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, thickness 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 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.
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. Radiative heat transfer flow is very important in manufacturing industries for the design
CHAPTER 1. INTRODUCTION
10
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 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.
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 manufactured 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 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.
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 physical 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 concentration gradient. Usually, in heat and mass transfer problems the variation of density
with temperature and concentration give rise to a combined buoyancy force under natural convection. The heat and mass transfer simultaneously affect each other that create
CHAPTER 1. INTRODUCTION
11
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 reactions 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 considering 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 equations by considering effects of thermal radiation, variable viscosity and thermal conductivity, 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 analysis, 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
CHAPTER 1. INTRODUCTION
12
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,
CHAPTER 1. INTRODUCTION
13
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
, when µ is the viscosity of the fluid. In common terms, this
∂y
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 NonNewtonian 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
1.3
Types of flows
1.3.1
Steady and Unsteady Flows
∂u
.
∂y
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.
CHAPTER 1. INTRODUCTION
14
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 pressure. 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
CHAPTER 1. INTRODUCTION
15
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 nondeformable) 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=
Vv
void volume
=
total volume
V
(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)
CHAPTER 1. INTRODUCTION
16
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.
κ
−
→
q = ∇P
µ
(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
k
−
→
→
q = − (∇p + ρ−
g ).
µ
(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.
CHAPTER 1. INTRODUCTION
17
(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
κ dP
−
→
Q =−
µ 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
→
→
√ b |−
q |−
q
obstacles which is comparable with the surface drag due to friction of the form ρC
in addition to linear drag
µ→
−
q.
κ
b
Thus according to Joseph et al. (1982) the appropriate
modification to Darcy’s equation is to replace (1.6) by
→
µ→
∂−
q
→
→
→
= −∇p − −
q − cb k −1/2 ρ|−
q |−
q + ρ−
q
∂t
κ
(1.8)
CHAPTER 1. INTRODUCTION
18
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
( →
∂−
q
)
µ → ρCb −
−
→
→
→
q − √ |→
q |→
q + ρ−
g.
(1.9)
+ (−
q .∇)−
q = −∇p − −
∂t
k
k
→
→
The term ρ(−
q .∇)−
q was first considered by Lapwood (1948). In the case of sparsely
ρ
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
)
µ → ρCb −
−
→
→
→
→
e 2−
q − √ |→
+ (→
q .∇)−
q = −∇p − −
q |−
q − µ∇
q + ρ−
g.
∂t
k
k
(1.10)
where µe 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
CHAPTER 1. INTRODUCTION
19
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
CHAPTER 1. INTRODUCTION
20
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).
CHAPTER 1. INTRODUCTION
1.8.1
21
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 = −k
1.9.1
dT
.
dx
(1.11)
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
CHAPTER 1. INTRODUCTION
22
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.
CHAPTER 1. INTRODUCTION
23
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
CHAPTER 1. INTRODUCTION
24
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 boundarylayer 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 40 C 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
CHAPTER 1. INTRODUCTION
25
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 thermal 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 saturated 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 solutions. Ramanaiah and Malarvizhi (1991) investigated the non-Darcy axisymmetric free
convection on permeable horizontal surfaces in a saturated porous medium. The problem 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 numerical study of buoyancy-aided mixed convection in an isothermally heated vertical channel
CHAPTER 1. INTRODUCTION
26
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 convection 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 convection 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 thermally 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
CHAPTER 1. INTRODUCTION
27
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 temperature 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 characteristics 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
CHAPTER 1. INTRODUCTION
28
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
CHAPTER 1. INTRODUCTION
29
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 metalworking 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
CHAPTER 1. INTRODUCTION
30
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 magnetic 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 vertical 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 effects 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 stretching 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
CHAPTER 1. INTRODUCTION
31
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 vertical 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 Newtonian 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 conductivity, 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 electrically 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-
CHAPTER 1. INTRODUCTION
32
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 conductivity 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 temperature 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 significant decrease in heat transfer is observed with inclusion of viscous dissipation effect.
CHAPTER 1. INTRODUCTION
33
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 dissipation 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 liquids 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 thermal 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 dissipation 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
CHAPTER 1. INTRODUCTION
34
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 applications (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 developments 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 radiation effect of an optically thin gray gas bounded by a stationary vertical plate. Plumb
CHAPTER 1. INTRODUCTION
35
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 convection flow with radiation. Gorla and Pop (1993) studied the effects of radiation on mixedconvection 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 vertical 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) considering 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 environment 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)
CHAPTER 1. INTRODUCTION
36
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 Alarabi (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 convection 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
CHAPTER 1. INTRODUCTION
37
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 radiation 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. temperature dependent heat source/sink) on heat transfer. If the final product that is obtained after cooling needs to be non-uniform in terms of properties warranted by an application, then variable PHF case is appropriate. Furthermore, heat generation/absorption
may be important in weak electrically conducting polymeric liquids due to the nonisothermal 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 absorption 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
CHAPTER 1. INTRODUCTION
38
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 stagnationpoint 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 concentration 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 convection 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 magnitude 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-
CHAPTER 1. INTRODUCTION
39
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
CHAPTER 1. INTRODUCTION
40
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 reaction, 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 reaction 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 effect 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 dissipation 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
CHAPTER 1. INTRODUCTION
41
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 incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching 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 Dufour 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 transfer 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) analyzed 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.