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CHAPTER 1

INTRODUCTION

1.1 HEAT TRANSFER

Heat transfer considerations are often crucial and important in

modem day engineering design. Equipment size in power production and

chemical processing are determined primarily by the attainable heat-transfer

rates. A considerable fact is that the cost of many devices is due to heat

exchangers for example, air-conditioners and refrigeration systems. In many

types of equipments a successful design is possible only if provision is made to

maintain reasonable temperatures by adequate heat transfer. Such prominent

modem devices are rocket nozzles, compact electronic components, high-speed

aircraft, and atmosphere re-entry vehicles.

The study of heat transfer includes the physical processes whereby

thermal energy is transferred as a result of a difference or gradient of

temperature. There are basically three different processes whereby energy is

transported: Conduction, Convection and Radiation.

The process of heat transfer which takes place between particles

immediately adjacent to one another or through molecular action, supplemented

by free flow of electrons from a high temperature region to the low temperature

region is called conduction. In the process of convection, the thermal energy is

affected by the relative motion within the fluid so the resultant heat transfer

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occurs between the layers of a fluid. All solid bodies as well as liquids and

gases have a tendency of radiating thermal energy in the form of

electromagnetic waves and of absorbing similar energy from the neighboring

bodies. This type of heat transfer is known as thermal radiation.

1.2 MASS TRANSFER

Many of our day-to-day experiences involve mass transfer. For

example, a lump of sugar added to a cup of black coffee eventually dissolves

and then diffuses uniformly throughout the coffee. Mass transfer plays an

important role in many industrial processes. The removal of pollutants from

plant discharge streams by absorption, the stripping of gases from wastewater,

neutron diffusion within nuclear reactors are typical examples.

When a system contains two or more components whose

concentrations vary from point to point, there is a natural tendency for mass to

be transferred, minimizing the concentration differences within the system. The

transport of one constituent from a region of higher concentration to that of a

lower concentration is called mass transfer.

1.3 NATURAL CONVECTION

In studies related to heat transfer, considerable effort has been directed

towards the convective mode, in which the relative motion of the fluid provides an

additional mechanism for the transfer of energy and of material, the later being a

more important consideration in cases where mass transfer, due to a concentration

difference, occurs. Convection is inevitably coupled with conductive mechanisms,

since, although the fluid motion modifies the transport process, the eventual

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transfer of energy from one fluid element to another in its neighborhood is

through conduction. Also, at the surface, the process is predominantly that of

conduction because the relative fluid motion is brought to zero.

A study of convective heat transfer therefore involves the

mechanisms of conduction and sometimes, those of radiative processes as well,

coupled with those of fluid flow. This makes the study of the mode of heat and

mass transfer very complex, although its importance in technology and in

nature can hardly be exaggerated.

The convective mode of heat transfer is divided into two basic

processes. If the motion of the fluid is caused by an external agent, such as the

externally imposed flow of a fluid stream over a heated object, the process is

termed forced convection. The fluid flow may be the result of, for instance, a

fan, a blower, the wind, or motion of the heated object itself. Such problems are

very frequently encountered in scientific technology where the heat transfer to

or from a body is often due to an imposed flow of a fluid at a different

temperature from that of the body. On the other hand, if no such externally

induced flow is provided the flow arises ‘naturally’ owing to the effect of a

density difference, resulting from a temperature or concentration difference in a

body force field, such as the gravitational field. The process is termed natural

convection or free convection. The density difference gives rise to buoyancy

effects, owing to which the flow is generated. A heated body cooled in ambient

air generates such a flow in the region surrounding it.

Similarly, the buoyant flow arising out of heat injections to the

atmosphere and to other ambient media, circulations arising in heated rooms,

heat transfer in the atmosphere, and many other such heat transfer processes in

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our environment, as well as in many technological applications, are included in

the field of natural convection. The flow may also arise owing to concentration

differences, such as those caused by salinity differences in the sea and by

composition differences in chemical processing units which causes a natural

convection mass transfer. Convective mass transfer plays an important role in

meteorological phenomena, burning of hey stacks, spray drying milk, fluidized

bed catalysis, cooling towers, design of chemical processing equipment,

formation and dispersion of fog, distribution of temperature and moisture over

agricultural fields of crops due to freezing and pollution of the environment.

1.4 FLOW AND HEAT TRASFER OVER HORIZONTAL

CYLINDER

The analysis of heat transfer through a laminar boundary layer in the

flow of a viscous fluid over a body of arbitrarily shaped and arbitrarily

specified surface temperature constitutes a very important problem in the field

of heat transfer. The prediction of heat transfer under such conditions

encompasses a wide range of technological applications, such as the calculation

of a projectile, aircraft or other body moving through the atmosphere, cooling

problems in turbine blades, etc.

The problem of heat transfer from a horizontal circular cylinder in a

laminar viscous and incompressible fluid has been successfully studied in the

past. To the best of our knowledge only little work has been conducted in

investigating the effect of heat transfer on free or forced convection boundary

layer flow past a circular cylinder in a various compressible fluid. Koh and

Price (1965) studied the boundary layer differential equations for free convective

flow over a horizontal cylinder which was solved by a perturbation method.

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They found that the functional relationship between heat transfer and the Prandd

number for a cylinder was essentially the same as that of a vertical plate.

The effects of variable viscosity on heat transfer and shear stress

distribution around the horizontal hollow cylinder was discussed by Yao and

Catton (1978). They observed that the variable viscosity effect could enhance

the heat transfer rate and stabilize the boundary layer flow.

A numerical analysis was carried out by Fujii et al (1979) about the

thick boundary layer of a steady laminar free convection around a horizontal

cylinder. The results for the average heat transfer coefficient for Pr = 0.7, 10 and 100 and 10"4 < Gr < 104 are expressed with accuracy. Sedahmed et al

(1986) used an electrochemical technique for natural convection mass transfer

over horizontal cylinders. Measurement of mass transfer distribution revealed

the fact that mass was transferred at the lower semi cylinder by a laminar flow

mechanism and at the upper semi cylinder by a turbulent flow mechanism.

Zahariades and Assael (1987) reported the local heat transfer

coefficients of a horizontal cylinder in air-solid fluidized beds. It was observed

that local heat transfer coefficient varied significantly with angular positions

around the horizontal cylinder. Neilson and Incropera (1988) studied the

problem of local heat transfer from a horizontal cylinder in a quiescent fluid.

They concluded that in an unstratified ambient the circumferential variation of

the local Nusselt number is characterised by a monotonic decay from a

maximum value associated with the lower stagnation point.

Yih (1999) analyzed numerically the heat and mass transfer

characteristics of free convection about a permeable horizontal cylinder

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embedded in porous media under the coupled effects of thermal and mass

diffusion. It was found that the dimensionless temperature and concentration

profiles decrease monotonically from the surface to the ambient.

1.5 FLOW AND HEAT TRANSFER ON VERTICAL CYLINDER

Heat transfer by natural convection along a vertical cylinder has been

analysed rather extensively by many investigators using different solution

methods. Elenbaas (1948) used Langmuir’s stagnant film model to evaluate the

heat transfer coefficient for a vertical cylinder with uniform wall temperature.

Sparrow and Gregg (1956) also used the stagnant film model along with a

series expansion to solve the isothermal vertical cylinder problem.

Millsaps and Pohlhausen (1958) treated the laminar free convective

fluid motion produced by a heated vertical circular cylinder for which the

thermal distribution on the outer surface varies linearly with the distance from

the leading edge. The exact solutions were obtained by Karmann momentum

method. They found that these types of problems could be solved by the

similarity transformation.

Yang (1960) presented the unsteady boundary layer equations for

free convection on vertical plates and cylinders to establish necessary and

sufficient conditions under which similarity solutions were possible. On the

basis of these conditions, all possible cases were derived, including those for

unsteady conditions.

Goldstein and Briggs (1964) studied the transient free-convection, a

heat transfer problem from vertical circular cylinders to a surrounding initial

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quiescent fluid. The transient was initiated by a change in wall temperature of

the cylinder. Subsequently, the power series expansion method was used by

Kuiken (1968) and Fujii and Uehara (1970) in their studies of natural

convection along vertical cylinders with axial variations in wall temperature

and surface heat flux. However, due to uncertain convergence characteristics of

power series, their series solutions were expected to be valid only for small

values of the cylinder curvature parameter.

To analyse the problems of slender cylinders, Cebeci and Na (1969,

1970) and Narain and Uberoi (1972) employed the similarity solution method

and obtained results for the cases of uniform wall temperature and uniform

surface heat flux, respectively. Later Minkowycz and Sparrow (1974) used the

local non-similarity solution method. The solution was carried out for Prandtl

number of 0.733 and for a range of cases extending from small deviations of a

flat plate to a cylinder.

The transient and steady state temperatures of thin vertical cylinders

suspended in various fluids were subjected to steps in internal heat generation

which has been measured by Dring and Gebhart (1966). The problem of

laminar boundary-layer flow and heat transfer over a long thin cylinder in

uniform flow has been analysed by Eshghy et al. (1967). Solutions were

obtained for small as well as large values of the curvature parameter.

An experimental and analytical study was reported by Evans et al.

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

cylinder was subjected to a uniform heat flux at the wall for the experimental

study. The temperature of the core fluid was assumed to vary in vertical

direction but not in horizontal direction. They also presented a simplified model

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using the integral forms of the momentum and energy equations. This model

seemed to provide good results for the temperature behavior over a vertical

cylinder.

An experimental investigation has been made by Jaluria and Gebhart

(1974) for the processes occurring during the natural transition from laminar to

turbulent flow of natural convection flow of water adjacent to a flat vertical

surface where the surface heat flux was uniform. An experimental investigation

was conducted by Hess and Miller (1979) using a Laser Doppler Velocitimeter

to measure the axial velocity of a fluid contained in a cylinder subject to

constant heat flux on the side walls.

Surma Devi et al (1986) presented the effect of axial heat conduction

on the steady, laminar, and incompressible, natural convection flow over a

vertical cylinder. The effect of the axial heat conduction on the heat transfer

was found to be more pronounced for small curvature. The skin friction was

found to be weakly dependent on the axial heat conduction parameter.

Nair and Shupe (1987) studied the generalized finite difference

solution for heat and mass transfer from a finite cylinder during quench.

Application of the generalized solution, which utilizes the numerical method of

finite differences with forward stepping, was illustrated by determining surface

heat transfer rate (both instantaneous and cumulative).

Lee et al (1988) reported the limiting case of steady natural

convection along slender vertical cylinders or needles that have non-uniform

surface temperature. They employed a cubic spline interpolation technique that

associated with the large surface curvatures to solve the transformed system of

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equations. They had shown that, local Nusselt number parameter became

independent of Pr when the curvature parameter became very large.

Heckel et al (1989) studied the mixed convection in steady laminar

boundary layer flow along vertical cylinders. The governing equations were

solved by a weighted finite difference method. The steady incompressible

laminar mixed convection boundary layer flow along a rotating slender vertical

cylinder with an isothermal walls had been studied by Pop et al (1989). The

transformed conservation equations of the non-similar boundary layers are

solved by an efficient and very accurate finite-difference method. It was shown

that the buoyancy force gives rise to an overshoot in the axial velocity profile

and the effect of the curvature and rotation of the cylinder reduces the velocity

overshoot.

Velusamy and Garg (1992) studied the numerical solution for

transient natural convection flow over heat generating vertical cylinders of

various thermal capacities and radii. The rate of propagation of the leading edge

effect was given special consideration. They found that this rate, predicted by

the one-dimensional conduction solution is slower than that resulting from the

boundary layer solution. The transient boundary layer thickness was found to

exceed its steady-state values.

Daskalakis (1993) examined the mixed free and forced convection in

the incompressible laminar boundary layer flow along a rotating vertical

cylinder with fluid injection. They proved that the fluid injection could

considerably reduce the skin friction and heat transfer at the wall.

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Gorla et al (1993) presented the mixed convection in an

axisymmetric stagnation flow over a vertical cylinder with arbitrary

temperature variations. They have shown that the skin friction and Nusselt

number increase or decrease with the buoyancy force parameter depending

upon the flow regime.

Chaplin et al (1997) analysed the results of two series of experiment

concerned with the response of a single vertical cylinder in the inertia regime in

steep non-breaking waves. Firstly they recorded the loading on a cylinder when

it was held stationary, and secondly, its response in the same waves when it was

pivoted just above the floor of the wave flume, and supported at the top by

springs in the horizontal plane. Fully developed laminar natural convection in

an open ended vertical concentric cylinders has been studied numerically by

Hadjadj et al (1999). Solutions have been obtained for Prandtl number 0.01 to 10, Rayleizh number 1 to 105 and cylinder aspect radius of 0.5 to 10.

The problem of pure and saline water natural convection along a

vertical isothermal cylinder has been investigated by Pantokratoras (2000). Results have been produced for the temperature range between 0° to 20° C. The

international equation of state for sea water was used for the buoyancy force.

The viscosity and thermal diffusivity have been considered variable during the

solution procedure.

Harries et al (2000) presented the development of the free convection

boundary layer flow of a viscous and incompressible fluid near the lower

stagnation point of a cylindrical body which was subjected to a sudden change

in surface temperature. Analytical solutions for both small (unsteady) and large

(steady) values of time had been obtained for the boundary layer equations.

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Calmelet-Eluhu and Rosenhaus (2001) considered the system of equations of

motion for a micropolar fluid inside a circular cylinder subjected to longitudinal

and rotational motion.

Venkatachalappa et al (2001) investigated the effects of buoyancy,

rotation and aspect ratio on the axisymmetric flow in a vertical cylindrical

annulus with the cylinders rotating at different angular velocities.

Computational results reveal that the rate of heat transfer at the hot cylinder is

suppressed when its speed of rotation was higher than that of the cooler

cylinder.

Natural convection flows arising from the combined buoyancies due

to thermal and chemical species diffusion have received considerable attention

because of their importance in wide-ranging applications related to

manufacturing process in industries. The similarity solutions were given by

Gebhart and Pera (1971) who made a general formulation of the vertical two-

dimensional boundary layer flows. Their work also dealt with laminar

instability.

Bottmanne (1971) presented an analysis similar to that of Gebhart

and Pera (1971), and experimental results of Bottmanne (1972) for Pr = 0.71

and Sc = 0.63 agreed well with his analysis. Pera and Gebhart (1972) extended

the results of Gebhart and Pera (1971) to flows above horizontal surfaces. Chen

and Yuh (1980) included the effect of mass transfer and obtained results for

Pr = 0.71 and 7.0 covering large values of the curvature parameter for both

uniform wall temperature and uniform surface heat flux cases.

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Yucel (1990) examined the combined heat and mass transfer in

natural convection along vertical cylinder in a saturated porous medium. The

flow field characteristics were analysed in detail for both cases where the

concentration gradients could either aid or oppose the thermal buoyancy forces.

The effect of curvature, the buoyancy parameter and the Lewis number on the

temperature, concentration and flow fields and on the surface heat and mass

transfer rates were discussed. Ganesan and Rani (2001) studied the effects of

variable surface temperature along a vertical cylinder by an implicit finite

difference scheme of Crank-Nicolson type.

1.6 HORIZONTAL MOVING CYLINDER

The problem of flow past an impulsively started horizontal

cylindrical surface has been studied extensively. Sakiadis (1961a, 1961b)

studied the growth of the two-dimensional velocity boundary layer over a

continuously moving horizontal plate emerging from a wide slot, at uniform

velocity. The problem was solved using a similarity transformation. The

velocity was found to grow in the direction of the motion of the flat plate or

cylindrical rod. However, his investigation has been restricted to the momentum

transfer in the boundary layer.

The work of Sakiadis who restricted his study only to momentum

transfer in the boundary layer on a continuous moving cylindrical surface was

extended by Tsou et al (1967), Rotte and Beek (1969), Bourne and Elliston

(1970) and Bourne and Dixon (1971), to include heat transfer also. The Karman

Pohlhausen integral technique was adopted in their analysis. A theoretical

investigation of the initial flow over a moving circular cylinder at finite

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Reynolds numbers consists notably of the work of Collins and Dennis (1973,

1974) and Bar-Lev and Yang (1975).

Collins and Dennis (1973) has extended the theory for an impulsively

started circular cylinder to finite values of Reynolds number by determining

corrections of second and higher orders valid large Reynolds number. These

extensions are based on the full Navier-Stokes equations rather than boundary

layer equations. All these expansions are, however, limited in validating to

small times.

Collins and Dennis (1974) made a numerical extension of the method

of expansion in powers of the time for an impulsively started circular cylinder

by using an implicit time-dependent numerical integration procedure. In this

way accurate numerical solutions of the Navier-Stokes equations were obtained

upto quite moderate times over a wide range of Reynolds number. The

calculated results were found to agree well with previous numerical and

experimental work.

Bar-Lev and Yang (1975) solved the vorticity equations by using the

method of matched asymptotic expansions. Inner (rotational flow) and outer

(potential flow) solutions were obtained to the third order in time and a

composite solution was formed. Both works provide extensive information for

flow quantities of interest (such as vorticity field, stream lines and body forces)

that are valid for short times.

The analysis of Bar-Lev and Yang for the problem of transient flow

past an impulsively started circular cylinder was extended by Takao Sano

(1978) who analysed a transient temperature field which was produced by

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sudden imposition of a constant temperature difference between the cylinder

and the fluid as the impulsive motion gets started. According to the findings in

some critical Prandtl number, the increase in Nusselt number begins in the

unseparated region in front of the separation point. For smaller values Pr, on the

other hand, the increase in Nusselt number begins after the flow separates and

the minimum Nusselt number occurs somewhere between the separation point

and the rear stagnation point.

Cebeci (1978) described the heat transfer from a circular cylinder

impulsively started from rest. The results shown that the new method can easily

cope with flow situations containing backflow. It was suggested that it may be

extended to three dimensional compressible steady flow problems with negative

cross flow.

The accuracy of their integral solutions were tested by Kamis and

Pechoc (1978). They obtained exact solutions of the boundary layer equations

on a continuously moving isothermal cylinder by a power series method. Ta

Phuoc Loc (1980) used a fourth-order scheme to solve Poisson's equation for

the vorticity transport equation. Ta Phuoc Loc presented computations for a

range of Reynolds numbers and detailed diagnostics and comparisons with

experimental results.

Choi (1982) has considered the boundary layer flow on a moving

longitudinal cylinder, taking into account the effect of the variable properties of air.

The solution has been obtained by both, the momentum integral method and the

finite difference scheme. All the aforementioned studies were related to moving

cylinders in a fluid at rest. In vortex methods the most notable studies are those of

Smith and Stansby (1988) and more recent one of Cheng and Chem (1991).

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Smith and Stansby used the method of random walks whereas Cheng

and Chern used a finite difference scheme on the grid which was used in the

Cloud-in-Cell method (CIC) to resolve the diffusion operator. Both works took

advantage of the stability properties of vortex methods to extend their

computations to very high Reynolds numbers. However, it appears that the

increase in the simulated Reynolds number was not followed by an adequate

increase in the results.

The unsteady nonsimilar forced convection flow over a longitudinal

cylinder which moves in the same direction or in the opposite direction to the

free stream has been investigated by Eswara and Nath (1992). The development

of a two-dimensional viscous incompressible flow generated from a circular

cylinder impulsively started into rectilinear motion was studied computationally

by Koumoutsakos and Leonard (1995) and Badr et al (1996).

1.7 MAGNETOHYDRODYNAMICS

As a branch of plasma physics, the field of magnetohydrodynamics

(MHD) consists of the study of a continuous, electrically conducting fluid under

the influence of electromagnetic fields. Originally, MHD included only the

study of incompressible fluids strictly (hence the inclusion of the syllable

‘hydro’), but today the terminology is applied to studies of partially ionized

gases as well. Other names have also been suggested, such as magnetofluid-

mechanics, or magnetohydrodynamics, but the original nomenclature has

persisted. The essential requirement for problems to be analysed under the laws

of MHD is that the continuum approach be applicable.

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Many natural phenomena and engineering problems were susceptible

to MHD analysis. It was useful in geophysics. Geophysicists encounter MHD

phenomena in the interactions of conducting fluids and magnetic fields that

were present in and around heavenly bodies. Engineers employ MHD principles

in, the design of heat exchangers, pumps and flowmeters, in space vehicle

propulsion, control and re-entry, in heating novel power generating systems,

and in developing confinement schemes for controlled fusion. The most

important application of MHD is in the generation of electrical power with the

flow of an electrically conducting fluid through a transverse magnetic field.

Cryogenic and super conducting magnets were required to produce very large

magnetic fields. Generation of MHD power on a smaller scale is of interest for

space applications.

Several authors have studied the natural convection boundary layer

flow of an electrically conducting fluid in the presence of magnetic field. The

natural convection boundary layer of an electrically conducting fluid over a hot

vertical wall in the presence of a strong magnetic field has been studied by

Sparrow and Cess (1961), Riley (1964) and Kuiken (1970) because of its

application in nuclear engineering in connection with the cooling of reactors.

Emerly (1963) studied the effect of a magnetic field upon the free convection of

conducting fluid.

An exact solution for the magnetohydrodynamic flow between two

rotating cylinders under radial magnetic field was studied by Arora and Gupta

(1972). Soundalgekar and Ali (1986) studied the flow of a viscous

incompressible electrically conducting fluid past an impulsively started infinite

vertical isothermal plate using finite difference technique.

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Hossain and Ahmed (1990) has analysed the combined effect of

forced and free convection with uniform heat flux in the presence of magnetic

fields. In this case, the effect of both viscous and Joule heating were neglected.

Simultaneous heat and mass transfer in free convection past horizontal

cylindrical electrodes was studied experimentally using electrochemical

limiting diffusion current technique by Sarac et al (1991). The results included

the use of a combined Grashof number to account for thermal and concentration

on buoyancy effects.

Thermal boundary layer on a continuously moving semi-infinite plate

in the presence of transverse magnetic field with heat flux has been examined

by Murty (1991). This investigation has indicated a fall in the temperature of

the thermal boundary layer with increase in magnetic field parameter. Hossain

(1992) presented the effect of heat on the flow of an electrically conducting and

viscous incompressible fluid past a semi-infinite plate in which temperature

varied linearly with the distance from the leading edge in the presence of a

uniform transverse magnetic field.

Numerical solutions were obtained for small Prandtl numbers,

appropriate for coolant liquid metal, in the presence of large magnetic field.

Pressure distribution measurements around a cylinder placed in a liquid

metal flow aligned with a constant magnetic field was investigated by

Josserand et al (1993). The pressure drag was found to be reduced by the

electromagnetic forces. It was also shown that, for a sufficient value of the

magnetic field, the Von Karman Street behind the cylinder was suppressed.

Takhar and Ram (1994) have considered the steady free and forced convection flow of water at 4°C through a porous medium bounded by an

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impermeable and semi-infinite vertical plate in the presence of a uniform

transverse magnetic field. They used shooting numerical techniques to solve

the coupled non-linear equations.

Pop et al (1994) presented a numerical solution of the problem of

forced convection layer flow of an electrically conducting incompressible fluid

past a semi-infinite flat plate in the presence of an external magnetic field.

Zebib (1996) conducted a theoretical study of the character and stability of

thermo magnetic flow in a microgravity environment. It can be shown that

convection was set as in a stable supercritical bifurcation. MHD mixed

convection flow about a vertical cylinder embedded in a porous medium was

considered by Aldoss (1996), using non-Darcian model. The magnetic field

was found to have different behaviour in the forced convection dominated

regime than that in the natural convection dominated regime.

The effect of suction and blowing on convection heat transfer from a

horizontal cylinder in cross-magnetohydrodynamic flow was investigated by

Aldoss and Ali (1997). Local non-similarity technique was used to solve the

transformed non-linear partial differential equations. Ji and Gardner (1997)

formulated an electromagnetic damping model and incorporated in K-e turbulence

model for a turbulent pipe flow in a transverse magnetic field. The complex

governing equations were solved by an implicit and non-iterative method.

The problem of free convection boundary-layer flow of an

electrically conducting fluid around a vertical flat plate embedded in a

thermally stratified porous medium in the presence of uniform magnetic field

was investigated by Chamkha (1997). It was found that both the skin-friction

coefficient and the local Nusselt number were decreased as a Hartmann number

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and the stratification parameter, or the medium inertia parameter was increased.

Elbashbeshy (1997) studied heat and mass transfer along a vertical plate with

variable surface tension and concentration in the presence of magnetic field. It

was noted that the local wall shear stress decreases with increase in the

magnetic parameter and increases with increasing Prandtl number. Jones et al

(1997) developed an implicit algorithm for solving the time dependent, non­

ideal magnetohydrodynamic equation.

Shankar and Kishan (1997) presented the effect of mass transfer on

the MHD flow past an impulsively started infinite vertical plate. Hakien et al

(1999) studied the effect of viscous and Joule heating on the flow of an

incompressible, electrically conducting micropolar fluid past a semi-infinite

plate whose surface temperature linearly varies with the distance from the

leading edge. The plate was subjected to a uniform transverse magnetic field.

Kumari and Nath (1999) studied the development of the asymmetric

flow of a viscous electrically conducting fluid in the forward stagnation point

region of a two-dimensional body over a stretching surface with an applied

magnetic field. It was found that the surfaces shear stresses corresponding to

symmetric asymmetric flows increase with the magnetic field and time. They

developed an implicit algorithm for magnetohydrodynamic equation. Acharya

et al (2000) analysed a steady two-dimensional free convection and mass

transfer flow of a viscous incompressible electrically conducting fluid through a

porous medium bounded by a vertical infinite surface with constant suction

velocity and constant heat flux in the presence of magnetic field.

Kim (2000) has examined the governing equations for unsteady,

incompressible fluid past a semi-infinite porous plate whose velocity was

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maintained at a constant value. It is embedded in a porous medium in the

presence of magnetic field. It was observed that, when the magnetic parameter

increases, the velocity decreases, whereas when the permeability parameter and

Grashof number increase the velocity increases.

The combined effects of frictional forces and magnetic field on the

thermal boundary layer near a flat plate has been considered by Singh et ai

(2000). They used a model of flat plate thermometer mounted on a moving

body as such in a flying aircraft. Yih (2000) investigated the laminar boundary

flow and heat transfer characteristics of MHD-natural convection over a

horizontal cylinder under the effect of uniform blowing/suction.

Heat and mass transfer characteristics and the flow behaviour of the

MHD flow past a vertical cylinder was reported by Ganesan and Rani (2000).

The non-dimensional governing equations were solved by an efficient, more

accurate, unconditionally stable and fast converging implicit finite difference

scheme.

Takhar et al (2001) reported the steady laminar incompressible flow

of an electrically conducting fluid over an infinite permeable disk in the

presence of an axial magnetic field. Unsteady three dimensional MHD

boundary layer flow due to the impulsive motion of a stretching surface has

been obtained by Takhar et al (2001). It was observed that the surface shear

stresses and the heat transfer increase with the stretching parameter and the

magnetic parameter, and there is a smooth transition from the short time

solution to the long time solution. Kumari and Nath (2001) has considered the

MHD flow and heat transfer of a non-Newtonian power-law fluid over a

continuously moving surface with a parallel free stream. They found that the

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heat transfer coefficient increases significantly with the Prandtl number. The

gradient of the velocity at the surface was negative when the wall velocity was

greater than the free stream velocity, and it was positive when the wall velocity

was less that the free stream.

1.8 RADIATION

Many investigators have studied two-dimensional laminar boundary

layer flow and convective heat transfer. Not much attention has been given,

however, to cases where thermal-radiation becomes an additional factor. Recent

developments in hypersonic flight, missile reentry, rocket combustion

chambers, power plants for interplanetary flight and gas cooled nuclear

reactors, has focused attention on thermal radiation as a mode of energy

transfer. It has emphasized the need for an improved understanding of radiative

transfer in these processes.

Studies with interaction of thermal radiation and free convection

were made by Arpaci (1968), Rapits (1998), Cheng and Ozisik (1972),

Bankston (1972), Cess (1992) and Hossian and Takhar (1996,1999). In all these

papers, the flow was considered steady. The unsteady flow past a moving plate

in the presence of free convection and radiation were studied by Cogley et al

(1968), Grief et al (1971), Monsour (1990), Das et al (1996) and Rapits and

Perdikis (1999). The combined radiation and free convection flow over a

vertical cylinder was studied by Yih (1999). In the literature very few authors

have studied the flow past a vertical or horizontal circular cylinder. Studies of

free convection flow along a vertical or horizontal cylinder were important in

the field of geothermal power generation and drilling operations where the free-

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stream and buoyancy-induced fluid velocities were of roughly the same order

of magnitude.

In the context of space technology and in processes involving high

temperatures, the effects of radiation were of vital importance. Novotny and

Kelleher (1967) presented a laminar free convection of an absorbing-emitting

gas in the region of stagnation point around a horizontal cylinder. Hossain et al

(1999) reported the radiation-conduction interaction on mixed convection from

a horizontal circular cylinder. They were using local non-similarity variables to

solve the problem. They observed that increase in the surface heat temperature

parameter leads to an increase in the value of local skin-friction as well as in the

local Nusselt number. The governing equations were solved numerically using

implicit finite difference scheme of Keller Box method.

The problem of radiation effects on free convection flow past a

moving vertical cylinder has important applications in the study of geological

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

assessment of aquifers, geothermal reservoirs and underground nuclear waste

storage sites. Here intrusive magma may be taken as an isothermal vertical

cylinder with an impulsive motion subjected to radiative flux. Hence, it was

proposed to study the radiation effects of heat and mass transfer on the natural

convection of an incompressible viscous fluid past a moving semi-infinite

isothermal vertical cylinder in the vertically upward direction.

1.9 CHEMICAL REACTION

Diffusion rates can be tremendously altered by chemical reactions.

The effects of chemical reactions can be codified as either heterogeneous or

23

homogeneous processes. This depends on whether they occur as an interface or

as a single-phase volume reaction. Many transport processes exist in nature and

in industrial applications, in which the simultaneous heat and mass transfer

occur as a result of combined buoyancy effects of thermal diffusion and

diffusion of chemical species. However, in nature, along with the free

convection currents caused by temperature differences, the flow was also

affected by the differences in concentration. It was found to be useful in

chemical processing industries such as food processing, polymer production

and cooling towers. During geophysical exploration, the temperature and mass

distribution around the intrusive magma plays an important role in the

geothermal resources. Hence the intrusive magma may be taken as vertical

cylinder with heat and mass flux boundary condition. The time required to set

in for the intrusive magma was very essential.

Chambre and Young (1958) studied the problem of first order

chemical reactions in the neighbourhood of a flat plate for destructive and

generative reactions. The apparent kinetics of zeroth order surface-catalysed

reactions were quantitatively investigated by Rosner (1966) for several

configurations involving laminar or turbulent boundary layer flow, with or

without pressure gradient.

Ramanamuthy and Govindarao (1971) have considered a first-order

isothermal and irreversible chemical reaction taking place on the surface of a

cylindrical catalyst pellet. Takhar et al (1977) analysed the laminar boundary

layer flow on a moving continuous flat sheet with suction and injection for

diffusion Equation, taking into account the homogeneous chemical reaction of

nth order. They observed that the diffusive mass flux at the moving plate

increases with an increase either in the reaction rate parameter or in the order of

24

the reaction. An experimental study on mass transfer with chemical reactions

was analysed by Ogawa (1987) in cross flow containing Oxygen past a porous

graphite cylinder.

Andersson et al (1994) reported the transfer of a chemically reactive

species in the laminar flow over a linearly stretching surface. Das et al (1994)

analysed the theoretical solutions of mass transfer effects on the flow past an

impulsively started infinite vertical plate with uniform heat flux, taking into

account the homogeneous chemical reaction of first order. Exacts solutions

were derived by the Laplace transform technique. They observed that an

increase in the reaction parameter leads to a decrease in the concentration of the

species.

Streamline upwind Petrov-Galerkin finite element method (SUPG

FEM) was considered by Park (1995) to solve the general chemical reactive

species flow system with convection, diffusion, and reaction. In this study the

fixed Pseudo-homogeneous bed catalytic reactor model with Dankwert

boundary condition in two-dimensional domain was selected as an illustrative

example. It was found that the velocity distribution (uniform or laminar flow)

plays an important role on the dynamic behaviours of the concentration and

thermal waves and their steady state solutions. Pan (1996) described the

problem of a chemical reaction-diffusion process in which the reaction takes

place only at some local sites, due to the presence of a catalyst. It was found

that chemical concentration was continuous, but the gradient of the

concentration had jumped at the local sites.

Das et al (1999) studied an exact solution for the flow of viscous

incompressible fluid past an impulsively started infinite vertical plate in the

25

presence of mass transfer and first order chemical reaction. They showed that

an increase in the chemical reaction parameter led to decrease in the velocity of

air and water. Takhar et al (2000) studied the flow and mass transfer

characteristics of a viscous electrically conducting fluid on a continuously

stretching surface with non-zero slot velocity. It was observed that the surface

with mass transfer for the first order reaction is more than that of the second or

higher order reaction. Recently, Ganesan and Rani (2000) discussed the

diffusion of chemically reactive species in convective flow along a vertical

cylinder. The dimensionless governing equations were solved by an implicit

finite difference scheme of Crank-Nicolson type. Air and water were the fluids

considered for this study.

1.10 GOVERNING EQUATIONS

The basic partial differential equations used to interpret and analyse

natural convection which results from the consideration of the conservation of

mass (equations of continuity), of force momentum (Navier-Stokes equation),

of energy (energy equation) and of molecular species (mass diffusion equation).

In free convection, the fluid motion arises solely from the buoyancy

forces. The buoyancy effect arises due to the interaction between the density

differences in a body of fluid and body force, usually gravitational force. The

density differences are due to the temperature differences or concentration

differences of the diffusing species or the combination of these two. So, both

thermal and mass diffusing processes must be considered simultaneously for all

the aspects of flow.

26

A two-dimensional boundary layer governing equations of laminar

free-convection flow of an incompressible viscous fluid past an impulsively

started semi-infinite vertical cylinder using boundary layer approximation and

axisymmetric (Schlichting 1968, Eckert and Drake 1987) are

d(ru) d(rv) d x dr

(1-1)

( du du du —+u—+v~- dt dx dr

= _pgdp p d f du

+dx r dr dr

(1-2)

BY dr BY a B ( BY)----- b u-----1- v = r dt' dx dr rdr^ dr ^ (1.3)

dc' dc' dC' D df BC)—- + u——+ v---- =------- r dt dx dr rdr dr\

(1.4)

The fluid properties are assumed to be constant except for the density

variations, which induces the buoyancy force. It is also assumed that the heat

due to viscous dissipation in the energy equation is negligibly small, which is

possible in the case of ordinary fluid flow like air or water under usual

gravitational force. However, when the gravitational force is intensive or when

the Prandtl number of the fluid is very high, the viscous dissipative effects

cannot be neglected. In the species equation, the following assumptions are

made, (i) The concentration of the diffusing species is very low compared to the

other chemical species present in the fluid and (ii) There is a first order

homogeneous chemical reaction between the diffusing species and the fluid.

27

The viscous effects are negligibly small, outside the boundary layer

(i.e. r —> °o), hence the momentum Equation (1.2) reduces along a streamline to

-|^“Poog = 0

dx(1.5)

Hence, the pressure is taken as practically constant in a direction

normal to the boundary. Subtracting Equation (1.5) from Equation (1.2), the

Equation of momentum reduces to

du du du Adt' + u f V-

dx dr=-g(p-PM)

dp u d. du,~ + “ ^-(r-r-)dx r dr dr

(1.6)

For small temperature and concentration differences, the density p in

the Equation (1.6) is considered to be constant except for the term (p - p„).

This approximation is first introduced by Boussinesq.

Since the flow is driven by the buoyancy forces arising from the

density differences due to both temperature and concentration difference, the

density differences can be expressed by the Equation

dp=ar dC'

(1.7)

Here the effect of buoyancy force will be expressed in terms of the

volumetric coefficient of thermal expansion (3 and a volumetric coefficient of

expansion with concentration P*.

28

The thermal expansion coefficient for a fluid is defined by the

equation

p=-S{as A3T' L\ /p,C

(1.8)

where S indicates the specific volume of the fluid.

Use of the fluid density p for the specific volume S of the fluid

results in

p = -1 / -j \

laT')P,c(1.9)

Similarly an expression for coefficient of mass transfer can be

defined through the equation

p*=I p s^ as ^ac' Jp,T'

(1.10)

Also, this co-efficient of expansion can be expressed in terms of fluid

density p as

if aP ^Jp,T'

p* = p ac(l.ii)

In view of the Equations (1.9) and (1.11) the Equation (1.7) becomes

dp = -p(p dT'+p* dC') (1.12)

29

Which can be written as

P-Poo =-p[p(T,-0 + p*(C'-C/oo)] (1.13)

Introducing the Equation (1.13) into the Equation (1.6) the

momentum equation reduces to

du du du ~+u—+v— at dx dr =Pgp(T-r )+Pgp (c'-c'c#)+£—

r dr

p d { du ^

V 8r/(1.14)

du du du a , , * , , v dt.e. —+u--+v— =gp(T -TJ+gp (C-Coo) + -~

dt dx dr 00 00 r dr( a,. \

r.dud7 (1.15)

Hence, the governing equations of free convection are Equations

(1.1), (1.15), (1.3) and (1.4). The boundary conditions are prescribed

appropriately when the problems are discussed.

1.11 FINITE DIFFERENCE METHOD

Extensive theoretical and experimental work has been done on free

convection. The theoretical work when applied with minor modification can be

used for practical purpose. Apparently many of the topics were chosen because

numerical methods have given us a new dimension of power in natural

convection as well as in many other fields. Numerical finite difference methods

of solving partial differential equations are of increasing interest and

importance, because of the great advances in computer technology.

30

The principle attraction of numerical method is that solutions are

possible for many problems, which resist analytical methods. Modem

computers paved way for the development of efficient and more general

numerical techniques, which may permit solution for the most difficult

problems of heat and mass transfer. From the numerical methods available for

solving the boundary layer equations, finite difference methods are more

frequently used and they provide numerical solutions in a simple and efficient

manner.

In finite difference methods, the region of integration of the

governing equations is divided into a system of rectangular meshes formed by

two sets of lines, parallel to the coordinate axes. The numerical values of the

dependent variables are obtained at the intersecting points, which are called

mesh points or nodal points. The philosophy of the finite difference methods is

to replace the partial derivatives appearing in the governing equations with

algebraic difference quotients, yielding a system of algebraic equations, which

can be solved for the flow-field variables at the specific discrete grid points in

the flow. Accuracy can be improved by increasing the number of grid points.

The two types of finite difference methods for time dependent

problem are explicit and implicit methods. A formula that expresses one

unknown nodal value directly in terms of known nodal values is called explicit

method. This method is very simple to set up a program but computationally

costly. An implicit formula involves more than one grid point at the advanced

time level. This procedure leads to set simultaneous equations. Whereas

implicit methods are more complicated to set up and program, but are

unconditionally stable.

800706

5\ P2-31

There is no guarantee that the solutions obtained by the finite

difference method will be very accurate or even stable. So any finite difference

method must satisfy the following three important properties:

i) Stability

ii) Compatibility

iii) Convergence.

The detailed explanations are as follows:

i) STABILITY: If it is possible to carry out the calculations to an

infinite number of decimal places and if the initial and boundary

values were specified exactly, the numerical calculations would

produce the exact solution of the difference equations. In practice, of

course, each calculation is carried out to a finite number of decimals

and hence round off errors are introduced. The solution thus

computed may not be the exact solution of the difference equation.

Thus, a set of finite difference equations is said to be stable when the

cumulative effect of all rounding error is negligible or bounded.

ii) COMPATIBILITY: Finite difference equations are derived using the

Taylor's series expansion for two variables, neglecting the higher order

terms in the series. These terms contribute a truncation error. It is

required that the truncation error should tend to zero as the mesh sizes

approach zero. Otherwise, the finite difference scheme is said to be

incompatible or inconsistent with the partial differential equation. In

the case of inconsistency, the finite difference solution is not likely to

approach the desired solution of partial differential equation.

32

iii) CONVERGENCE: The term convergent is understood to mean that

the exact solution of the finite-difference problem (in the absence of

round-off error) tends to the solution of the partial differential

equation as the step sizes in time and distance tend to zero.

More about these criteria are given in Carnahan et al (1969), Mitchell

(1969), Wolf (1983) and Smith (1986).

1.12 AIM AND SCOPE OF THE THESIS

It can be clearly observed from the literature review that all the

investigators who have studied the free convective flow over a cylinder have

solved steady state partial differential equations theoretically or measured the

steady state solutions experimentally. But the effect of buoyancy forces on flow

over an impulsively started semi-infinite vertical cylinder with heat and mass

transfer has not received the attention of any researcher. Because of this, the

author has made an attempt to study the two-dimensional, laminar, unsteady

natural convective flow past an impulsively started semi-infinite vertical

cylinder with heat and mass transfer. The effects of magnetic field, radiation

and chemical reaction are also studied.

The main aim of the present thesis is to solve some problems of

natural convective flow past an impulsively started semi-infinite vertical

cylinder and to study the flow variables, skin-friction coefficients, heat and

mass transfer in the transient period for various values of the parameters.

The non-dimensional governing boundary layer equations are

unsteady, coupled and non-linear. No analytical method is available to solve

33

such a problem. The numerical method, particularly finite difference method,

paves the way to solve such problems. In the present work, an implicit finite

difference scheme of Crank-Nicolson type has been employed to solve the

problem because the scheme is unconditionally stable and is more accurate.

1.13 ORGANISATION OF THE THESIS

The thesis is divided into five chapters. Chapter 1 is a brief

introduction of natural convection and its application. It deals with the

systematic development of the literature survey of work done in the field of

natural convection.

Chapter 2 deals with the transient free convection boundary layer

flow of an incompressible viscous fluid past an impulsively moving semi­

infinite vertical cylinder with heat and mass transfer. The temperature and

concentration on the cylinder surface are taken to be uniform. The unsteady,

nonlinear and coupled governing equations of the flow are solved using an

implicit finite difference scheme. The finite difference scheme is

unconditionally stable and accurate. The stability of the finite difference

scheme is discussed in detail. Numerical results are presented with various sets

of parameters for both air and water. Transient effects of velocity, temperature

and concentration profiles are analyzed. Local and average skin friction, rates

of heat and mass transfer are shown graphically.

The problem of an unsteady, two-dimensional free convection MHD

flow of an incompressible, viscous fluid past an impulsively moving semi­

infinite vertical cylinder with constant heat flux has been considered in

Chapter 3. The parabolic partial differential equations governing the unsteady

34

flow have been solved numerically using an implicit finite difference scheme.

The effects of physical parameters such as magnetic parameter, Prandtl number

and Grashof number on the velocity, temperature, skin-friction and the rate of

heat transfer are discussed. Numerical results are presented graphically.

The interaction of free convection with thermal radiation of a viscous

incompressible unsteady flow past a moving vertical cylinder with heat and

mass transfer is analyzed in Chapter 4. The fluid is a gray, absorbing-emitting

but non-scattering medium and the Rosseland approximation is used to describe

the radiative heat flux in the energy equation. The governing equations are

solved using an implicit finite difference scheme of Crank-Nicolson type.

Numerical results for the transient velocity, temperature, concentration, local as

well as average skin-friction, Nusslet number and Sherwood number are shown

graphically.

Chapter 5 deals with heat and mass flux effects on a moving vertical

cylinder with chemically reactive species diffusion. The heat supplied and mass

diffused from the cylinder to the fluid are at a uniform rate. There is a first

order homogeneous chemical reaction between the diffusing species and the

fluid. The dimensionless governing equations are solved by an implicit finite

difference scheme of Crank-Nicolson type. Numerical results are computed for

both generative and destructive reactions for various physical parameters such

as chemical reaction parameter, thermal Grashof number, mass Grashof

number, Schmidt number and Prandtl number. The effects of velocity,

temperature, concentration, shear stress, rate of heat transfer and rate of mass

transfer are presented in the form of graphs.