (1.8) represents inertia.

1.4.6 Darcy-Lapwood-Forchheimer Model

In a densely packed porous medium of large velocity, inertia term ρ(−→q .∇)−→q is added to

the Darcy-Forchheimer equation (1.8). It is called Darcy-Lapwood-Forchheimer equation

given by

ρ(∂−→q∂t

+ (−→q .∇)−→q)= −∇p− µ

k−→q − ρCb√

k|−→q |−→q + ρ−→g . (1.9)

The term ρ(−→q .∇)−→q was first considered by Lapwood (1948). In the case of sparsely

packed porous media the porosity, ϵ, is large but less than unity so one has to take into

account of boundary large effect.

1.4.7 Darcy-Lapwood-Forchheimer Brinkman Model

Brinkman (1947) was the first to propose the momentum equation with boundary layer

effect, known as Darcy-Lapwood-Forchheimer-Brinkman equation namely,

ρ(∂−→q∂t

+ (−→q .∇)−→q)= −∇p− µ

k−→q − ρCb√

k|−→q |−→q − µ̃∇2−→q + ρ−→g . (1.10)

where µ̃ is called effective viscosity or Brinkman viscosity and all other quantities are

defined earlier.

1.5 Stretching Sheet Flow

The flow produced due to stretching of an elastic flat sheet which moves in its plane with

velocity varying with the distance from a fixed point due to the application of a stress

are known as stretching flow. The production of sheeting material arises in a number of

industrial manufacturing processes and includes both metal and polymer sheets. In the

manufacturing of the latter, the material is in a molten phase when thrust through an

extrusion die and then cools and solidifies some distance away from the die before arriving

at the cooling stage. The tangential velocity imported by the sheet induces motion in

the surrounding fluid, which alters the convection of the sheet. Similar situation prevails

during the manufacture of plastic and rubber sheets where it is often necessary to blow

a gaseous medium through the not yet solidified material, and where the stretching force


depends upon time. Another example that belongs to this class of problems is the cooling

of a large metallic plate in a bath, which may be an electrolyte. In this class the fluid flow

is induced due to shrinking of the plate. Glass blowing, continuous casting and spinning

of fibers also involve the flow due to stretching surface. Due to very high viscosity of the

fluid near the sheet, one can assume that the fluid is affected by the sheet but not vice

versa.

1.6 Boundary Layer Flow

At the beginning of the 20th century L. Prandtl has given a new dimension to fluid

mechanics by introducing viscosity in the fluid. It was noted by him that in the thin region

near the solid boundary, the viscous interactions have a significant effects on fluid motion,

whereas far away from the solid boundary, viscous interactions were not that significant

in order to determine the flow field. Before this the viscosity effects were completely

ignored in ideal flow solutions and the equations describing viscous interaction were very

complex. The Navier-Stokes equations behave well for small Reynold’s number whereas

for higher values of Reynold’s number the non-linear term are insignificant. The flow

past a body can be divided into a thin region very near to the body called the boundary

layer where the viscosity is important and the remaining region where the viscosity is

insignificant. These equations are highly non-linear, second order and elliptic in space so

there arises great mathematical difficulties in the solution of the boundary layer equations.

By assuming a thin boundary layer, several terms are negligible and the elliptic equation

become parabolic.

1.7 Magnetohydrodynamic (MHD) Flow

It is concerned with the study of the motions of electrically conducting fluids and their

interactions with magnetic fields. Magnetohydrodynamics (MHD) is relatively new and

important branch of fluid dynamics. When a conducting fluid moves through a magnetic

field, an electric field and consequently current may be induced and in turn the current

interacts with the magnetic field to produce a body force. According to Faraday, when

a conductor carrying an electric current moves in a magnetic field, it experiences a force

tending to move it at right angles to the electric field and conversely, when a conductor

moves in a magnetic field, a current is induced in the conductor in a direction mutually at


right angles to both the field and the direction of motion. In the case when the conductor

is electromagnetic forces of the same order of magnitude as the hydrodynamical and

inertial forces. Thus these electromagnetic forces are taken into account in the equation

of motion in addition to other forces. The set of equations which describe MHD are a

combination of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of

electromagnetism. These differential equations are required to be solved simultaneously.

The interaction of moving conducting fluids with electric and magnetic fields provides for

a rich variety of phenomena associated with electro-fluid-mechanical energy conversion.

Effects from such interactions can be observed in liquids, gases, two-phase mixtures, or

plasmas. Numerous scientific and technical applications exist, such as heating and flow

control in metals processing, power generation from two-phase mixtures or seeded high

temperature gases, magnetic confinement of high-temperature plasmas even dynamos that

create magnetic fields in planetary bodies. Several terms have been applied to the broad

field of electromagnetic effects in conducting fluids, such as magneto-fluid mechanics,

magneto-gas-dynamics, and the more common one used here magnetohydrodynamics, or

MHD.

1.8 Convection

Convection is the movement of molecules within fluids (i.e. liquids, gases). It cannot

take place in solids, since neither bulk current flows nor significant diffusion can take

place in solids. Convection is one of the major modes of heat transfer and mass transfer.

Convective heat and mass transfer take place through both diffusion the random Brownian

motion of individual particles in the fluid and by advection, in which matter or heat is

transported by the larger-scale motion of currents in the fluid. In the context of heat and

mass transfer, the term ”convection” is used to refer to the sum of advective and diffusive

transfer. Convection also includes fluid movement both by bulk motion (advection) and

by the motion of individual particles (diffusion). However in some cases, convection is

taken to mean only advective phenomena. For instance, in the transport equation, which

describes a number of different transport phenomena, terms are separated into convective

and diffusive effects. Convective heat transfer is a mechanism of heat transfer occurring

because of bulk motion (observable movement) of fluids. Heat is the entity of interest

being advected (carried), and diffused (dispersed).


1.8.1 Natural Convection

Natural convection, or free convection, occurs due to temperature differences which affect

the density, and thus relative buoyancy, of the fluid. Heavier (more dense) components will

fall while lighter (less dense) components rise, leading to bulk fluid movement. Natural

convection occurs, only in a gravitational field. Natural convection is more likely and/or

more rapid with a greater variation in density between the two fluids and a larger distance

through the convecting medium. Convection will be less rapid with more rapid diffusion

(thereby diffusing away the gradient that is causing the convection) and a more viscous

(sticky) fluid.

1.8.2 Forced Convection

When the density difference is created by some means due to which circulation takes place

then it is known as forced convection. In forced convection, also called heat advection, fluid

movement results from external surface forces such as a fan or pump. Forced convection

is typically used to increase the rate of heat exchange. Many types of mixing also utilize

forced convection to distribute one substance within another. Forced convection also

occurs as a by-product to other processes, such as the action of forced convection may

produce results more quickly than free convection. For instance, a convection oven works

by forced convection, as a fan which rapidly circulates hot air forces heat into food faster

than would naturally happen due to simple heating without the fan.

1.9 Conduction

Conduction is the transfer of heat between two bodies or two parts of the same body

through molecules. This type of heat transfer is governed by Fourier’s Law which states

that Rate of heat transfer is linearly proportional to the temperature gradient. For 1-D

heat conduction

qk = −kdTdx. (1.11)

1.9.1 Thermal Conductivity

Thermal conductivity, κ, is the property of a material that indicates its ability to conduct

heat. It appears primarily in Fourier’s Law for heat conduction. Conduction is the

most significant means of heat transfer in a solid. By knowing the values of thermal


conductivities of various materials, one can compare how well they are able to conduct

heat. The higher the value of thermal conductivity, the better the material is at conducting

heat. On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms

and molecules interact with neighbouring atoms and molecules, transferring some of their

energy (heat) to these neighboring atoms. In insulators the heat flux is carried almost

entirely by phonon vibrations.

1.9.2 Thermal Radiation

Thermal radiation is electromagnetic radiation from an object that is simply caused by its

temperature. It rapidly increases in power, and also increases in frequency, with increasing

temperature. A black body is an object that absorbs all electromagnetic radiation that

falls onto it. For this case there are formulas for the power as a function of temperature,

etc. For example, spacecraft may have thermal radiators, also called heat radiators to

lose excess heat. They tend to be reflective to avoid absorption of solar radiation energy.

e.g. the space shuttle has heat radiators mounted on the inner surfaces of the payload

bay doors, and so are kept open while the Shuttle is in orbit. Examples of thermal

radiation are an incandescent light bulb emitting visible-light, infrared radiation emitted

by a common household radiator or electric heater, as well as radiation from hot gas in

outer space. A person near a raging bonfire feels the radiated energy of the fire, even

if the surrounding air is very cold. Thermal radiation is generated when thermal energy

is converted to electromagnetic radiation by the movement of the charges of electrons

and protons in the material. Sunlight is solar electromagnetic radiation generated by

the hot plasma of the Sun, and this thermal radiation heats the Earth by the reverse

process of absorption, generating kinetic, thermal energy in electrons and atomic nuclei.

The Earth also emits thermal radiation, but at a much lower intensity and different

spectral distribution because it is cooler. The balance between heating by incoming

solar radiation and cooling by the Earth’s outgoing radiation is the primary process that

determines Earth’s overall temperature. If a radiation-emitting object meets the physical

characteristics of a black body in thermodynamic equilibrium, the radiation is called

black body radiation. The emitted frequency spectrum of the black body radiation is

described by a probability distribution depending only on temperature given by Planck’s

law of black-body radiation. Wien’s displacement law gives the most likely frequency of

the emitted radiation, and the Stefan-Boltzmann law determines the radiant intensity.


In engineering, thermal radiation is considered one of the fundamental methods of heat

transfer, although it does not involve the transport of heat. The characteristics of thermal

radiation depends on various properties of the surface it is emanating from, including

its temperature, its spectral absorptivity and spectral emissive power, as expressed by

Kirchhoff’s law.

1.10 Literature Review

1.10.1 Effects of Mixed Convection in a Porous Medium

The mixed (combined forced and free) convection arises in many natural and technological

processes, depending on the forced flow direction, the buoyancy forces may aid (aiding

or assisting mixed convection) or oppose (opposing mixed convection) the forced flow,

causing an increase or decrease in heat transfer rates. The problem of mixed convection

resulting from the flow over a heated vertical plate is of considerable theoretical and

practical interest. Convection heat transfer and fluid flow through porous medium is a

phenomenon of great interest from both theoretical and practical point of view because

of its applications in many engineering and geophysical fields such as geothermal and

petroleum resources, solid matrix heat exchanges, thermal insulation drying of porous

solids, enhanced oil recovery, cooling of nuclear reactors and other practical interesting

designs. The possible use of porous media adjacent to surfaces to enhance heat transfer

characteristics have lead to extensive research in heat transfer and flows over bodies

embedded in a porous media. Physically, the problem of mixed convection flow past

a stretching sheet embedded in porous medium arise in some metallurgical processes

which involve the cooling of continuous strips or filaments by drawing them through

quiescent fluid and the rate of cooling can be better controlled and final product of

desired characteristics can be achieved if the strips are drawn through porous media. The

possibility of obtaining similarity solutions for mixed convection boundary-layer was first

considered by Sparrow et al. (1959) who showed that the boundary-layer equations could

be reduced to a system of ordinary differential equations. Sparrow and Lee (1976) were

the first to study the problem of mixed convection boundary layer flow about a horizontal

circular cylinder. In view of this, Cheng and Minkowycz (1977) presented similarity

solutions for free thermal convection from a vertical plate embedded in a fluid-saturated

porous medium for situations where the wall temperature is a power-law function of the


distance along the plate. Rudraiah and Veerabhadraiah (1978) studied effect of buoyancy

force on the free surface flow past a permeable bed. Bejan and Khair (1985) studied the

heat and mass transfer by natural convection in a porous medium. Lai (1991) investigated

coupled heat and mass transfer by mixed convection from an isothermal vertical plate in

a porous medium. The effect of wall fluid blowing on the coupled heat and mass transfer

boundary layer flow over a vertical plate was investigated by Lai and Kulacki (1991).

Comprehensive reviews of the convection through porous media have been reported

by Nield and Bejan (1992) and by Ingham and Pop (1998). Bejan et al. (1995) analyzed

the nonsimilar solutions for mixed convection on a wedge embedded in a porous medium.

Gorla and Kumari (1996) studied the mixed convection in non-Newtonian fluids along a

vertical plate in a porous medium. Yih (1998) studied the uniform lateral mass flux effect

on natural convection of non-Newtonian fluids over a cone in a porous media. Kumari

et al. (2000) studied the mixed convection flow over a vertical wedge embedded in a

highly porosity porous medium. Merkin and Pop (2002) obtained similarity solutions of

mixed convection boundary-layer flow over a vertical semi-infinite flat plate in which the

free stream velocity is uniform and the wall temperature in inversely proportional to the

distance along the plate. Aly et al. (2003) examined the mixed convection boundary-

layer flow over a vertical surface embedded in a porous medium. Rudraiah et al. (2003)

presented a review work on nonlinear convection in porous media. In (2006), Guedda

studied the multiple solutions of mixed convection boundary-layer approximations in a

porous medium. Shivakumara et al. (2006) investigated onset of surface-tension-driven

convection in superposed layers of fluid and saturated porus medium. Ling et al. (2007)

have studied the numerical solutions of steady mixed convection boundary layer flow over

a vertical impermeable flat plate in a porous medium saturated with water at 40C when

the temperature of the plate varies as xm and the velocity outside boundary varies as

x2m, where x measures the distance from leading edge of the plate. Ishak et al. (2008)

presented the problem of mixed convection boundary layer flow over a vertical surface

embedded in a thermally stratified porous medium assuming that the external velocity

and surface temperature to vary as xm, where x is measured from the leading edge of

the vertical surface and m is a constant. Shivakumara et al. (2009) analyzed natural

convection in a vertical cylindrical annulus using a non-Darcy equation.

Most of the earlier studies on porous media have used the Darcy’s law which states

that the volume averaged velocity is proportional to the pressure gradient and is limited

to relatively low velocities and small porosity. However, for relatively high velocity flow


situations, the Darcy’s law is inadequate for representing the flow behavior correctly since

it does not account for the resulting inertia effects of the porous medium. In this situation,

the pressure drop has a quadratic relationship with the volumetric flow rate. The high flow

situations is established when the Reynolds number based on the pore size is greater than

unity. Thus to model a real physical situation such as a non-uniform porosity distribution,

it is, therefore, necessary to include the non-Darcian effects in the analysis of convective

transport in a porous medium. The inertia effect is expected to be important at higher

flow rate and it can be accounted for through the addition of a velocity-squared term in

the momentum equation, which is known as the Forchheimer extension. The Brinkman

extension is usually used to shed light on the importance of boundary effects. Brinkman

(1947, 1948) combines the viscous penetration dominated flow (Stoke’s flow) with the

Darcy flow. These non-Darcian effects include nonuniform porosity distribution and ther-

mal dispersion. Vafai and Tien (1981) arrive at a semi-empirical momentum equation.

Vafai and Tien (1982) discussed the importance of these two effects in flows over surfaces

embedded in a porous media. The Darcy-Forchheimer (DF) model is probably the most

popular modification to Darcian flow utilized in similarity inertia effects. A numerical

study based on the Forchheimer-Brinkmann-extended Darcy equation of motion has also

been reported recently by Beckermann et al. (1986). Hong et al. (1987) investigated

the effects of non-Darcian and nonuniform on vertical plate natural convection in porous

media. Inertia effect is accounted through the inclusion of a velocity squared term in the

momentum equation, which is known as Forchheimer’s extension.

An analysis of the Brinkman equation as a model for flow in porous media is given

by Durlofsky and Brady (1987). Non-Darcian convection in cylindrical packed beds was

studied by Hunt and Tien (1988). Nakayama et al. (1989) presented a similarity solution

for the non-Darcy free convection from a non isothermal curved surface in a fluid satu-

rated porous medium. Flow transitions in buoyancy-induced non-Darcian convection in

a porous medium heated from below was analyzed by Kladias and Prasad (1990). For

the problem of mixed convection flow over a vertical plate embedded in a non-Newtonian

fluid saturated porous medium, Wang et al. (1990) obtained similar and integral solu-

tions. Ramanaiah and Malarvizhi (1991) investigated the non-Darcy axisymmetric free

convection on permeable horizontal surfaces in a saturated porous medium. The prob-

lem of non-Darcy mixed convection along a vertical wall in a saturated porous medium

was analyzed by Lai and Kulacki (1991). Hadim and Chen (1993) carried out a numeri-

cal study of buoyancy-aided mixed convection in an isothermally heated vertical channel


filled with a fluid saturated porous medium. Chen et al. (1996) analyzed the non-Darcy

mixed convection along non isothermal vertical surfaces in porous media. The effects of

non-Darcian surface tension on free surface transport in porous media was studied by

Chen and Vafai (1997). They employed the Darcy- Brinkman-Forchheimer model. The

problem of mixed convection heat and mass transfer in a fluid-saturated porous media was

studied by Rami et al. (2001) considering the Darcy-Forchheimer model. Elbashbeshy

(2003) studied the mixed convection along a vertical plate embedded in non-Darcian

porous medium with suction and injection. The non-similar non-Darcy mixed convection

flow over a non-isothermal horizontal surface which covers the entire regime of mixed con-

vection flow starting from pure forced convection to pure free convection flow has been

studied by Kumari and Nath (2004). Pal and Shivakumara (2006) studied the mixed con-

vection heat transfer from a vertical heated plate embedded in a sparsely packed porous

medium.

1.10.2 Flow Over a Stretching Sheet

During past several years considerable interest has been evinced in the study of steady

flows of a viscous incompressible fluid driven by a linearly stretching surface through a

quiescent fluid. Such flow situations are encountered in a number of industrial processes

e.g. the cooling of metallic plates in a cooling bath, the aerodynamic extrusion of plastic

sheets, polymer sheet extruded continuously from a dye and heat-treated materials that

travel between feed and wind-up rolls or on a conveyer belt possesses the characteristics

of a moving continuous surfaces. During the manufacturing of these sheets, the mixture

which is issued from a slit is subsequently stretched to achieve the desired thickness.

Finally, this sheet solidifies while it passes through effectively controlled cooling system

in order to acquire the top-grade final product. Apparently, the quality of such a sheet

is definitely influenced by heat and mass transfer between the sheet and fluid. During

its manufacturing process, a stretched sheet interacts with the ambient fluid both ther-

mally and mechanically. Sakiadis (1961) introduced in his pioneering work, the study of

boundary layer flow over a continuous solid surface moving with constant velocity. It is

usually assumed that the sheet is inextensible, but in some different studies such as in the

polymer industry it is necessary to deal with the stretching sheet as mentioned by Crane

(1970). The heat and mass transfer of viscous fluids over an isothermal stretching sheet

with suction or blowing have been extended by Gupta and Gupta (1977). Rajagopal


et al. (1984) studied the flow of a viscoelastic fluid over a stretching sheet. Dutta et

al. (1985) have investigated the temperature distribution in the flow over a stretching

sheet with uniform wall heat flux. Chen and Char (1988) studied this linearly stretching

sheet problem with suction or blowing for a power-law surface temperature as well as a

power-law surface heat flux.

It is worth mentioning that there are several practical applications in which significant

temperature differences between the body surface and the ambient fluid exist. The tem-

perature differences cause density gradients in the fluid medium and free convection effects

become more important in the presence of gravitational force. There arise some situations

where the stretching sheet moves vertically in the cooling liquid. In this situation, the

fluid flow and the heat transfer characteristic are determined by two mechanisms namely,

the motion of stretching sheet and the buoyancy force. The thermal buoyancy generated

due to heating/cooling of a vertically moving stretching sheet has a large impact on the

flow and heat transfer characteristics than when it is moving horizontally. Mahaparta and

Gupta (2002) analyzed Heat transfer in stagnation-point flow towards a stretching sheet.

Ali and AI-Yousef (2002) studied the laminar mixed convection boundary-layers induced

by a linearly stretching permeable surface. Vajravelu and Cannon (2006) studied the fluid

flow over a nonlinearly stretching sheet. Liu (2006) analyzed the flow and heat transfer

of viscous fluids saturated in porous media over a permeable non-isothermal stretching

sheet. Cortell (2007) gave a numerical analysis of momentum and mass transfer charac-

teristics in two viscoelastic fluid flows influenced by a porous stretching sheet, namely,

second-grade and second-order non-Newtonian. Prasad et al. (2010) analyzed the mixed

convection heat transfer over a non-linear stretching surface with variable fluid properties.

1.10.3 Effects of Variable Viscosity and Thermal Conductivity

Newton’s law of viscosity states that shear stress is proportional to velocity gradient.

Thus the fluids that obey this law are known as Newtonian fluids. Numerous work has

been undertaken in recent past. However, it is well known that the physical properties

of fluid may change significantly with temperature. For lubricating fluids, heat generated

by the internal friction affects the viscosity of the fluid, thus the fluid viscosity can no

longer be assumed constant. The increase of temperature leads to a local increase in the

transport phenomena by reducing the viscosity across the momentum boundary layer due

to which heat transfer at the wall is also affected. Thus in order to predict most accurately


the flow behaviour, it is important and necessary to take into account the variation of

viscosity with temperature. In mixed convection heat transfer takes place under conditions

when there are large temperature differences within the fluid thus it becomes necessary to

consider the effects of variable fluid properties. The effect of variation of viscosity to study

the instability of flow and temperature fields are discussed by Kassoy and Zebib (1975),

Gray et al. (1982). Lai and Kulacki (1990) analyzed the effects of variable viscosity

on convective heat transfer along a vertical surface in a saturated porous medium. A

theoretical investigation of the temperature-dependent fluid viscosity influence for the

forced convection flow through a semi-infinite porous medium bounded by an isothermal

flat plate was presented by Ling and Dybbs (1992). Pop et al. (1992) studied the effect of

variable viscosity on flow and heat transfer to a continuous moving flat plate. Kafoussian

and Williams (1995) investigated on free forced convective boundary layer flow past a

vertical isothermal flat plate considering temperature-dependent viscosity of the fluid.

Elbashbeshy and Bazid (2000) studied the effect of a temperature-dependent viscosity

on heat transfer over a continuous moving surface. The effect of variable viscosity on

non-Darcy, free or mixed convection flow on a horizontal surface in a saturated porous

medium was studied by Kumari (2001).

The case of visco-elastic fluid flow and heat transfer over a stretching sheet with

variable viscosity studied by Abel et al. (2002). Pantokratoras (2002, 2004) studied the

effects of variable viscosity on the laminar heat transfer flow of Newtonian fluids along a

vertical/flat plate for various flow conditions. Recently, Ghaly and Seddeek (2004) have

studied the Chebyshev finite difference method for the effects of chemical reaction, heat

and mass transfer on laminar flow along a semi infinite horizontal plate with temperature

dependent viscosity. The influence of variable viscosity on forced convection heat transfer

over a flat plate in a porous medium is examined by Seddeek (2005). Pantokratoras (2006)

made a theoretical study to investigate the effect of variable viscosity on the classical

Falkner-Skan flow with constant wall temperature and obtained results for wall shear

stress and the wall heat transfer for various values of ambient Prandtl numbers varying

from 1 to 10000. Jayanthi and Kumari (2007) studied the effect of variable viscosity on

non-Darcy free or mixed convection flow on a vertical surface in a non-Newtonian fluid

saturated porous medium. Hayat and Ali (2008) studied the effect of variable viscosity on

the peristaltic transport of a Newtonian fluid in an asymmetric channel. Palani and Kim

(2010) analyzed the numerical study on a vertical plate with variable viscosity and thermal

conductivity. Hassanien and Rashed (2011) analyzed the non-Darcy free convection flow


over a horizontal cylinder in a saturated porous medium with variable viscosity, thermal

conductivity and mass diffusivity. Botong Li et al. (2011) analyzed the heat transfer in

pseudo-plastic non-Newtonian fluids with variable thermal conductivity.

1.10.4 Effects of Magnetic Field and Electric Field

The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid due to

the stretching of the sheet is of considerable interest in modern metallurgical and metal-

working processes. Many metallic materials are manufactured after they have been refined

sufficiently in the molten state. The study of the flow and heat transfer in an electrically

conducting fluid permeated by a transverse magnetic field is of special interest and has

many practical applications in manufacturing processes in industry. Study of MHD heat

transfer field can be divided into two classes, in the first class the electromagnetic fields

use to control the heat transfer as in the convection flows and aerodynamic heating,

while in the second class the heating is produced by electromagnetic fields for example

in generators, pumps, etc. In the present study the first class is used. The study of

flow and heat transfer of an electrically conducting fluid in the presence of magnetic field

i.e. magnetohydrodynamic flow past a heated surface have applications in manufacturing

processes such as the cooling of the metallic plate, nuclear reactor, extrusion of polymers,

etc. In many metallurgical processes such as drawing of continuous filaments through

quiescent fluids, and annealing and tinning of copper wires, the properties of the end

product depend greatly on the rate of cooling involved in these processes. Therefore, it

is central problem in metallurgical chemistry to study the heat transfer on liquid metal

which is perfect electric conductor. Thus a careful examination of the needs in the system

suggests that it is advantageous to have a controlled cooling system. An electrically

conducting liquid can be regulated by external means through a variable magnetic field.

Liquid metals have high thermal conductivity and are used as coolants in addition to it

they have high electrically conductivity hence are susceptible to transverse magnetic field.

Many practical applications of convective heat transfer exist, for examples, in chemical

factories, in heaters and coolers of electrical and mechanical devices, in lubrication of

machine parts, etc.

Recently, several researchers have focused their attention to the problem of combined

heat and mass transfer in an MHD free convection flow due to the fact that free convection

induced by a simultaneous action of buoyancy forces resulting from thermal and mass


diffusion is of considerable interest in nature and in many industrial applications. The

magnetohydrodynamic problem was first studied by Pavlov (1974) who investigated the

MHD flow over a stretching wall in an electrically conducting fluid, with an uniform

magnetic field. Rudraiah et al. (1975) studied Hartmann flow of a conducting fluid past

a permeable bed in the presence of a transverse magnetic field with an interface at the

surface of the permeable bed. Chakrabarti and Gupta (1979) studied the hydromagnetic

flow and heat transfer over a stretching sheet. Vajravelu and Nayfeh (1992) studied

the hydromagnetic flow of a dusty fluid over a stretching sheet. Vajravelu and Rollins

(1992) studied heat transfer in an electrically conducting fluid over a stretching surface

taking into account the magnetic field only. Malashetty and Leela (1992) have studied

the Hartmann flow characteristic of two fluids in horizontal channel.

Keeping in mind some specific industrial applications such as in polymer processing

technology, numerous attempts have been made to analysis the effect of transverse mag-

netic field on boundary layer flow characteristics (Andresson (1992), Char (1994) and

Lawrence and Rao (1995)). Takhar et al. (1996) studied the radiation effects on MHD

free convection flow for a non gray-gas past a semi-infinite vertical plate. Bakier and Gorla

(1996) investigated the effect of thermal radiation on mixed convection from horizontal

surfaces in saturated porous media. Magnetohydrodynamic mixed convection from a ver-

tical plate embedded in a porous medium was presented by Aldoss et al. (1995). Aldoss

and Ali (1997) studied mixed convection from a horizontal circular cylinder embedded

in electrically conducting fluid and exposed to a transverse magnetic field in a porous

medium. The study of two phase flow and heat transfer in an inclined channel has been

made by Malashetty and Umavathi (1997). Chamkha (1998) presented an analysis on

unsteady hydromagnetic flow and heat transfer from a non-isothermal stretching sheet

in a porous medium. Seddeek (2001) studied the thermal radiation and buoyancy ef-

fects on MHD free convection heat generation flow over an accelerating permeable surface

with temperature dependent viscosity. Seddeek (2002) analyzed the effects of magnetic

field, variable viscosity and non-Darcy effects on forced convection flow about a flat plate

with variable wall temperature in the porous medium. Abo-Eldahab and Abd El Aziz

(2004) studied the effect of Ohmic heating on mixed convection boundary layer flow of

a micropolar fluid from a rotating cone with power-law variation in surface temperature.

Abo-Eldahab and Abd El-Aziz (2005) studied MHD three-dimensional flow over a stretch-

ing sheet in a non-Darcian heat generation or absorption effects. Mukhopadhyay et al.

(2005) studied the effects of variable viscosity on the MHD boundary layer flow over a


heated stretching surface. Ali (2006) studied the effect of variable viscosity on mixed

convection heat transfer along a vertical moving surface. Ishak et al. (2006) studied

magnetohydrodynamic stagnation point flow towards a stretching vertical sheet. In view

of this, Damesh et al. (2006) analyzed magnetohydrodynamics (MHD) forced convection

heat transfer from radiative surfaces in the presence of a uniform transverse magnetic field

with conductive fluid suction or injection from a porous plate. Afify (2007) studied the

effects of variable viscosity on non-Darcy MHD free convection along a non-isothermal ver-

tical surface in a thermally stratified porous medium. Salem (2007) studied the problem of

flow and heat transfer of all electrically conducting visco-elastic fluid having temperature

dependent viscosity as well as thermal conductivity fluid over a continuously stretching

sheet in the presence of a uniform magnetic field for the case of power-law variation in

the sheet temperature.

In all above works effect of electric field has been neglected which is also one of the

important parameters to alter the momentum and heat transfer characteristics in a New-

tonian boundary layer flow. Aydyin and Kaya (2007) analyzed the mixed convection

of a viscous dissipating fluid about a vertical flat plate. Mahmoud (2007) studied the

thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface

with variable thermal conductivity. Abel and Mahesha (2008) studied the heat transfer

in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductiv-

ity, non-uniform heat source and radiation. Pal (2008) studied the MHD flow and also

heat transfer past a semi-infinite vertical plate embedded in a porous medium of variable

porosity. The problems of coupled heat and mass transfer in MHD two-dimensional flow,

the effects of Ohmic heating have not been studied by previous authors. However, it is

more realistic to include this effect to explore the impact of the magnetic field on the

thermal transport in the boundary layer. Abel et al. (2008) studied momentum and heat

transfer characteristics in an incompressible electrically conducting viscoelastic boundary

layer flow over a linear stretching sheet in the presence of viscous and Ohmic dissipations.

The use of magnetic field that influences heat generation/absorption process in electri-

cally conducting fluid flows has important engineering applications. Kumar and Gupta

(2009) considered the unsteady MHD and heat transfer of two viscous immiscible fluids

through a porous medium in a horizontal channel. Prasad et al. (2009) examined the

influence of variable fluid properties on the hydromagnetic flow and heat transfer over

a nonlinearly stretching sheet. Rahman and Salahuddin (2009) have studied effects of

a variable electric conductivity and temperature-dependent viscosity on magnetohydro-


dynamic heat and mass transfer flow along a radiative isothermal inclined surface with

internal heat generation. Hsiao (2010) studied the heat and mass mixed convection for

MHD visco-elastic fluid past a stretching sheet with ohmic dissipation. Pal (2010) studied

the mixed convection heat transfer in the boundary layers on an exponentially stretching

surface with magnetic field. Sharma and Singh (2010) analyzed the effects of variable

thermal conductivity, viscous dissipation on steady MHD natural convection flow of low

Prandtl number fluid on an inclined porous plate with Ohmic dissipation. Mohamed

Abd El-Aziz (2010) studied the temperature dependent viscosity and thermal conduc-

tivity effects on combined heat and mass transfer in MHD three-dimensional flow over a

stretching surface with Ohmic heating. Prasad et al. (2010) studied the mixed convection

heat transfer over a non-linear stretching surface with variable fluid properties. Makinde

and Onyejekwe (2011) analyzed numerical study of MHD generalized Couette flow and

heat transfer with variable viscosity and electrical conductivity fluid. Recently, Kumar

and Gupta (2011) studied the MHD free-convective flow of micropolar and Newtonian

fluids through porous medium in a vertical channel.

1.10.5 Effects of Viscous Dissipation

Viscous dissipation plays a significant role in natural convection in various devices that

are subjected to large variations of gravitational force or that operate at high rotational

speeds (Gebhart (1962)). Gebhart and Mollendorf (1969) analyzed the effect of viscous

dissipation in external natural convection considering exponential variation of wall tem-

perature through a similarity solution. A comment was made by Fand and Brucker (1983)

that the effect of viscous dissipation might become significant in case of natural convection

in porous medium in connection with their experimental correlation for the heat transfer

in external flows. The validity of the comment was tested for the Darcy model by Fand

et al. (1986), both experimentally and analytically while estimating the heat transfer

coefficient from a horizontal cylinder embedded in a porous medium. Viscous dissipation

acts as a heat source and generates appreciable temperature in the medium. Nakayama

and Pop (1989) considered the effect of viscous dissipation on the Darcian free convection

over a non-isothermal body of arbitrary shape embedded in porous media. Murthy and

Singh (1997) studied viscous dissipation on non-Darcy natural convection from a vertical

flat plate in a porous media saturated with Newtonian fluid. They concluded that a sig-

nificant decrease in heat transfer is observed with inclusion of viscous dissipation effect.


El-Amin (2003) analyzed the combined effect of viscous dissipation and Joule heating on

MHD forced convection over a non-isothermal horizontal cylinder embedded in a fluid

saturated porous medium. In the porous medium, it is interpreted as the rate at which

mechanical energy is converted into heat in a viscous fluid per unit volume was studied

by Bejan (2004). The mathematical analysis is confined to studying the dissipation effect

using a steady, 1-D energy equation, on the basis of the equation form analogy given by

Bejan (2004) for the inclusion of viscous dissipation effects.

The effect of viscous dissipation in natural convection along a heated vertical plate

studied by Pantokratoras (2005). Seddeek (2006) studied the influence of viscous dissi-

pation and thermophoresis on DarcyForchheimer mixed convection in a fluid saturated

porous media. Duwairi et al. (2007) investigated viscous dissipation and Joule heating

effects over an isothermal cone in a saturated porous media. Many non-Newtonian liq-

uids are highly viscous such that the irreversible work due to viscous dissipation can, in

some instances, becomes quite important, this motivated researchers to study the viscous

dissipation phenomena in non-Newtonian fluid saturated porous media. Aydin and Kaya

(2007) studied the mixed convection of a viscous dissipating fluid about a vertical flat

plate. Cortell (2008) analyzed the effects of viscous dissipation and radiation on the ther-

mal boundary layer over a nonlinearly stretching sheet. Kairi and Murthy (2011) studied

the effect of viscous dissipation on natural convection heat and mass transfer from vertical

cone in a non-Newtonian fluid saturated non-Darcy porous medium. Abel et al. (2011)

studied the MHD flow, and heat transfer with effects of buoyancy, viscous and Joules dis-

sipation over a nonlinear vertical stretching porous sheet with partial slip. Cortell (2011)

analyzed the suction, viscous dissipation and thermal radiation effects on the flow and

heat transfer of a power-law fluid past an infinite porous plate.

1.10.6 Effects of Thermal Radiation

A new dimension is added to the study of mixed convection flow past a stretching sheet

embedded in a porous medium by considering the effect of thermal radiation. Thermal

radiation effect plays a significant role in controlling heat transfer process in polymer

processing industry. The quality of the final product depends to a certain extent on

heat controlling factors. Also, the effect of thermal radiation on flow and heat transfer

processes is of major important in the design of many advanced energy convection systems

which operate at high temperature. Thermal radiation occurring within these systems is


usually the result of emission by the hot walls and the working fluid. Thermal radiation

effects become more important when the difference between the surface and the ambient

temperature is large. Thus thermal radiation is one of the vital factors controlling the

heat and mass transfer. Another important effect of considering thermal radiation is

to enhance the thermal diffusivity of the cooling liquid in the stretching sheet problem.

Thus the knowledge of radiation heat transfer in the system can perhaps lead to a desired

product with sought characteristics. In many new engineering areas processes (such as

fossil fuel combustion energy processes, solar power technology, astrophysical flows, and

space vehicle re-entry) occur at high temperatures so knowledge of radiation heat transfer

beside the convective heat transfer play very important role and cannot be neglected.

Also, thermal radiation on flow and heat transfer processes is of major importance in

the design of many advanced energy conversion systems operating at high temperature.

The Rosseland approximation is used to describe the radiative heat flux in the energy

equation. Also, the effect of thermal radiation on the forced and free convection flows are

important in the content of space technology and processes involving high temperature.

Viskanta and Grosh (1962) have considered boundary layer flow in thermal radiation

absorbing and emitting media. If the radiation is taken into account in some industrial

applications such as glass production and furnace design and in space technology appli-

cations (such as propulsion system, plasma physics, cosmical flight aerodynamics rocket

and geophysics) then the governing equations become quite complicated and hold to be

solved. However, Cogley et al. (1968) showed that, in the optically thin limit, the fluid

does not absorb its own emitted radiation, but the fluid does absorb radiation emitted

by the boundaries. In the processes involving high temperatures and in the context of

space technology, the effects of radiation are of vital importance. Also, recent develop-

ments in hypersonic flights, missile reentry, rocket combustion chambers, power plants

for inter-planetary flights, gas cooled nuclear reactors, nuclear power plants, gas turbines,

propulsion devices for air-craft, satellites and space vehicles have focused attention on

thermal radiation as a mode of energy transfer and emphasize the need for improved

understanding of radiative heat transfer in these processes. In addition, radiative heat

and mass transfer flow plays an important role in manufacturing industries in the design

of reliable equipment, nuclear power plants, gas turbines and various propulsion devices

for air-craft, satellites and space vehicles as well as many other astrophysical and cosmic

studies. Based on these applications, England and Emery (1969) studies the thermal ra-

diation effect of an optically thin gray gas bounded by a stationary vertical plate. Plumb


et al. (1981) was the first to examine the effect of horizontal cross-flow and radiation

on natural convection from vertical heated surface in saturated porous media. Ali et

al. (1984) have considered natural convection-radiation interaction in the boundary layer

flow over semi-infinite horizontal surface considering grey-gas that emits and absorbs but

does not scatter thermal radiation. Ibrahim and Hady (1990) have investigated mixed

convection-radiation interaction in boundary layer flow over a horizontal surface.

Rosseland diffusion approximation had been utilized in this investigation of convec-

tion flow with radiation. Gorla and Pop (1993) studied the effects of radiation on mixed-

convection flow over vertical cylinders. Hossain and Takhar (1996) have investigated the

radiation effect on mixed convection boundary layer flow of an optically dense viscous

incompressible fluid along a vertical plate with uniform surface temperature. The effects

of thermal dispersion and lateral mass flux on non-Darcy natural convection over a ver-

tical flat plate in a fluid saturated porous medium were studied by Murthy and Singh

(1997). Mansour (1997) analyzed combined forced convection and radiation interaction

heat transfer in the boundary layer flow over flat plate immersed in porous medium of

variable viscosity. Raptis (1998) analyzed radiation and free convection flow through a

porous medium using Rosseland approximation for the radiative heat flux. Raptis and

Perdikis (1999) solved analytically the governing equations to study the effects of thermal

radiation and free convection flow past a moving vertical plate. The effect of radiation

on the free convection heat transfer problem was studied by Hossain et al. (1999) con-

sidering suction boundary condition and used Rosseland approximation to describe the

radiative heat flux in the energy equation. Mansour and El-Amin (1999) studied the

effects of thermal dispersion on non-Darcy axisymmetric free convection in a saturated

porous medium with lateral mass transfer. Mohammadein and Ei-Amin (2000) studied

the problem of thermal dispersion-radiation effects on non-Darcy natural convection in

a fluid saturated porous medium. Israel-Cookey et al. (2003) investigated the influence

of viscous dissipation and radiation on the problem of unsteady magnetohydrodynamic

free convection flow past an infinite vertical heated plate in an optically thin environ-

ment with time-dependent suction. Pop et al. (2004) investigated theoretically steady

two-dimensional stagnation-point flow of an incompressible fluid over a stretching sheet

by taking into account of the thermal radiation effects.

Abel et al. (2005) performed analysis to study the effect of buoyancy force and thermal

radiation in MHD boundary layer visco-elastic fluid flow over a continuously moving

stretching surface embedded in a porous medium. Siddheshwar and Mahabaleswar (2005)


studied the MHD flow and also heat transfer in a viscoelastic liquid over a stretching

sheet in the presence of radiation. They have assumed that stretching of the sheet be

proportional to the distance from the slit. They used Keller box method to solve the

nonlinear equations. Rashad (2007) studied thermal radiation effects on free convection

flow of Newtonian fluid-saturated porous medium in the presence of pressure work and

viscous dissipation using regular three-parameter perturbation analysis. The effect of

chemical reaction and thermal radiation absorption on unsteady MHD free convection

flow past a semi-infinite vertical permeable moving surface with heat source and suction

was analyzed by Ibrahim et al. (2008). Pal (2009) analyzed heat and mass transfer in

two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching

vertical sheet in the presence of buoyancy force and thermal radiation. Hassanien and Al-

arabi (2009) studied the non-Darcy unsteady mixed convection flow near the stagnation

point on a heated vertical surface embedded in a porous medium with thermal radiation

and variable viscosity. Ali et al. (2011) studied the unsteady MHD natural convection

from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical

reaction and radiation effects. Singh et al. (2011) analyzed the effects of thermophoresis

on hydromagnetic mixed convection and mass transfer flow past a vertical permeable plate

with variable suction and thermal radiation. Turkyilmazoglu (2011) studied the thermal

radiation effects on the time-dependent MHD permeable flow having variable viscosity.

Recently, Hayat et al. (2011) analyzed the radiation effects on MHD flow of Maxwell fluid

in a channel with porous medium.

1.10.7 Effects of Non-uniform Heat Source/Sink

Yih (1998) studied the heat source/sink effect on MHD mixed convection in stagnation

flow on a vertical permeable plate in porous media. Kamel (2001) studied the unsteady

MHD convection through porous medium with combined heat and mass transfer with

heat source/sink. Chamkha (2000) analyzed the thermal radiation and buoyancy effects

on hydromagnetic flow over an accelerating permeable surface with heat source or sink.

Yih (2000) studied the viscous and Joule heating effects on non -Darcy MHD natural con-

vection flow over a permeable sphere in porous media with internal heat generation. Emad

et al. (2004) have included the effect of non-uniform heat source with suction/blowing,

but confirm to the case of viscous fluid only. Eldahab and Aziz (2004) have included the

effect of non-uniform heat source with suction/blowing, but confirm to the case of viscous


fluid only. In most of the investigations involving the heat transfer, we observe that either

the prescribed constant surface temperature (PST) or the prescribed constant wall heat

flux (PHF) boundary condition is assumed. Seddeek (2007) analyzed the heat and mass

transfer on a stretching sheet with a magnetic field in a visco-elastic fluid flow through

a porous medium with heat source or sink. Pal and Malashetty (2008) have presented

similarity solutions of the boundary layer equations to analyze the effects of thermal ra-

diation on stagnation point flow over a stretching sheet with internal heat generation or

absorption. The study of heat source/sink effects on heat transfer is very important in

view of several physical problems.

Aforementioned studies include only the effect of uniform heat source/sink (i.e. tem-

perature dependent heat source/sink) on heat transfer. If the final product that is ob-

tained after cooling needs to be non-uniform in terms of properties warranted by an appli-

cation, then variable PHF case is appropriate. Furthermore, heat generation/absorption

may be important in weak electrically conducting polymeric liquids due to the non-

isothermal situation and also due to the cation/anion salts dissolved in them. Ali (2007)

analyzed the effect of lateral mass flux on the natural convection boundary layer induced

by a heated vertical plate embedded in a saturated porous medium with an exponential

decaying heat generation. Layek et al. (2007) investigated the structure of the boundary

layer stagnation-point flow and heat transfer over a stretching sheet in a porous medium

subject to suction or blowing and in the presence of internal heat generation or absorp-

tion by using a similarity analysis. Abel et al. (2007) have investigated heat transfer

in a viscoelastic fluid past a stretching sheet with non-uniform heat source. Abel et al.

(2007) investigated on non-Newtonian boundary layer flow past a stretching sheet taking

into account of non-uniform heat source and frictional heating. Abel and Mahesha (2008)

studied the magnetohydrodynamic boundary layer flow and heat transfer characteristic

of a non-Newtonian viscoelastic fluid over a flat sheet with variable thermal conductivity

in the presence of thermal radiation and non-uniform heat source. They have reported

that the combined effect of variable thermal conductivity, radiation and non-uniform heat

source have significant impact in controlling the rate of heat transfer in the boundary layer

region. Abel et al. (2009) studied the effect of non-uniform heat source on MHD heat

transfer in a liquid film over an unsteady stretching sheet. Pal and Chatterjee (2010)

analyzed the heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over

a stretching sheet embedded in a porous media with non-uniform heat source and thermal

radiation. Zheng et al. (2011) studied the analytic solutions of unsteady boundary flow


and heat transfer on a permeable stretching sheet with non-uniform heat source/sink.

Mahantesh et al. (2011) analyzed the heat transfer in MHD viscoelastic boundary layer

flow over a stretching sheet with thermal radiation and non-uniform heat source/sink.

Recently, Pal (2011) studied the combined effects of non-uniform heat source/sink and

thermal radiation on heat transfer over an unsteady stretching permeable surface.

1.10.8 Effects of Soret and Dufour

The effect of diffusion-thermo and thermal-diffusion of heat and mass has been developed

by Chapman and Cowling (1952) and Hirshfelder et al. (1954) from the kinetic theory

of gases. Sparrow et al. (1964) have considered diffusion-thermo effects in stagnation-

point flow of air with injection of gases of various molecular weights into the boundary

layer. Thermal diffusion or Soret effect corresponds to species differentiation developing

in an initial homogeneous mixture submitted to a thermal gradient. On the other hand,

diffusion-thermo or Dufour effect corresponds to the energy flux caused by a concentra-

tion gradient in a binary fluid or mixture. The Dufour effect was found to be of order

of considerable magnitude such that it cannot be ignored (Eckert and Drake, (1972)).

Kafoussias and Williams (1995) considered the boundary layer flows in presence of Soret

and Dufour effects associated with thermal diffusion and diffusion thermo for the mixed

forced-natural convection problem. Anghel et al. (2000) analyzed the Dufour and Soret

effects on free convection boundary-layer over a vertical surface embedded in a porous

medium. Singh and Kumar (2001) studied the MHD free convection and mass transfer

flow with heat source and thermal diffusion. Postelnicu (2004) has examined Soret and

Dufour effects on combined heat and mass transfer in natural convection boundary layer

flow in a Darcian porous medium in the presence of transverse magnetic field. Alam and

Rahman (2005, 2006) analyzed the Dufour and Soret effects on mixed and free convec-

tion heat and mass transfer flow past a vertical porous flat plate embedded in a porous

medium in absence/presence of variable suction, respectively. In many studies, Dufour

and Soret effect are neglected on the basis that they are of a smaller order of magni-

tude than the effects described by Fourier’s and Fick’s. Thermal-diffusion (Soret) and

diffusion-thermo (Dufour) effects have been found to appreciably influence the flow field

in mixed convection boundary-layer over a vertical surface embedded in a porous medium.

Alam and Rahman (2006) investigated the Dufour and Soret effects on mixed convection

flow past a vertical porous flat plate with variable suction. Alam et al. (2006) have stud-


ied the Dufour and Soret effects on steady free convection and mass transfer flow past a

semi-infinite vertical porous plate in a porous medium. Alam et al. (2007) studied the

diffusion-thermo and thermal-diffusion effects on free convective heat and mass transfer

flow in a porous medium with time dependent temperature and concentration.

Chamkha and Ben-Nakhi (2008) considered the mixed convection flow with thermal

radiation along a vertical permeable surface immersed in a porous medium in the presence

of Soret and Dufour effects. Mohamed Abo El-Aziz (2008) have investigated the combined

effects of thermal-diffusion and diffusion-thermo on MHD heat and mass transfer over

a permeable stretching surface with thermal radiation. El-Aziz (2008) investigated the

combined effects of thermal-diffusion and diffusion-thermo on MHD heat and mass transfer

over a permeable stretching surface with thermal radiation. Maleque (2009) studied Soret

effect on convective heat and mass transfer past a rotating porous disk and he neglected

the Dufour effect. Rani and Kim (2009) studied a numerical study of the Dufour and

Soret effects on unsteady natural convection flow past an isothermal vertical cylinder.

Ahmed (2009) investigated the Dufour and Soret effects on free convective heat and mass

transfer over a stretching surface considering suction or injection. Recently, numerical

study of free convection magnetohydrodynamic heat and mass transfer due to a stretching

surface under saturated porous medium with Soret and Dufour effects was also discussed

by Anwar Beg et al. (2009). Postelnicu (2010) analyzed the heat and mass transfer

by natural convection at a stagnation point in a porous medium considering Soret and

Dufour effects. Hayat et al. (2010) analyzed the heat and mass transfer for Soret and

Dufours effect on mixed convection boundary layer flow over a stretching vertical surface

in a porous medium filled with a viscoelastic fluid. Recently, Pal and Chatterjee (2011)

investigated mixed convection MHD heat and mass transfer past a stretching sheet with

Ohmic dissipation Soret and Dufour effects considering micropolar fluid. Recently, Anjali

and Uma (2011) analyzed the Soret and Dufour effects on MHD slip flow with thermal

radiation over a porous rotating infinite disk.

1.10.9 Effects of Chemical Reaction

The chemical reaction can be codified as either a heterogeneous or a homogeneous process.

This depends on whether it occurs at an interface or as a single-phase volume reaction.

A few representative fields of interest where combined heat and mass transfer with a

chemical reaction and thermal radiation plays an important role are design of chemical


processing equipment, cooling towers, etc. In many transport processes existing in nature

and industrial applications in which heat and mass transfer is a consequence of buoyancy

effects caused by diffusion of heat and chemical species. The study of such processes is

useful for improving a number of chemical technologies, such as polymer production and

food processing. In nature, the presence of pure air or water is impossible. Some foreign

mass may be present either naturally or mixed with the air or water. The presence of a

foreign mass in air or water causes some kind of chemical reaction. During a chemical

reaction between two species, heat is also generated. In most cases of chemical reac-

tion, the reaction rate depends on the concentration of the species itself. Anjalidevi and

Kandasamy (1999) studied the effects caused by chemical-diffusion mechanisms and the

inclusion of a general chemical reaction of order n on the combined forced and natural

convection flows over a semi-infinite vertical plate immersed in an ambient fluid. They

stated that the presence of pure air or water is impossible in nature and that some foreign

mass may be present either naturally or mixed with air or water. Mulolani and Rahman

(2000) studied laminar natural convection flow over a semi-infinite vertical plate under the

assumption that the concentration of species along the plate follows some algebraic law

with respect to chemical reaction. They obtained similarity solutions for different order

of reaction rates and Schmidt number. Muthucumaraswamy and Ganesan (2001) studied

the effect of the chemical reaction and injection on flow characteristics in an unsteady

upward motion of an isothermal plate. Prasad et al. (2003) studied the influence of re-

action rate on the transfer of chemically reactive species in the laminar, non-Newtonian

fluid immersed in porous medium over a stretching sheet. They concluded that the effect

of chemical reaction is to reduce the thickness of concentration boundary layer and to

increase the mass transfer rate from the sheet to the surrounding fluid and that this ef-

fect is more effective for zero and first-order reactions than second-order and third-order

reactions.

A reaction is said to be first-order if the rate of reaction is directly proportional to

concentration itself. The problem of combined heat and mass transfer of an electrically

conducting fluid in MHD natural convection adjacent to a vertical surface is analyzed

by Chen (2004) by taking into account the effects of Ohmic heating and viscous dissi-

pation but neglected chemical reaction of the species. Ghaly and Seddeek (2004) have

investigated the effect of chemical reaction, heat and mass transfer on laminar flow along

a semi-infinite horizontal plate with temperature dependent viscosity. Kandasamy et al.

(2005) investigated the effects of chemical reaction, heat source and thermal stratification


on heat and mass transfer in MHD flow over a vertical stretching surface. The problem

of diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous

medium over a stretching sheet was considered by Akyildiz et al. (2006). Raptis and

Perdikis (2006) considered the problem of the steady two-dimensional flow of an incom-

pressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretch-

ing sheet in the presence of a chemical reaction and under the influence of a magnetic field.

Postelnicu (2007) studied the influence of chemical reaction on heat and mass transfer

by natural convection from vertical surfaces in porous media considering Soret and Du-

four effects. Afify (2007) analyzed the effects of Temperature-Dependent Viscosity with

Soret and Dufour Numbers on Non-Darcy MHD Free Convective Heat and Mass Transfer

Past a Vertical Surface Embedded in a Porous Medium. Kandasamy and Palanimani

(2007) carried out an analysis on the effects of chemical reactions, heat, and mass trans-

fer on non-linear magnetohydrodynamic boundary layer flow over a wedge with a porous

medium in the presence of Ohmic heating and viscous dissipation. Seddeek et al. (2007)

analyzed the effects of chemical reaction and variable viscosity on hydromagnetic mixed

convection heat and mass transfer for Hiemenz flow through porous media with radiation.

El-Amin et al. (2008) studied the effects of chemical reaction and double dispersion on

non-Darcy free convection heat and mass transfer. Alam et al. (2009) studied transient

magnetohydrodynamic free convective heat and mass transfer flow with thermophoresis

past a radiative inclined permeable plate in the presence of a variable chemical reaction

and temperature-dependent viscosity. Mohamed and Abo-Dahab (2009) presented for the

effects of chemical reaction and thermal radiation on hydromagnetic free convection heat

and mass transfer for a micropolar fluid via a porous medium bounded by a semi-infinite

vertical porous plate in the presence of heat generation. Pal and Talukdar (2010) an-

alyzed the buoyancy and chemical reaction effects on MHD mixed convection heat and

mass transfer in a porous medium with thermal radiation and Ohmic heating. Das (2011)

analyzed the effect of chemical reaction and thermal radiation on heat and mass transfer

flow of MHD micropolar fluid in a rotating frame of reference. Recently, Pal and Talukdar

(2011) studied the combined effects of Joule heating and chemical reaction on unsteady

magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate

in porous media with thermal radiation.

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