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CHAPTER 1

Introduction &

Basic Concepts

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1.1. INTRODUCTION

Flow of fluids bounded by one or two planes aroused the dragging interest of

the mathematicians and provoked extensive study in the literature. In view of the fact

that there are ideal models of practical importance and interest in exploring the

general features of fluid forces, like viscous force, electromagnetic force, viscoelastic

force, gravitational force and corialis force. The flow of fluids bounded by one or two

vertical planes is particularly important in the area of heat transfer. The following are

some of the important areas in fluid dynamics in which flows through such

geometries are explored by several authors.

1.1.1. Newtonian fluids

1.1.2. Non-Newtonian fluids

We make a brief survey of the fundamental aspects concerning these areas of

study which in turn will provide a foundation for formulating the problem of this

dissertation.

1.1.1. NEWTONIAN FLUIDS

The classical Navier-Stokes equation of motion is derived by assuming a

linear relationship between stress tensor and rate of strain in the fluid. Fluids which

obey this relationship are known as Newtonian fluids. They possess single rheological

property called viscosity. Water, air, mercury and engine oil are some of the examples

of Newtonian fluids.

1.1.2. NON-NEWTONIAN FLUIDS

Many important industrial fluids are Non-Newtonian in their flow

characteristics. These include paints, various suspensions, glues, pulps, printing inks,

food materials, soap and detergent slurries polymer solutions, medical cream

manufacturing, medical biotechnological areas, chemical engineering process and

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many others. Because such fluids have more complicated equations that relate the

stress to the velocity gradient that is the case with Newtonian fluids, new branches in

the field of fluid mechanics and heat transfer are developed.

Another important characteristic of such fluids, because of their large apparent

viscosities, they have a tendency towards low Reynolds and Grashof numbers and

high Prandtl numbers. Thus laminar flow situations are encountered more often in

practice than with Newtonian fluids. The subject of thermo physical properties and

their measurements is an important one when dealing with Non-Newtonian fluids.

The Non-Newtonian fluids can in turn be divided into purely viscous and viscoelastic

fluids. The purely viscous time-dependant fluids are defined as those whose shear

stress depends only upon some function of the shear rate, and sometimes an initial

yield stress. Viscoelastic fluids are those which possess properties of both viscosity

and elasticity. We list below various Non-Newtonian fluids and some examples.

Type Examples

Newtonian

Pseudo plastic

Dilatants

Bingham plastic

Thixotrophic

Rheopactic

Viscoelastic

(Non-Newtonian)

Paints, glues, blood, suspensions

Water, air, mercury, engine oil

Wet sand, sugar and borax solutions

certain emulsions land paints

Printing inks, food materials, paints

Clay suspensions

Medical creams, coaxial mixers, blood oxygenators, Pulps

Polymer solutions (ex: polyox water)

4 1.2. NATURAL (FREE) CONVECTION

The fluid motion may be caused by external mechanical forces is called forced

convection. Forced convection is a mechanism, or type of transport in which fluid

motion is generated by an external source (like a pump, fan, suction device, etc.).

Whereas free convection flow occurs frequently in nature. Free convection occurs not

only due to the temperature difference but also due to the concentration differences

(or) combination of these two. In natural convection, fluid surrounding a heat source

receives heat, becomes less dense and rises. The surrounding, cooler fluid then moves

to replace it. This cooler fluid is then heated and the process continues, forming

convection current; this process transfers heat energy from the bottom of the

convection cell to top. The driving force for natural convection is buoyancy, a result

of differences in fluid density.

• Buoyancy forces are responsible for the fluid motion in natural convection.

• Viscous forces oppose the fluid motion.

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1.2.1. UNSTEADY FREE CONVECTION

• The unsteadiness in the flow field is caused either by time dependent motion of the

external stream (or the surface of the body) or by Impulsive motion of the external

stream (or the body surface).

• When the fluid motion over a body is created impulsively, the inviscid flow over

the body is developed instantaneously but the viscous layer near the body is

developed slowly and it becomes fully developed steady state viscous flow after

certain instance of time.

• The study of unsteady boundary layer is useful in several physical problems such

as flow over a helicopter in translation motion, flow over blades of turbines and

compressors, flow over the aerodynamic surfaces of vehicles in manned flight, etc.

The study of unsteady boundary layer is useful in several physical problems

such as flow over a helicopter in translation motion, flow over blades of turbines and

compressors, flow over the aerodynamic surfaces of vehicles in manned flight, etc.

The unsteadiness in the flow field is caused either by time dependent motion of the

external stream (or the surface of the body) or by impulsive motion of the external

stream (or the body surface). When the fluid motion over a body is created

impulsively, the inviscid flow over the body is developed instantaneously but the

viscous layer near the body is developed slowly and it becomes fully developed

steady state viscous flow after certain instance of time.

1.3. HEAT TRANSFER

Heat transfer is a phenomena associated with both Newtonian and Non-

Newtonian fluids. The problems concerning heat transfer have many applications in

science and engineering such as the design of cooling systems for motors, generators

and transformers. Chemical engineers are concerned with the evaporation,

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condensation, heating and cooling of fluids. Heat transfer is associated with the

process of transmission of internal energy (Kinetic energy) from one region to another

as a result of temperature difference between them. It is customary to categorize the

various heat transfer processes into three basic modes viz. conduction, convection and

radiation. This flow of energy or heat passes from the high energy molecules to the

lower energy ones.

Convection is the term applied to the heat transfer mechanism which occurs in

a fluid by the mixing of one portion of the fluid with another portion due to gross

moments of the mass of fluid, but the energy may be transported from one point in

space to another by the displacement of the fluid itself is still one type of conduction.

In the presence of a temperature distribution in the fluid, the velocity field and

temperature field mutually interacts, which means that the temperature and the

velocity depend on one another. In special case when buoyancy force 휌푔̅ may be

disregarded, and the properties of the fluid may be assumed to be independent of

temperature, mutual interaction ceases, and the velocity field no longer depends on

the temperature field, although the converse dependence of the temperature on

velocity field still persists. This happens at large velocities (large Reynolds numbers)

and small temperature differences, such flows being termed forced. The process of

heat transfer in such flows is described as Forced Convection. Flows in which

buoyancy forces are dominant are called free, the respective heat transfer being

known as Free Convection.

1.4. HEAT AND MASS TRANSFER OF VISCOELASTIC FLUIDS

Heat and mass transfer from a vertical flat plate is encountered in various

applications such as heat exchangers, cooling systems and electronic equipment. In

addition, Non-Newtonian fluids such as (viscoelastic) molten plastics, polymers,

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medical cream manufacturing, formation and dispersion of fog, distribution of

temperature and moisture over agricultural fields and in drying process of paper,

glues, ink, pulps, foodstuffs or slurries are increasingly used in various

manufacturing, industrial and engineering applications especially in the chemical

engineering processes. Many transport processes exist in Geophysics, Aeronautical

Engineering and Industrial applications in which the transfer of heat and mass occurs

simultaneously. The problem of heat transfer in the boundary layer on a continuous

moving surface has many practical applications in manufacturing process industry.

The Walters-B viscoelastic model was developed to simulate viscous fluids

possessing short memory elastic effects and can simulate accurately many complex

polymeric, biotechnological and tribiological fluids. The standard boundary-layer

equations play a central role in many aspects of fluid mechanics as they describe the

motion of a slightly viscous fluid close to a surface. An unsteady two-dimensional

laminar natural convection radiative flow of a viscoelastic fluid past a vertical plate is

considered here. The x-axis is taken along the plate in the upward direction and

y- axis is taken normal to it. The physical models are show in figs.1 and 1(a).

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1.5. SIGNIFICANT PARAMETERS IN CONVECTIVE HEAT AND MASS

TRANSFER

1.5.1. NUSSELT NUMBER

In heat transfer at a boundary (surface) within a fluid, the Nusselt number is

the ratio of convective to conductive heat transfer across (normal to) the boundary. In

this context, convection includes both advection and conduction, named after

Wilhelm Nusselt, it is a dimensionless number. The conductive component is

measured under the same conditions as the heat convection but with a (hypothetically)

stagnant (or motionless) fluid a Nusselt number close to one, namely convection and

conduction of similar magnitude, is characteristic of "slug flow" or laminar flow. A

larger Nusselt number corresponds to more active convection, with turbulent flow

typically in the 100– 1000 range. The convection and conduction heat flows are

parallel to each other and normal to the boundary surface, and are all perpendicular to

the mean fluid flow in the simple case.

푁푢 = ℎ퐿푘 =

퐶표푛푣푒푐푡푖표푛ℎ푒푎푡푡푟푎푛푠푓푒푟푐표푒푓푓푖푐푖푒푛푡퐶표푛푑푢푐푡푖푣푒ℎ푒푎푡푡푟푎푛푠푓푒푟푐표푒푓푓푖푐푖푒푛푡

퐿 = Characteristic length

kf = Thermal conductivity of the fluid

ℎ = Convective heat transfer coefficient

Selection of the characteristic length should be in the direction of growth (or

thickness) of the boundary layer. Some examples of characteristic length are: the

outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder

axis), the length of a vertical plate undergoing natural convection, or the diameter of a

sphere. For complex shapes, the length may be defined as the volume of the fluid

body divided by the surface area. The thermal conductivity of the fluid is typically

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(but not always) evaluated at the film temperature, which for engineering purposes

may be calculated as the mean-average of the bulk fluid temperature and wall surface

temperature. For relations defined as a local Nusselt number, one should take the

characteristic length to be the distance from the surface boundary to the local point of

interest.

DERIVATION

The Nusselt number may be obtained by a non dimensional analysis of the

Fourier's law since it is equal to the dimensionless temperature gradient at the surface:

풒 = −풌훁푻

Where q is the heat flux, k is the thermal conductivity and T is the fluid temperature.

Indeed if: ∇ = −퐿∇ and 푇 =

We arrive at: −∇ 푇 = −( )

,푞 =

Then we define:푁푢 =

So the equation becomes: 푁푢 = ∇ 푇

Dimensionless parameters are often used to correlate convective transfer data.

In momentum transfer Reynolds number and friction factor play a major role. In the

correlation of convective heat transfer data, Prandtl and Nusselt numbers are

important. Some of the same parameters, along with some newly defined

dimensionless numbers, will be useful in the correlation of convective mass-transfer

data. The molecular diffusivities of the three transport process (momentum, heat and

mass) have been defined as follows

푀표푚푒푛푡푢푚푑푖푓푓푢푠푖푣푖푡푦휈 = 휇휌

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푇ℎ푒푟푚푎푙푑푖푓푓푢푠푖푣푖푡푦훼 = 푘휌퐶

푀푎푠푠푑푖푓푓푢푠푖푣푖푡푦퐷

It can be shown that each of the diffusivities has the dimensions of ; hence, a

ratio of any of the two of these must be dimensionless. The ratio of the molecular

diffusivity of momentum to the molecular diffusivity of heat (thermal diffusivity) is

designated as the Prandtl number

푀표푚푒푛푡푢푚푑푖푓푓푢푠푖표푛푡ℎ푒푟푚푎푙푑푖푓푓푢푠푖표푛 = Pr =

휈훼 =

퐶 휇푘

The analogous number in mass transfer is Schmidt number given as

푀표푚푒푛푡푢푚푑푖푓푓푢푠푖푣푖푡푦푀푎푠푠푑푖푓푓푢푠푖푣푖푡푦 = 푆푐 =

휈퐷 =

휇휌퐷

1.5.2. SKIN-FRICTION

Skin friction arises from the friction of the fluid against the "푠푘푖푛" of the

object that is moving through it. Skin friction arises from the interaction between the

fluid and the skin of the body, and is directly related to the wetted surface, the area of

the surface of the body that is in contact with the fluid. As with other components of

parasitic drag, skin friction follows the drag equation and rises with the square of the

velocity. The skin friction coefficient 퐶 is defined by:

퐶 = 푇

12 휌푢∞

Where is 푇 the local wall shear stress, 휌 is the fluid density and 푢∞ is the free-

stream velocity (usually taken outside of the boundary layer or at the inlet).

For comparison, the turbulent empirical relation known as the 1/7 Power Law

(derived by Theodore von Karman) is: Skin friction is caused by viscous drag in the

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boundary layer around the object. The boundary layer at the front of the object is

usually laminar and relatively thin, but becomes turbulent and thicker towards the

rear. The position of the transition point depends on the shape of the object. There are

two ways to decrease friction drag: the first is to shape the moving body so that

laminar flow is possible, like an airfoil. The second method is to decrease the length

and cross-section of the moving object as much as is practicable. To do so, a designer

can consider the fineness ratio, which is the length of the aircraft divided by its

diameter at the widest point (L/D).

1.5.3. SHERWOOD NUMBER

The Sherwood number, (also called the mass transfer Nusselt number) is a

dimensionless number used in mass-transfer operation. It represents the ratio of

convective to diffusive mass transport, and is named in honour of Thomas Kilgore

Sherwood. It is defined as follows

푆ℎ = . =

Where

L - is a characteristic length (m)

D - is mass diffusivity (m2.s−1)

K - is the mass transfer coefficient (m.s−1)

1.5.4. SCHMIDT NUMBER

Schmidt number is a dimensionless number defined as the ratio of momentum

diffusivity (viscosity) and mass diffusivity, and is used to characterize fluid flows in

which there are convection processes like simultaneous momentum and mass

diffusion. It was named after the German engineer Ernst Heinrich Wilhelm Schmidt

(1892-1975). Schmidt number is the ratio of the shear component for diffusivity

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viscosity/density to the diffusivity for mass transfer퐷. It physically relates the relative

thickness of the hydrodynamic layer and mass transfer boundary layer. It is defined as

푆푐 =휈휌퐷 =

푉푖푠푐표푢푠푑푖푓푓푢푠푖표푛푟푎푡푒푚표푙푒푐푢푙푎푟(푚푎푠푠)푑푖푓푓푢푠푖표푛푟푎푡푒

Where:

휈 − The kinematic viscosity or ( ) in units of (m2/s)

퐷 − The mass diffusivity (m2/s).

휇 −Is the dynamic viscosity of the fluid (Pas or N s/m² or kg/ms)

휌 −Is the density of the fluid (kg/m³).

1.5.5. PRANDTL NUMBER

The Prandtl number is a dimensionless number; the ratio of momentum

diffusivity (kinematic viscosity) to thermal diffusivity. It is named after the German

physicist Ludwig Prandtl. It is defined as:

푃푟 =휈훼 =

푉푖푠푐표푢푠푑푖푓푓푢푠푖표푛푟푎푡푒푇ℎ푒푟푚푎푙푑푖푓푓푢푠푖표푛푟푎푡푒 =

퐶 휇푘

Where:

휈 : Kinematic viscosity, 휈 = , (SI units : m2/s)

훼 : Thermal diffusivity, 훼 = , (SI units : m2/s)

휇 : Dynamic viscosity, (SI units: Pa s = (N s)/m2)

푘: Thermal conductivity, (SI units: W/ (m K))

퐶 : Specific heat, (SI units: J/ (kg K))

휌 : Density, (SI units: kg/m3).

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Note that whereas the Reynolds number and Grashof number are subscripted

with a length scale variable, the Prandtl number contains no such length scale in its

definition and is dependent only on the fluid and the fluid state. As such, the Prandtl

number is often found in property tables alongside other properties such as viscosity

and thermal conductivity.

Typical values for 푃푟 are:

(Low 푃푟 - conductive transfer strong)

around 0.015 for mercury

around 0.16-0.7 for mixtures of noble gases or noble gases with hydrogen

around 0.7-0.8 for air and many other gases,

between 4 and 5 for R-12 refrigerant

around 7 for water (At 20 degrees Celsius)

13.4 and 7.2 for seawater (At 0 and 20 degrees Celsius respectively)

between 100 and 40,000 for engine oil

Around 1×1025 for Earth's mantle.

(High 푃푟 - convective transfer strong):

For mercury, heat conduction is very effective compared to convection

thermal diffusivity, which is dominant. For engine oil, convection is very effective in

transferring energy from an area, compared to pure conduction (momentum

diffusivity). In heat transfer problems, the Prandtl number controls the relative

thickness of the momentum and thermal boundary layers. When 푃푟 is small, it

means that the heat diffuses very quickly compared to the velocity (momentum). This

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means that for liquid metals the thickness of the thermal boundary layer is much

bigger than the velocity boundary layer.

1.6. RADIATION

All substances (solids, liquids and gases) at normal and especially at elevated

temperatures emit energy in the form of radiation and are also capable of absorbing

such energy. This shows that all heat transfer processes are accompanied by a heat

exchange by radiation. However in some cases heat exchange by radiation may be

very small fraction of the total quantity of heat exchanged, as such it may be

neglected. In case significant amount of heat transfer occurs by radiation, then use

may be made of the various laws of radiation.

The relative importance of the various modes of heat transfer differs

considerably with the temperature. Heat transfer by conduction and convection

depends basically on the temperature difference and is little affected by the

temperature level. For example, other factors remaining constant, heat transfer by

conduction or convection from a body at 10000 c to a body at 2000 c remains the same

as that from same body at 9000 c to a body at 1000 c. In case of radiation this,

however, does not hold good. There may be about 35% more heat transfer at higher

temperature even for the same temperature difference assuming all other factors as

constant. Another difference between the radiation and the other modes of heat

transfer lies in the fact that radiation heat transfer does not require any intermediate

medium where as in case of conduction and convection medium for heat transfer is

essentially required. Radiative heat and mass transfer may occur from a hot body,

through a cold non-absorbing medium leaving it unaffected, and then reach a

‘푤푎푟푚푒푟’ body.

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Several theories have been proposed to explain the transport of energy by

radiation. Whichever theory is used, radiant energy is the same type of wave motion

as radio waves, X-rays and light waves except for the wavelength. In fact there is a

whole spectrum of electromagnetic radiation in which the various arbitrary divisions

are referred to by names reflecting the methods of origin or some characteristic

quantity. All forms have the same velocity of propagation but different wavelengths

and sources of origin. All forms produce heat when absorbed. Nevertheless it is only

the electromagnetic radiation produced by the virtue of the temperature of the emitter

that we call thermal radiation. The following Table gives the approximate ranges of

wavelength of some forms of radiation. The amount of thermal radiation emitted by a

body depends on its temperature and surface condition. Radiant energy emitted by a

hot body is not confined to the visible range of wave length.

CHARACTERISTIC WAVELENGTHS OF RADIATION

NAME WAVELENGTH RANGE IN MICRONS*

Cosmic rays up to ( 10 )

Gamma rays 1( 10 ) to 140( 10 )

X-rays 6( 10 ) to 100,000( 10 )

Ultraviolet rays 0.014 to 0.4

Visible or light rays 0.4 to 0.8

Infrared rays 0.8 to 400

Radio 10( 10 ) to 30,000( 10 )

*1 micron = ퟏퟎ ퟔ meter

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But extends itself on both sides somewhat beyond this region. A thermometer

placed in a dark or invisible region beyond the red end of a solar spectrum will detect

a temperature rise.

1.6.1. LAWS OF THERMAL RADIATION

(a). Plank’s law

In 1900, Max Plank developed the quantum theory of electromagnetic waves

and with the help of this he has suggested the following formula for the

monochromatic emissive power (퐸 ) of a block-body which is based on his

theoretical analysis, given as

퐸 = 푐 휆

[(푒) / − 1]

Where the values of 퐶 and 퐶 are given as follows:

퐶1 = 3.21푥10 푘푐푎푙 −푚 /ℎ푟

= 3.21푋10

푋푐푚

퐴푠1휇 = 10 푐푚

∴ 1푐푚 = 휇

10

∴ 푐푚 =

Substituting this in the above equation

퐶 =3.21x10

10 푘푐푎푙– 휇푚 − ℎ푟 푋푐푚

퐶 = 3.21x10 푘푐푎푙 −휇푚 − ℎ푟

and 퐶 = 1.438 푐푚 − 푘

= 14380 휇 − 푘

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(b). Rayleigh - Jean’s law

Describe the spectral radiance of electromagnetic radiation at all wavelengths

from a black body at a given temperature through classical arguments. The Rayleigh–

Jeans law agrees with experimental results at large wavelengths (or, equivalently, low

frequencies) but strongly disagrees at short wavelengths (or high frequencies). This

inconsistency between observations and the predictions of classical physics is

commonly known as the ultraviolet catastrophe and its resolution was a foundational

aspect of the development of quantum mechanics in the early 20th century.

Planck’s law has two limiting cases one of which is that when the product 휆푇

is large compared with the constant 퐶 . With this provision we can confine ourselves

to only two terms of the exponential function (1) expanded into a series with the

퐶 /휆푇 exponent:

푒 / = 1 + 11!

푐휆푇 +

12!

푐휆푇 + − −− −− −− −−

The equation (1) becomes

퐸 = 2휋푐 푇푐 휆

This relationship expresses Rayleigh-Jean’s law

(c). Wien’s law of deviation

The second extreme case corresponds to a small value of the product 휆푇 as

compared to constant 퐶 . Then, the unity present in the denominator of equation

(5.11) can be neglected and the relationship becomes Wien’s law (1893).

퐸 = 2휋푐휆 푒 /

Coordinates of the maximum values of the emissive power can be obtained from the

extreme value. For this purpose the derivative of the function is found for the wave

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length. Equating the derivative to zero, we obtain the following transcendental

equation:

푒 + 푐

5휆 푇 − 1 = 0

Whose solution is,

푐5휆 푇 = 4.965

From which 휆 푇 = 2.8978x10

Where 휆 is the wave length corresponding to a maximum intensity of

radiation, the product 휆 푇 is measured in units of m .k. As Earth warms from

solar heating it radiates heat back into space more efficiently. Eventually it radiates

back as much energy as it receives and the temperature stops changing.

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The figure at below shows that a heated object radiates energy (E) across a

broad spectrum of wavelengths. Emission drops sharply towards the short

wavelengths but spreads broadly through the longer wavelengths. A simple equation

called Wien’s Law describes the position of lambda max, the wavelength at which the

most energy is radiated. As shown by Wien’s Law, a hot body radiates energy at

shorter wavelengths than does a cooler body. This can be very important.

Earth’s atmosphere is transparent to the short waves that transmit the majority

of the Sun’s energy. These rays strike and warm the ground surface. The heated

ground radiates energy back into Space, but at longer wavelengths because the ground

is much cooler than the Sun. These longer, infrared rays are absorbed by 퐻 표, 퐶표

and 퐶퐻 in the atmosphere. This greenhouse effect warms the atmosphere by about

33o C (60o F).

(d). Stefan-Boltzmann’s Law

A hot body radiates energy (퐸 ) much more efficiently than a cooler body.

Ideally the relationship is given by the Stefan-Boltzmann’s Law, relates the total

amount of radiation emitted by an object to its temperature:

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퐸 = 퐸 푑휆 = 휎 푇 (표푟)퐸 = 휎 푇

Where:

퐸 = Total amount of radiation emitted by an object per square meter (Watts m-2)

휎 = Is a constant called the Stefan-Boltzmann constant = 5.67 x 10-8 Watts m-2 K-4

푇 = Is the temperature of the object in K.

To facilitate practical calculations, the above equation usually presented as

퐸 = 푐 푇

100

Where 푐 = 5.6687 = 5.67 is the radiation constant of a block-body.

(e). Kirchhoff’s Law of Radiation

In 1859 Robert Gustav Kirchhoff (1824 Mar 12–1887 Oct 17) set the science

of radiation thermodynamics onto a proper mathematical foundation. Imagine a tube

whose inner wall neither emits nor absorbs radiation. Two cylindrical slugs occupy

opposite ends of that tube and contact with appropriate heat reservoirs maintains them

both at the same temperature. Slug-1 emits radiation with a presumed intensity of 퐸

watts per square centimeter and Slug-2 absorbs a fraction 퐴 of that emission.

Likewise, Slug-2 emits radiation with an intensity of 퐸 watts per square centimeter

and Slug-1 absorbs a fraction 퐴 of it, reflecting the rest.

The second law of thermodynamics (in Rudolf Clausius’s version: net heat

will not, of itself, flow from a cold body to a warmer body) necessitates that the

amount of radiation coming from one slug must equal the radiation coming from the

other if that did not happen, then one slug would send more radiation to the other slug

than the other slug sends back to it, become cooler as a result, and thereby destroy the

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equilibrium. The amount of radiation coming off Slug-1 equals the sum of 퐸 and the

amount of 퐸 reflected from the slug, both multiplied by the area of the slug’s face.

The analogous sum describes the radiation coming off the face of Slug-2. Equating

those two sums and, because both slugs’ faces have the same area, dividing out the

area yields:

퐸 + 퐸 (1 − 퐴 ) = 퐸 + 퐸 (1 − 퐴 )

Subtracting 퐸 + 퐸 from both sides of that equation and dividing the result by

minus, the above equation yields

퐸 퐴 = 퐸 퐴

Finally, dividing that equation by the product A1A2 yields

퐸퐴 =

퐸퐴 = −− −−−= 퐸 = 푓(푇)

That equation expresses the fact that the emissivity of a given body stands in

direct proportion to that body’s absorptivity. To the extent that a body’s emissivity

depends upon certain properties of the body, to the same extent and in precisely the

same way the absorptivity will depend upon those properties. Put more simply, a good

absorber of radiation is also a good emitter of radiation.

1.7. BUOYANCY FORCES

Anybody completely or partially submerged in a fluid is buoyed up by a force

equal to the weight of the fluid displaced by the body. Everyone has experienced

Archimedes' principle. As an example of a common experience, recall that it is

relatively easy to lift someone if the person is in a swimming pool whereas lifting that

same individual on dry land is much harder. Evidently, water provides partial support

to any object placed in it. The upward force that the fluid exerts on an object

submerged in it is called the buoyant force. The Archimedes' principle: the magnitude

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of the buoyant force always equals the weight of the fluid displaced by the object. The

buoyant force acts vertically upward through what was the centre of gravity of the

displaced fluid.

퐵 = 푊

Where ‘퐵’ is the buoyant force and ‘푊’ is the weight of the displaced fluid. The

units of the buoyant force and weight are Newton (N) in SI and "pound force" (lbf) in

British engineering units. The buoyant force acting on the steel is the same as the

buoyant force acting on a cube of fluid of the same dimensions. This result applies for

a submerged object of any shape, size, or density. When an object is placed in a fluid,

the fluid exerts an upward force we call the buoyant force. The buoyant force comes

from the pressure exerted on the object by the fluid. Because the pressure increases as

the depth increases, the pressure on the bottom of an object is always larger than the

force on the top - hence the net upward force.

• Buoyancy forces are responsible for the fluid motion in natural convection.

• Viscous forces oppose the fluid motion.

• Buoyancy forces are expressed in terms of fluid temperature differences through the

volume expansion coefficient.

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1.8. GOVERNING EQUATIONS

Based on the experimental research, presented an unsteady two-dimensional

laminar natural convection radiative flow of a Walters-B viscoelastic fluid past a

semi- infinite (or impulsively started) vertical plate. The x-axis is taken along the

plate in the upward direction and y- axis is taken normal to it. Initially assumed that

the plate and the fluid are at the same temperature 푇 and concentration level퐶

everywhere in the fluid. At time 푡 > 0 the temperature of the plate and

concentration level near the plate are raised to 푇 and 퐶 respectively and

maintained constantly thereafter. It is assumed that the concentration 퐶 of the

diffusing species in the mixture is very less in comparison to the other chemical

species, which are present and hence the Soret and Dufour effects are negligible. It is

also assumed that there is no chemical reaction between the diffusing species and the

fluid. Under the above assumptions, the governing boundary layer equations with

Boussinesq’s approximation are

The equation of continuity

휕푢휕푥 +

휕푢휕푦 = 0

The equation of Linear momentum

휕푢휕푡 + 푢

휕푢휕푥 + 푣

휕푢휕푦 = 휈

휕 푢휕푦 + 푔훽(푇 − 푇 ) + 푔훽∗(퐶 − 퐶 ) − 푘

휕 푢휕푦 휕푡

The equation of Energy

휕푇휕푡 + 푢

휕푇휕푥 + 푣

휕푇휕푦 = 훼

휕 푇휕푦 −

1휌푐

휕푞휕푦

The equation of Diffusion

휕퐶휕푡 + 푢

휕퐶휕푥 + 푣

휕퐶휕푦 = 퐷

휕 퐶휕푦

Whereu, v are velocity components along the x, y directions respectively,

푡 -the time, 푔 -the acceleration due to gravity, 훽 -the volumetric coefficient of

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thermal expansion, 훽∗ -the volumetric coefficient of expansion with concentration,

푇 - the temperature of the fluid in the boundary layer, 퐶 ′- the species concentration in

the boundary layer, 푇 - the wall temperature, 퐶 - the concentration at the plate,

푇 - the free stream temperature of the fluid for away from the plate, 퐶∞′ - the species

concentration in the fluid for away from the plate, 휈- the kinematic viscosity, 훼- the

thermal diffusivity, 휌- the density of the fluid, 푐 - the specific heat at constant

pressure, 푞 -the radiation heat flux, 퐶 - the dimensionless concentration, 퐷 - Mass

diffusion coefficient, 푘 -Thermal conductivity, 푃푟 -the Prandtl number, 푇 -

Dimensionless temperature, 푡 - Dimensionless time, 푘 - Walters-B viscoelasticity

parameter,푢 - velocity of the plate, 푤- Conditions on the wall, ∞- Free stream

condition. By using the Rosseland’s approximation, the radiative heat flux 푞 is

given by

푞 = −4휎3푘

휕푇휕푦

Where 휎 is the Stefan - Boltzmann constant and 푘 is the mean absorption

coefficient. It should be noted that by using the Rosseland approximation, the present

analysis is limited to optically thick fluids. Under the initial and boundary conditions

and introducing the non-dimensional quantities the above equations reduces into non-

dimensional form. In order to solve these unsteady, non-linear coupled equations we

use an implicit finite difference scheme of The Crank- Nicolson type.

1.9. CONSTITUTIVE EQUATIONS FOR THE WALTERS-B VISCOELASTIC FLUID

Walters [10] in 1962 has developed a physical accurate and mathematically

amenable model for the rheological equation of state of a viscoelastic fluid of short

memory. This model has been shown to capture the characteristics of actual

viscoelastic polymer solutions, hydro-carbons, paints and other chemical engineering

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fluids. The Walters-B model generates highly non–linear flow equations which are an

order higher than the classical Navier–Stokes (Newtonian) equations. It also

incorporates elastic properties of the fluids are important in extensional behaviour of

polymers the constitute equations for a Walters–B liquid in tensorial form may be

presented as follows.

푝 = −푝푔 + 푝∗ (1.9.1)

푝∗ = 2 Ψ(푡 −푡 )푒( )(푡∗)푑푡∗ (1.9.2)

Ψ(t − t∗) = N(τ)τ

e( ∗)

dτ (1.9.3)

Where 푝 is the stress tensor, p is arbitrary isotropic pressure, 푔 is the

metric tensor of fixed coordinate system푋 , 푒( ) is the rate of strain tensor and

N(휏) is the distribution function of relaxation time 휏 . The following generalized

form of (1.9.2) has been shown by Walters-B model to be valid for all classes of

motion and stress.

푝∗ (푥, 푡) = 2 Ψ(푡 − 푡∗)휕푥휕푥∗

휕푥휕푥∗ 푒

( ) (푥∗푡∗)푑푡∗ (1.9.4)

In which 푥∗ = 푥∗(푥, 푡, 푡∗) denotes the position at time 푡∗ of the element

which is instantaneously at the position 푥 , at time t. Liquids obeying the relations

(1.9.1) and (1.9.4) are of the Walters-B type. For such fluids with short memory,

i.e. low relaxation times, equation (1.9.4) may be simplified to:

푝∗ (푥, 푡) = 2휂 푒( ) − 2푘 휕푒( )

휕푡 (1.9.5)

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In which 휂 = ∫ 푁(휏)푑휏∞ defines the limiting Walters-B viscosity at low

shear rates, 푘 = ∫ 휏∞ 푁(휏)푑휏 is the Walters-B viscoelasticity parameter and is

the convective time derivative. This rheological model is very versatile and robust,

gives a relatively simple mathematical formulation, which easily incorporated into

boundary layer theory.