1

CHAPTER-1

INTRODUCTION

1.1 Introduction

Curiosity about the fluids and fluid motion are old as the human

civilization. Ancient civilizations had some practical knowledge about the fluid

motion and apply them for betterment of their lives, for example, building canals,

sailing ships and irrigation systems. Understanding and knowledge about the fluids

were grown with time but it was not systematic until the 15th century. Great Italian

artist and experimentalist, Leonardo da vinci first studied fluid motion and

described carefully about eddies, waves, interference of waves. Also, the author

has given an idea about conservation of mass. Later Isaac Newton (17th century)

continued and forwarded by contributing postulates about conservation of

momentum and the law of viscosity. Applying conservation of mass and

momentum, Leonhard Euler derived equation of motion for inviscid fluids. But,

experimentalists and engineers of that time rejected mathematical theories due to

its non-realistic nature (real fluids are viscous). The works were limited to fluid

flows related with pipe flows, waves, channel flows. Later, French engineer

Claude Louis Marie Henry Navier and Irish scientist George Gabriel Stokes added

term of viscous dissipation in to the Euler equations and it become famous Navier-

Stokes equations. Navier-Stokes theory had to be re-examined due to the rapid

changing in science technology.

The present century examine a thorough understanding of the principles of

fluid mechanics and knowledge of how to apply them to many practical problems,

like aeronautical, biomedical, civil, marine and mechanical engineers as well as

astrophysicist, geophysicists, space researchers, meteorologists, physical

oceanographers, physicists and mathematicians have used this knowledge to tackle

a multitude of complex flow phenomena. The typical complex flows encountered

by these researchers often comprise of two or more phase in which the interaction

between them plays a prominent role in controlling the transport process such as

Introduction

2

heat mass transfer and reaction kinetics. A quantitative study through a proper

theory is essential to understand the physics of the complex flow behavior and

also to obtain invaluable scale-up information for industrial applications. The last

three to four decades of the last century also witnessed a great surge in the use of

liquid system involving Newtonian as well as non-Newtonian liquid as a working

medium. This encompasses in its real complex liquids like polymeric suspensions,

animal blood, liquid crystals, which have very small sized suspended particles of

different shapes. These particles may change shape, may shrink and expand, and

more over, they may execute rotation independent of the rotation and movement of

the liquid.

The present chapter covers the literature survey pertaining to my research

work and the recent developments in the area of applied fluid mechanics. The basic

concepts pertaining to the work is discussed in detail in the next section.

1.2 Basic concepts

FluidFluid mechanics is the branch of physics that studies behaviour of fluids

(liquids, gases, and plasmas) and the forces on them. In physics, a fluid is a

substance that continually deforms (flows) under an applied shear stress. Fluids are

subset of the phases of matter and include liquids, gases, plasmas and, to some

extent, plastic solids. For example, ‘brake fluid’ is hydraulic oil and will not

perform its required function if there is gas in it. This colloquial usage of the term

is also common in medicine and in nutrition. Liquids form a free surface (that is, a

surface not created by the container) while gases do not. The distinction between

solid and fluid is not entirely obvious. The distinction is made by evaluating the

viscosity of the substance. Silly Putty can be considered to behave like a solid or a

fluid, depending on the time period over which it is observed.

Fluids display the following properties such as:

not resisting deformation, or resisting it only lightly (viscosity), and

the ability to flow (also described as the ability to take on the shape of the

container).This also means that all fluids have the property of fluidity.

These properties are typically a function of their inability to support a shear stress

in static equilibrium. In contrast and obvious that, ideal fluids can only be

Introduction

3

subjected to normal, compressive stress which is called pressure, where as real

fluids display viscosity and so are capable of being subjected to low levels of shear

stress. Depending on the relationship between shear stress and the rate of strain and

its derivatives, fluids can be characterized as:

Newtonian fluids

Non-Newtonian fluids

Newtonian fluid In continuum mechanics, a fluid is said to be Newtonian if the viscous

stresses that arise from its flow, at every point, are proportional to the local strain

rate i.e., the rate of change of its deformation over time. That is equivalent to

saying, the forces are proportional to the rate of change of the fluid velocity vector

as one move away from the point in various directions. More precisely, a fluid is

Newtonian if and only if the tensors that describe the viscous stress and the strain

rate are related by a constant viscosity tensor that does not depend on the stress

state and velocity of the flow. If the fluid is isotropic (its mechanical properties are

the same along any direction), the viscosity tensor reduces to two real coefficients,

describing the fluid's resistance to continuous shear deformation and compression

or expansion, respectively. Newtonian fluids are the simplest mathematical models

of fluids that account for viscosity. While no real fluid fits the definition perfectly,

many common liquids and gases, such as water and air, can be assumed to be

Newtonian for practical purpose under ordinary conditions.

Newtonian fluids are named after Isaac Newton, who first derived the

relation between the rate of shear strain rate and shear stress for such fluids in

differential form. For an incompressible and isotropic Newtonian fluid, the viscous

stress is related to the strain rate by the simpler equation du dy , where

is the shear stress (drag) in the fluid, is a scalar constant of proportionality, the

shear viscosity of the fluid du dy is the derivative of the velocity component that

is parallel to the direction of shear, relative to displacement in the perpendicular

direction. More generally, in a non-isotropic Newtonian fluid, the coefficient

that relates internal friction stresses to the spatial derivatives of the velocity field is

replaced by a viscosity tensor . This can be decomposed into an antisymmetric

part that expresses the fluids resistance to shearing flow and a symmetric part that

Introduction

4

shows its resistance to compression or expansion flows. Water, air, most aqueous

solutions, oil, corn syrup, glycerin, air and other gases are examples of Newtonian

fluids, which include and exhibit the Weissenberg effect.

Non-Newtonian fluid Many solid-liquid and liquid-liquid suspensions are considered to be non-

Newtonian, examples are solutions of macromolecules, molten plastics, oil

slurries, mammalian whole blood, the synovial fluid, synthetic lubricants and

lubricant with additives. Apart from these, many commonly found substances such

as ketchup, custard, toothpaste, starch suspensions, paint, blood and shampoo are

the examples for non-Newtonian fluid. Hence the study of the non-Newtonian fluid

flows gains immense interest in present century. For the non-Newtonian fluids,

viscosity at a given temperature and pressure is not constant but depends on other

factors such as the rate of shear in the fluid etc. However, there are some non-

Newtonian fluids with shear-independent viscosity that exhibit normal stress-

differences or non-Newtonian behavior.

In a non-Newtonian fluid, the relation between the shear stress and the

shear rate is different, and can even be time-dependent. Therefore, a constant

coefficient of viscosity cannot be defined. For all fluids, the flow curve

xy vs u y is not linear through the origin at a given temperature and pressure

are said to be non-Newtonian. These materials are commonly divided in to three

broad areas, although in reality these classifications are often by no means distinct

or sharply defined.

Time-independent fluids are those for which the rate of shear at a given

point is solely dependent upon the instantaneous shear stress at that point.

Time-independent fluids are those for which the shear rate is a function of

both the magnitude and the duration for shear and possibly of the time lapse

between consecutive applications of shear stress.

Viscoelastic fluids are those that show partial elastic recovery upon the

removal of a deforming shear stress. Such materials possess properties of

both fluids and elastic solids.

Non-Newtonian fluids are further classified into three distinct types depending on

the nature, namely;

Introduction

5

Psedoplastic fluid,

Dialatent fluid,

Bingham plastic,

Pseudoplastic fluid or shear thinning fluid

Shear thinning is an effect where the fluid's viscosity (measure of a fluid's

resistance to flow) decreases with an increasing rate of shear stress. Another name

for a shear thinning fluid is pseudoplastic and found in certain complex solutions,

such as lava, ketchup, whipped cream, blood, paint, and nail polish. It is also a

common property of polymer solutions and molten polymers. Pseudo plasticity can

be demonstrated by the manner in which squeezing a bottle of ketchup, a Bingham

plastic, causes the contents to undergo a change in viscosity. The force causes it to

go from being thick like honey to flowing like water. A simple example is that, if

one were to hold a sample of hair gel in one hand and a sample of corn syrup or

glycerin in the other hand, they would find that the hair gel is much harder to pour

off the fingers (a low shear application), but that it produces much less resistance

when rubbed between the fingers (a high shear application).

Dilatant fluid or shear thickening fluid

A dilatant (also termed as shear thickening) material is one in which

viscosity increases with the rate of shear strain. This behavior is only one type of

deviation from Newton’s Law and it is controlled by such factors as particle size,

shape and distribution. The properties of these suspensions depend on Hamaker

theory and Van der Waals forces and can be stabilized electro statically or

sterically. Shear thickening behavior occurs when a colloidal suspension

transitions from a stable state to a state of flocculation. Such behavior is currently

being researched for use in body armor applications. A large portion of the

properties of these systems are due to the surface chemistry of particles in

dispersion, known as colloids. Another example for shear thickening is an

uncooked paste of cornstarch and water. Under high shear the water is squeezed

out from between the starch molecules, which are able to interact more strongly.

While not strictly a dilatant fluids, Silly Putty is an example of a material that

shears the viscosity characteristics. Another use in a viscous coupling in which if

both ends of the coupling are spinning at the same (rotational) speed, the fluid

Introduction

6

viscosity is minimal, but if the ends of the coupling differ greatly in speed, the

coupling fluid becomes very viscous. Such couplings have applications as a light

weight, passive mechanism for passenger automobile to automatically switch from

two-wheel drive such as when the vehicle is stuck in snow and the primary driven

axle starts to spin due to loss of traction under one or both tires.

Table 1.1. Comparison of Newtonian, non-Newtonian and viscoelastic properties.

Fluid Material Characteristics Examples

Kelvin material"Parallel"linearstic combination of elastic and viscous effects.

Some lubricants,

whipped cream

Viscoelastic

ThixotropicApparent viscosity decreases with duration of stress.

Some clays, some drilling mud, many paints, synovial fluid

Shearthickening(dilatant)

Apparent viscosity increases with increased stress.

Suspensions of corn starch or sand in water

Shear thinning

(pseudoplastic)

Apparent viscosity decreases with increased stress.

Paper pulp in water, latex paint, ice, blood, syrup, molasses

Time-

independent

viscosity Generalized

Newtonian

fluids

Viscosity is constant Stress depends on normal and shear strain rates and also the pressure applied on it.

Blood plasma,

custard, water

Bingham plastic fluid

There are fluids which have a linear shear stress/shear strain relationship

which require a finite yield stress before they begin to flow (the plot of shear stress

against shear strain does not pass through the origin). The graph shows shear stress

on the vertical axis and shear rate on the horizontal one. (Volumetric flow rate

depends on the size of the pipe, shear rate is a measure of how the velocity changes

with distance. It is proportional to flow rate, but does not depend on pipe size). As

before, the Newtonian fluid flows and gives a shear rate for any finite value of

shear stress. However, the Bingham Plastic does not exhibit any shear rate (no flow

and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the

slope of this line is the viscosity, which is the only parameter needed to describe

Introduction

7

the flow. By contrast the Bingham plastic requires two parameters, the yield stress

and the slope of the line, known as the plastic viscosity. The physical reason for

this behavior is that the liquid contains particles (example, clay) or large molecules

(example, polymers) which have some kind of interaction, creating a weak solid

structure, formerly known as a false body and a certain amount of stress is required

to break this structure. Once the structure has been broken, the particles move with

the liquid under viscous forces. If the stress is removed, the particles associate

again. Examples are clay suspensions, drilling mud, toothpaste, mayonnaise,

chocolate, and mustard.

Fig 1.1. Graphical representation of fluids

Power-Law fluid or Ostwald-de Waele fluid

A Power-law fluid or the Ostwald-de Waele fluid (Scott et al. [1939]), is a

type of generalized Newtonian fluid for which the shear stress is given by

nu y , where is a scalar constant of proportionality, the shear

viscosity of the fluid, u y is the shear rate or the velocity gradient perpendicular

to the plane of shear (SI unit S-1) and n is the flow behavior index (dimensionless).

The quantity 1n

eff u y represents an apparent or effective viscosity as a

function of the shear rate (SI unit Pa•S). This mathematical relationship is useful

because of its simplicity, but only approximately describes the behavior of a real

Introduction

8

non-Newtonian fluid. For example, if n were less than one, the power law predicts

that the effective viscosity would decrease with increasing shear rate indefinitely,

requiring a fluid with infinite viscosity at rest and zero viscosity as the shear rate

approaches infinity, but a real fluid has both a minimum and a maximum effective

viscosity that depend on the physical chemistry at the molecular level. Therefore,

the power law is only a good description of fluid behavior across the range of shear

rates to which the coefficients were fitted. There are a number of other models that

better describe the entire flow behavior of shear-dependent fluids, but they do so at

the expense of simplicity, so the power law is still used to describe fluid behavior,

permit mathematical predictions, and correlate experimental data. Power-law fluids

can be subdivided into three different types of fluids based on the value of their

flow behavior index; For 1n , the fluid is called shear thinning or pseudo plastic;

1n , fluid is said to be dilatant or shear thickening and for 1n the fluid is

simply the Newtonian fluid. After knowing the basic concepts pertaining to

different types of fluids, the next section incorporates how these fluids are applied

to different physical concepts namely, Ludwig Prandtl boundary layer theory using

boundary layer approximation and the behavior over a continuous moving surface.

1.3 Prandtl boundary layer theory

In physics and fluid mechanics, a boundary layer is a layer of the fluid in

the immediate vicinity of a bounding surface where the effects of viscosity are

significant. In the Earth's atmosphere, the planetary boundary layer is the air layer

near the ground affected by thermal heat, moisture or momentum transfer from the

surface. The boundary layer theory is an essential and obvious part of engineering

applications. On an aircraft wing the boundary layer is the part of the flow close to

the wing, where viscous forces distort the surrounding non-viscous flow. The

deduction of the boundary layer equations was one of the most important advances

in fluid dynamics. Using an order of magnitude analysis, the well known

governing Navier–Stokes equations of viscous fluid flow can be greatly simplified

within the boundary layer. Notably, the characteristic of the partial differential

equations (PDE) becomes parabolic, rather than the elliptical form of the full

Navier–Stokes equations, which greatly simplifies the solution of the equations. By

making the boundary layer approximation, the flow is divided into an inviscid

Introduction

9

(which is easy to solve by a number of methods) and the boundary layer, which is

governed by an easier to solve PDE. The behavior of fluids can be described by the

Navier–Stokes equations, to a set of partial differential equations which are based

on:

continuity (conservation of mass),

conservation of linear momentum,

conservation of angular momentum,

conservation of energy.

An important contribution to the science of fluid motion was made by Ludwig

Prandtl in 1904 when he clarified the essential influence of viscosity in flows at

high Reynolds numbers and showed how the Navier-Stokes equations could be

simplified to yield approximate solutions for this case. We shall explain these

simplifications with the aid of an argument which preserves the physical picture of

the phenomenon and it will be recalled that in the bulk of the fluid inertia forces

predominate, the influence of viscous forces being vanishingly small. The simplest

example of the application of the boundary-layer equations is afforded by the flow

along a very thin flat plate. Historically this was the first example illustrating the

application of Prandtl’s boundary-layer theory; it was later discussed by Blasius.

Fig 1.2. Description of the boundary layer phenomenon.

Boundary-layer thickness

The definition of the boundary-layer thickness is to a certain extent

arbitrary because transition from the velocity in the boundary to that outside it

takes place asymptotically. This is, however, of no practical importance, because

the velocity in the boundary layer attains a value which is very close to the external

Introduction

10

velocity already at a small distance from the wall. It is possible to define the

boundary layer thickness as that distance from the wall where the velocity differs

by one percent from the external velocity. Instead of the boundary layer thickness,

another quantity, the displacement thickness H, is sometimes used, Fig.1.3. It is

defined by the equation0

( )U H U u dy . The displacement thickness indicates the

distance by which the external streamlines arc shifted owing to the formation of the

boundary layer.

Fig 1.3. Displacement thickness H in a boundary layer.

Laminar boundary layers can be loosely classified according to their structure and

the circumstances under which they are created.

Stokes boundary layer : The thin shear layer which develops on an

oscillating body.

Blasius boundary layer: Refers to the well-known similarity solution near

an attached flat plate held in an oncoming unidirectional flow.

Stokes boundary layer

In fluid dynamics, the Stokes boundary layer or oscillatory boundary layer

refers to the boundary layer close to a solid wall in oscillatory flow of a viscous

fluid. Or, it refers to the similar case of an oscillating plate in a viscous fluid at rest

with the oscillation direction(s) parallel to the plate. For the case of laminar flow at

low Reynolds numbers over a smooth solid wall, George Gabriel Stokes derived an

analytic solution is one of the few exact solutions for the Navier–Stokes equations.

In turbulent flow, this is still named a Stokes boundary layer, but now one has to

rely on experiments, numerical simulations or approximate methods in order to

obtain useful information on the flow.

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11

Fig 1.4. Stokes boundary layer in a viscous fluid due to the harmonic oscillation of a plane rigid

plate. Velocity (curve line) and particle excursion (dot line) as a function of the distance to the wall.

Blasius boundary layer

In physics and fluid mechanics, a Blasius boundary layer (named after Paul

Richard Heinrich Blasius) describes the steady two-dimensional boundary layer

that forms on a semi-infinite plate which is held parallel to a constant

unidirectional flow U. The solution to the Navier–Stokes equation for this flow

begins with an order of magnitude analysis to determine what terms are important.

Within the boundary layer the usual balance between viscosity and convective

inertia is struck, resulting in the scaling argument 2 2U L U H , where H is

the boundary-layer thickness, L is the characteristic length and is the kinematic

viscosity.

Fig 1.5. A schematic diagram of the Blasius flow profile. The stream wise velocity component

( ) ( )u U x is shown, as a function of the stretched co-ordinate .

Boundary layer behavior on continuous moving surface

Consider a long continuous sheet which issues from a slot as shown in Fig

1.6. The moving continuous sheet is taken up by a wind-up roll. The slot and the

Introduction

12

wind-up roll are at a finite distance apart and constitute the boundaries of the

system. The assumption is also made that a certain time has elapsed after the

initiation of motion so that steady state conditions prevail. Any flow disturbance

created by the roll is neglected. An observer fixed in space will note that the

boundary layer on the sheet originates at the slot and grows in the direction of the

motion of the sheet. The boundary layer behavior here appears to be different from

what would be expected if the sheet is considered as a moving flat plate of finite

length on which the boundary layer would grow in the direction opposite to the

direction of motion of the plate or away from the leading edge of the plate. The

difference in the boundary layer on a continuous moving surface and on a surface

of finite length raises the question whether the results of investigations of boundary

layer behavior on moving surfaces of finite length are applicable to the moving

continuous surfaces.

Fig 1.6. Boundary layer behavior on continuous moving surface.

Comparison with flat surfaces of finite length Consider steady two-dimensional incompressible flow around a continuous

solid surface moving in a fluid medium at rest as shown in the figure. An observer

fixed in space will note that the boundary layer on the solid surface, which

originates at the slot, grows in the direction of motion of the surface. At the solid

surface the fluid moves in the x- direction with a velocity ( u- component) equal to

the velocity of the solid surface, where as at increasing distance from the surface

the velocity of the fluid in the x-direction approaches to zero asymptotically. The

fluid velocity in the y- direction (v- component) varies from zero at the solid

surface to some finite value at the edge of the boundary layer. The essential

Introduction

13

physical characteristics of the boundary layer on a continuous surface is that the

origin and the termination of the boundary layer around such a surface are not

identified with any part of the solid surface but are determined by the boundaries

of the system. By contrast of the boundary layer limits on a moving surface of

finite length are definitely identified with specific parts of the surface: namely

leading and trailing edges of the surface.

For an instance, in technological industry, fluid mechanical application is

found in polymer extrusion processes where the object on passing between two

closely placed solid blocks is stretched in to a liquid region. The stretching imparts

an unidirectional orientation to the extrudate; thereby improving its mechanical

properties. The liquid is basically meant to cool the stretching sheet whose

property as a final product depends greatly on the rate at which it is cooled. It is

imperative therefore to consider two important aspects in this physically interesting

problem:

Proper choice of cooling liquid.

Regulation of the flow of cooling liquid, due to the stretching sheet, to

achieve a desired rate of cooling appropriate for successfully arriving at a

sought final product.

Fig 1.7: Schematic of an extrusion processes.

The cooling liquid in earlier times was chosen to be the abundantly available water

but this has the drawback of rapidly quenching the heat leading to sudden

solidification of the plate. From the standpoint of desirable properties of the final

product (solidified stretching sheet) water does not seem to be the ideal cooling

Introduction

14

liquid. A careful examination of the needs in the system suggests that it is

advantageous to have a controlled cooling system. An electrically conducting

liquid seems to be a good candidate for such an application situation because its

flow can be regulated by external means through a magnetic field. Further, this

arrangement does not involve any moving parts and does not tamper with the flow

that we are investigating theoretically. The problem is a prototype for many other

practical problems also, similar to the polymer extrusion process (Fig 1.7), like

drawing, annealing and tinning of copper wires.

continuous stretching, rolling and manufacturing of plastic film and

artificial fibers.

extrusion of a material and heat- treated materials that travel between feed

and wind-up rollers or on conveyers or belts.

cooling of an infinite metallic sheet.

boundary-layer along a liquid film in condensation process etc.

The delicate nature of the problem dictates the fact that the magnitude of the

stretching rate has to be small. This also ensures that the stretching material

released from between the two solid blocks in to the liquid continues to be a plane

surface rather than a curved one.

Flow over a stretching sheet

This type of work was initiated initially by Sakiadis [1961 a, b, c], in his

series of papers, the authors investigated the two-dimensional, axisymmetric

boundary layer flow over a stretched surface moving constant velocity. Both exact

and approximate solutions were presented for laminar flow with the latter being

obtained by the integral method. Cane [1970] extended the work of Sakiadis

[1961] to the flow generated due to the stretching of an elastic flat sheet which

moves in its plane with a velocity varying with the distance from a fixed point due

to the application of a stress are known as the flow due to the stretching sheet.

This flow was of Blasius type, in which the boundary layer thickness increased

with the distance from the slit. An extension to this problem is that of a stretching

sheet whose velocity is proportional to the distance from the slit. This occurs in the

drawing of plastic films. The flow in this case has certain similarities with Hiemenz

[1911] the boundary layer flow near a stagnation point in which the main velocity

in the outer flow is proportional to the distance from the stagnation point.

Introduction

15

Fig 1.8. Schematic of the stretching sheet problem.

Just as in the Hiemenz flow the boundary layer thickness is constant and a solution

of the boundary layer equations of the form yfxu , . Here (x, y) are the

rectangular Cartesian coordinates with origin at the slit, x is measured along the

sheet in the direction of the motion and (u, v) are the corresponding velocity

components. One important difference is that since u=0 at the edge of the

boundary layer the outer pressure is constant; this leads to the homogeneous

Hiemenz type equation which has a exponential solution in f. the velocity

components are exp( ),u x y 1 exp( )v y where

, is the co efficient of kinematic viscosity , x is the velocity of the

sheet and is a constant. At the edge of the boundary layer there is a transverse

component of velocity . Later Erickson et al. [1966] extended the work of

Sakiadis [1961] to study mass transfer at the stretched sheet surface. In the next

section, literature survey pertaining to different types of Newtonian/ non-

Newtonian fluids over a stretching sheet.

1.4 Stretching sheet problem involving Newtonian fluids

Regulation of the stretching sheet boundary layer flow of Newtonian fluids

is important from the practical point of view. An analytical form was presented by

Crane [1970] for steady boundary layer flow of an incompressible viscous liquid

caused solely by the linear stretching of an elastic flat sheet which moves in its

own plane with velocity varying linearly with distance from a fixed point. Gupta

and Gupta [1977] investigated the heat and mass transfer in the flow over a

stretching surface (with suction and blowing) issuing from thin silt. A non-

Introduction

16

isothermal moving sheet was dealt with and the temperature and concentration

distribution profiles for that situation were obtained. Banks [1983] examined a

class of similarity solution of the boundary layer equations for the flow due to

stretching surface. The ordinary differential equation that arises admits of one-

parameter family of solutions in which the same way the Falkner-skan equation

does. The equation is integrated numerically for a number of parameter values and

various results are presented. Analytical solution is also presented for a couple of

values of the parameters and these, together with perturbation solutions, support

the numerical results. Dutta et al. [1985] analyzed the temperature distribution in a

flow over a stretching sheet uniform heat flux. The governing differential equation

transformed to a confluent hypergeometric differential equation and solution was

obtained in terms of incomplete gamma function. It was shown that temperature at

a point decreased with the increase of Prandtl number. Dutta and Gupta [1987]

solved the coupled heat transfer problem involving the stretching sheet. Variation

of the sheet temperature with distance from the silt was found for several values of

the Prandtl number and stretching speeds. It was shown that for a fixed Prandtl

number, the surface temperature decreases with increase in the stretching speed.

Dutta [1988] presented an analytical solution of the heat transfer problem for

cooling of a thin stretching sheet in a viscous flow in the presence of suction or

blowing. The local velocity of the sheet material was assumed to be proportional to

the distance from the silt. The convergence criteria of the solution were also

established. Chen and Char [1988] explores the effects of both power law surfaces

temperature and power law heat flux variations on the heat transfer characteristics

of a continuous, linearly stretching sheet subjected to suction or blowing. Soewono

et al. [1992] analyzed the existence of solutions of a nonlinear boundary value

problem, arising in flow and heat transfer over a stretching sheet with variable

thermal conductivity and temperature- dependent that sources or sinks. Karahalios

[1992] obtained an exact similarity solution of the time- dependent Navier-Strokes

equation when a flat surface stretches radially. Maneschy et al. [1993] examined

the flow of a second grade fluid over a porous elastic sheet due to stretching and

gave a brief review of the previous works. Maneschy et al. [1993] considered the

heat transfer over a stretching sheet with suction at the surface. The velocity

components were expressed in a power series in time up to the second-order

Introduction

17

approximation. Vajravelu [1994] carried out an analysis of convective flow and

heat transfer in a viscous heat-generating liquid near an infinite vertical stretching

surface. The effects of free convention and suction or injection on the flow and

heat transfer were considered. The equations of conservation of momentum, mass

and energy, which govern the flow and heat transfer were solved numerically by

using a variable order, variable step size finite-difference method. The numerical

results obtained for the flow and heat transfer characteristics revealed many an

interesting behaviour. Kumaran and Ramanaiah [1996] for the first time studied

the viscous boundary layer flow over a quadratically stretching sheet. The plots of

skin friction and streamline pattern as a function of the stretching parameters were

discussed. Magyari and Keller [1999] examined both analytical and numerically,

the heat and mass transfer in the boundary layer on an exponentially stretching

continuous surface with as exponential temperature distribution. Magyari and

Keller [2000] studied the steady boundary layer flow induced by permeable

stretching surfaces with variable temperature distribution under Reynold’s analogy.

Reynolds’ analogy makes use of the advantage of all the exact analytic solutions of

the momentum and energy equations. Magyari and Keller [2001] analyzed the free

laminar jets of classical hydrodynamics that may be identified with certain

boundary-layer flow induced by continuous surfaces immersed in quiescent

incompressible liquids and stretched with well defined velocities. By presenting an

analytical solution of the flow problem, it was shown that in the limiting case of a

vanishing lateral mass flux, this stretching-induced flow goes over, by an adequate

scaling transformation to the well known wall jet. Wang [2002] investigated the

flow due to stretching flat boundary with partial slip and gave an exact solution of

the Navier-Strokes equation. Andersson [2002] presented a slip flow past a linearly

stretching sheet using the Navier slip condition at the sheet. An exact analytical

solution of the Navier-Strokes equation that is formally valid for all Reynolds

numbers was found. Magyari et al. [2002, 2003] examined the self-seminar

boundary-layer flow of Newtonian liquid over a permeable continuous plane

surface stretching with inverse linear velocity. It was shown that in order to obtain

from pseudo-similarity the correct similarity problem, in this case, the usual

expression of the stream function a logarithmic term in the wall coordinate x must

be added. The new analytical solution of a well-known boundary value problem

Introduction

18

shows that the hyperbolic-tangent solution of this problem belongs to a one-

parameter family of multiple solutions that can be expressed in terms of Airy’s

function. Mahapatra and Gupta [2003] examined an exact similarity solution of

the Navier-Strokes equation. The solution represents steady asymmetric

stagnation-point flow towards a stretching surface. It is known that the flow

displays a boundary layer structure when the stretching velocity of the surface is

less than the free stream velocity. On the other hand, an inverted boundary layer is

formed when the surface stretching velocity exceeds the free stream velocity.

Temperature distribution in the flow is found when the surface is held at a constant

temperature. It turns out that when the surface temperature exceeds the ambient

temperature, heat flows from the surface to the liquid near the stagnation point but

further away from the stagnation point heat flows from the liquid to the stretching

surface. Partha et al. [2004] presented the mixed convention flow and heat transfer

from an exponentially stretching vertical surface in a quiescent liquid using a

similarity solution. They found that the wall temperature and stretching velocity

can have a specific exponential form. The influence of buoyancy along with

viscous dissipation on the convective transport in the boundary layer region was

analyzed in both aiding and opposing flow situations. Liao [2003, 2005], Liao and

Pop [2004] and Xu [2005] have used the homotopy analysis method for nonlinear

problems arising in a moving sheet. Two rules, namely, the rules of solution

expression and rule of coefficient ergodicity, were proposed which play prominent

role in the fame of the homotopy analysis method. An explicit analytic solution is

given for the first time, with recursive formulae for coefficient. Abd El-Aziz and

Salem [2007] worked on MHD mixed convection and mass transfer through a

vertical stretching sheet with diffusion of chemically reactive species and space or

temperature dependent heat source. Makinde and Ogulu [2008] studied the effect

of thermal radiation on the heat and mass transfer flow of a variable viscosity fluid

past a vertical porous plate tempered by a transverse magnetic field. Liu and

Andersson [2008] studied on the heat transfer on an unsteady stretching sheet.

Ishak et al.[2009] investigated the boundary layer flow and heat transfer over an

unsteady stretching vertical surface. Abd El-Aziz [2010] worked on unsteady fluid

and heat flow induced by a stretching sheet with mass transfer and chemical

reaction. Elbashbeshy and Aldawody [2011] studied the effects of thermal radiation

Introduction

19

and magnetic field on unsteady mixed convection flow and heat transfer over a

porous stretching surface in the presence of internal heat generation/absorption.

1.5 Stretching sheet problem involving non-Newtonian fluids

Flow of non-Newtonian fluids over a stretching sheet has been studied

extensively in the recent years. This problem is of interest when a polymer sheet

is extruded continuously from a die. Keeping these in view, Fox et al. [1969]

studied the flow of power law fluid past an inextensible flat surface moving with

constant velocity in its own plane. This model, however, does not exhibit certain

non-Newtonian liquid properties like normal stress difference. Rajagopal et al.

[1984] analyzed the flow of a second order liquid over a stretching sheet without

heat transfer and presented a perturbation solution for the velocity distribution.

Siddappa and Abel [1985, 1986] studied the flow of Walter’s liquid B past a

stretching sheet, and obtained analytical solutions to the flow equation.

Rajagopal et al. [1987] studied the boundary layer flow of second order liquid

over a stretching sheet with uniform free stream and obtained some interesting

results. Bujurke et al. [1987] explores the effect of heat transfer in the flow of a

second order liquid, obeying Coleman and Noll’s constitutive equation, over a

stretching sheet. Chen and Char [1988] investigated the temperature distribution

in a Walter’s liquid B model over a horizontal stretching plate. The velocity of

the sheet was assumed to be proportional to the distance from the slit and the

sheet subjected to a variable heat flux. The solution of the heat transfer equation

was expressed in terms of Kummer’s function. Several closed-form solution for

specify conditions were considered. The effect of the viscoelastic parameter and

the heat flux on the temperature field was also studied. Dandapat and Gupta

[1989] studied flow of second order liquid and heat transfer affected by a

stretching sheet. The influence of viscoelasticity on flow behavior and heat

transfer characteristics was examined. An analytical solution was also presented

for velocity and temperature distributions along with numerical results. Chen et

al. [1990] analyzed temperature distribution in the flow of Walters’ liquid B over

a horizontal stretching plate with constant surface temperature/uniform surface

heat flux. It is shown that temperature at a point decreases with decrease in the

value of viscoelastic parameter. The dimensionless heat transfer coefficient and

the temperature viscoelastic parameter. Rollins and Vajravelu [1991] analyzed

Introduction

20

the heat transfer in a second order liquid over a continuous stretching surface

with power-law surface temperature/power law heat flux including the effects of

internal heat generation. The solution and heat transfer characteristics were

obtained in terms of parabolic cylinder function with a boundary layer of width

reciprocal of the Prandtl number in both the PST and PHF cases. Also it was

shown that no boundary layer type solution exists for small Prandtl number. Sam

and Rao [1992] examined the problem of heat transfer in a second order liquid

over a stretching sheet. An expression was obtained for skin friction and heat

transfer coefficient. Sam and Rao [1993] provided two closed form solution of

the momentum equation for certain range of values of viscoelastic parameter.

This brought in questions of uniqueness of the solution. Siddappa et al. [1995]

investigated an oscillatory motion of viscoelastic liquid past a stretching sheet.

The solution of the equation of motion was obtained by power series method.

They found that the effects of unsteadiness on wall velocity and skin friction

were appreciable. Bhatnagar et al. [1995] considered the flow of an Oldroyd-B

liquid, occupying the space over an elastic sheet, due to the stretching of the

sheet in the presence of a constant free steam velocity. By introducing similarity

transformation for the velocity field as well as the components of the stress

tensor, the governing equations were reduced to a system of coupled non-linear

ordinary differential equations. The resulting equations were solved numerically

and by means of perturbation in the Weissenberg number. A comparison of the

two solutions showed good arrangement. Ariel [1997] investigated the

generalized Gear’s method for computing the flow of a second order liquid. A

fourth-order predictor-corrector method was used for obtaining the numerical

solution of a class of singular boundary value problems in which the coefficient

of the highest derivative is a small parameter. The two-dimensional stagnation

point flow of the viscoelastic liquid was computed. Subhas Abel and Veena

[1998] studied the Walters’ liquid B flow and heat transfer in a saturated porous

medium over an impermeable stretching surface with frictional heating and

internal heat generation or absorption. Two cases were considered; namely PST

and PHF. Exact solution for the velocity field and the skin friction were

obtained. Also, the solutions for the temperature and heat transfer characteristics

were obtained in terms of Kummer’s function. Ariel [2001] studied the steady

Introduction

21

laminar flow of a second grade liquid over a radially stretching sheet. The

viscoelasticity of the liquid was shown to give rise to boundary value problem in

which the order of the differential equations exceeds the number of boundary

conditions. It was shown that the solution exists for all values of viscoelastic

parameter. A perturbation, valid for small viscoelastic parameter and an

asymptotic solution, valid for large viscoelastic parameter, were also obtained.

Elbashbeshy and Bazid [2003] presented a similarity solution for the boundary

layer equations, which describe the unsteady flow and heat transfer over a

stretching sheet. Massoudi and Maneschy [2004] investigated the numerical

solution to the flow of a second grade liquid over a stretching sheet using the

method of quasi-linearization. This problem was studied using a perturbation

scheme by Rajgopal and Gupta [1984] and Rajgopal et al. [1984]. Prasad and

Datti [2008] studied non-Newtonian power law fluid flow and heat transfer in a

porous medium over a non-isothermal stretching sheet. Abd El-Aziz [2009]

extended the work of Elbashbeshy and Bazid [2003] for some physical realistic

phenomena of radiation effect on the flow and heat transfer over an unsteady

stretching sheet. Mukhopadhyay [2009] analyzes the effect of variable fluid

properties on the unsteady fluid flow and heat transfer over a stretching sheet in

the presence of suction. Vajravelu et al. [2011] worked on Diffusion of a

chemically reactive species of a power-law fluid past a stretching surface. Prasad

[2012] studied the heat transfer in a Ostwald-De-Waele fluid over a stretching

sheet with prescribed heat flux. Prasad et al. [2012] worked on non- Newtonian

Power law Fluid Flow and Heat Transfer over a non-Linearly Stretching Surface.

Prasad et al. [2013] studied on the influence of internal heat generation/

absorption, thermal radiation, magnetic field, variable fluid property and viscous

dissipation on heat transfer characteristics of a Maxwell fluid over a stretching

sheet. The next section, deals with brief review on the magnetohydrodynamic

(MHD) flow over a stretching sheet.

1.6 Magnetohydrodynamic flow over a stretching sheet

Magnetohydrodynamics (MHD) is the study of the motion of an electrically

conducting fluid in the presence of external electromagnetic fields. It is the

combination of two branches viz, hydrodynamics and electromagnetism. The

Introduction

22

dictionary meaning of hydro is water but hydrodynamics includes study of all

liquids as well as gases. Hence in MHD we study dynamical behavior of

electrically conducting medium which may be a liquid or an ionized gas in

presence of magnetic field. Both plasma and conducting fields are related in

common theory by assuming plasma as a continuous fluid for which the kinetics

theory of gases still holds true. In MHD induced electric current produces

mechanical force which in turn modified the motion in the fluid. Hence, study of

electrically conducting fluid flow in the presence of transverse magnetic field

assures significance. The following information about the early development of

MHD is worth mentioning, Hartmann in 1937, studied the motion of electrically

conducting fluids in presence of magnetic field. Chapman and Ferraro developed

the theory of magnetic storms during 1930-1935. The systematic study of MHD

dates from 3rd October 1942, the date of issue of ‘Nature’ in which Alfven of the

Royal institute of Technology at Stochelm, Swedon published an article describing

the prediction of new type of wave. By combining Maxwell’s equations with the

fundamental equations of hydrodynamics, Alfven established the theorems of

Frozen fluid i.e., in a highly conducting fluid, the magnetic lines of force are

frozen in to the fluid. The subject of MHD had its origin in the study of magnetism

of cosmic problems, interior of the sun, problems of earth, the stars, the inter stellar

space, etc. MHD deals with the problems such as cooling of nuclear reactor by

liquid sodium in the extraction of electrical energy directly from hot plasma

through a powerful magnetic field. The study of uniform magnetic field on the

motion of a electrically conducting fluid over a stretching sheet finds its

application in various engineering disciplines such as polymer technology, where

one deals with stretching plastic sheet and metallurgy, where hydrodynamic

techniques have recently been used. It may be pointed out that many metallurgical

processes involve the cooling of continuous strips or filaments by drawing them

through a quiescent fluid. In the process of drawing, these strips are sometimes

stretched. Drawing annealing and tinning of copper wires may be mentioned in this

regard. In these cases the properties of final product depend to a great extent on the

rate of cooling. By drawing such strips in an electrically conducting fluid, the rate

of cooling can be controlled and final product of desired characteristics might be

achieved. Another important application of MHD flow in metallurgy is the

Introduction

23

purification process of molten metal from non-metallic inclusions using magnetic

field. Motivated by these applications, Sarpakaya [1961] had pointed out that some

liquids such nuclear fuel slurries, liquid metals, mercury amalgams, biological

liquids, plastic extrusions, paper coatings and lubricating oil greases have

applications in many areas both in the absence as well in the presence of a

magnetic field. Since cooling liquids most applications are known to be

electrically conducting, the application of magnetic field provides a rheostatic

effect on the flow. Pavlov [1974] studied the flow of an electrically conducting

fluid caused solely by stretching of an elastic sheet in presence of a uniform

transverse magnetic field and obtained a similarity solution of this problem.

Chakrabarti and Gupta [1979] extended the work of Pavlov to study temperature

distribution in MHD boundary layer flow in the presence of uniform suction,

Soundalgekar and Takhar [1977] investigated the effects of uniform transverse

magnetic field on forced and free convection flow past a semi-infinite plate taking

in to of viscous dissipation and stress work. They discussed the effects of different

physical parameters on MHD flow and heat transfer characteristics. Raptis and

Tzivandis [1983] carried out analytical investigations on free convective flow past

an infinite vertical surface when the fluid is electrically conducting in the presence

of an external transverse uniform magnetic field. Hydromagnetic flow of

Newtonian fluid and heat transfer over a continuous moving flat surface with

uniform suction was been studied by Vajravelu and Nayfeh [1983]. Mahesh

Kumari et al. [1990] studied the effects of induced magnetic field source/sink on

flow and heat transfer characteristics over a stretching surface. Andersson et al.

[1992] studied the boundary layer flow of an electrically conducting compressible

power-law liquid in the presence of a transverse magnetic field. Gorla et al [1993]

investigated the effects of magnetic field strength on mixed convective flow arising

from an infinitely long horizontal line source of heat when the ambient fluid

considered was a non-Newtonian power-law fluid having moderately large values

of Grashoff number. Na and Pop [1996] investigated the boundary layer flow over

a moving continuously flat plate in an electrically conducting ambient fluid with a

step change in applied magnetic field. Elbashbeshy [1997] investigated heat and

mass transfer phenomena along a vertical plate under the combined buoyancy

effects of thermal and species diffusions in presence of a magnetic field. Vajravelu

Introduction

24

and Hadjinicolaou [1997] carried out the investigations of free convection and

internal heat generation on flow and heat transfer characteristics in an electrically

conducting fluid near an isothermal sheet. Chaim [1997] presented an analytical

solution of the energy equation for a boundary layer flow of an electrically

conducting fluid under the influence of transverse magnetic field over a linearly

stretched non-isothermal flat sheet. Ali and Chamkha [1997] obtained the

similarity solutions of laminar boundary layer equations describing the steady

hydromagnetic two-dimensional flow and heat transfer in a stationary electrically

conducting and heat generating fluid driven by a continuously moving porous

surface immersed in a fluid saturated porous medium. Howell et al. [1997]

examined the momentum and heat transfer in the laminar boundary layer on a

continuously moving and stretching surface in a power law fluid. Elbashbeshy

[2000] studied the flow of a viscous incompressible fluid along heated vertical

plate, taking in to account the variation of viscosity and thermal diffusion with

temperature in the presence of magnetic field. Subhas Abel et al. [2001] studied the

effect of magnetic field on viscoelastic fluid flow and heat transfer over a

stretching sheet with internal heat generation/ absorption. Emad et al. [2004]

analysed the MHD free convection flow non-Newtonian power law fluid near a

stretching surface with uniform surface heat flux and included the effects of Hall

current. Prasad and Vajravelu [2009] studied on heat transfer in the MHD flow of

a Power law fluid over a non-isothermal stretching sheet. Prasad et al. [2009]

worked on the MHD flow and heat transfer in the flow of a power law fluid over a

non-isothermal stretching sheet. Prasad et al. [2010 a] studied the hydromagnetic

flow and heat transfer of a non-Newtonian Power law fluid over a vertical

stretching sheet. Prasad et al. [2010 b] studied the effect of variable viscosity on

MHD visco-elastic fluid flow and heat transfer over a stretching sheet. The study

of non-Newtonian fluid is more important in constructing electromagnetic flow

meters, in understanding the principles of the method and its application to blood

flow measurements. When conducting of non-Newtonian fluids flow in the

presence of external magnetic field, the non-Newtonian and the magnetic forces

effects will be coupled with in the flow field. Thus it would be possible to

influence the flow of these conducting fluids.

Introduction

25

The science of thin liquid films has developed rapidly in recent years, with

applications to coating flows, biofluids, microfluidic engineering, and medicine.

These recent developments open up many opportunities for substantial

contributions to this field, using a combination of mathematical modeling, analysis

and numerical simulation, coupled to carefully chosen quantitative experiments.

The brief literature survey on thin liquid film has been discussed in the next

section.

1.7 Thin liquid film flow over a stretching sheet

A layer of liquid over a solid/ porous substrate with free surface is called a

thin film and film thickness is small compared to all relevant length scales parallel

to the substrate. Thin liquid films are ubiquitous in nature and technology, so an

understanding of their mechanics is important in many technological applications.

A typical thin film coating flow consists of an amount of liquid over a solid/

porous surface or substrate to coat that surface by thin layer (thin film) of the

liquid. (examples like, the flow of a (thin) raindrop down a windowpane under the

action of gravity, ink-jet printing, painting the wall etc). Thin film coating flow

appears in many industrial process for example manufacturing colour television

screen, hard drives, CD, plastic sheets etc. Due to its wide range of applications

and having many allied interesting physical phenomena like rapture, fingering,

wetting, de-wetting etc, it attracts scientists from engineering, applied mathematics

and physics. The coating process requires a smooth surface for the best product

appearance (properties as low friction, transparency and strength). The quality of

product such as extrusion processes depends considerably on the flow and heat

transfer characteristics of a thin liquid film over a stretching sheet, an analysis of

momentum and heat transfer in such processes is essential. In many practical thin

film models surface tension plays a significant role. Many liquids (e.g., water)

have a surface tension that can be varied by the addition of surface active

substances (surfactants). These are substances that tend to congregate in the upper

layers of the liquid, changing its surface energy and hence its surface tension. For

example, soap lowers the surface tension of water, which is why we use it while

washing. Surfactants introduce the possibility of generating a surface tension

gradient, resulting in a shear stress on the liquid free surface. The most correct

approach to modeling such flows is via the macroscopic momentum equation (e.g.,

Introduction

26

the Stokes or Navier Stokes equations in the case of a Newtonian liquid) and this

approach invariably involves detailed numerical computation. The approach taken

here will be to exploit the existence of the small aspect ratio ( H L ) to expand

the momentum equations in a perturbation series in powers of .

Keeping these applications in view, Wang [1990] was the first among the

others to consider such a flow situation over an unsteady stretching surface. Ma

and Hwang [1990] examined the effect of air shear on the flow thin liquid film

over a rough rotating disk. Usha and Sridharan [1993] considered a similar

problem of axisymmetric flow in a liquid film. Ray and Dandapat [1994]

presented the flow of thin liquid film on a rotating disk in the presence of

transverse magnetic filed. Later Andersson et al. [1996] examined numerically the

behavior of a liquid film of an incompressible non-Newtonian fluid obeying a

power-law model due to unsteady stretching surface. Ray and Dandapat [1998]

carried out the effect of thermocapilarity on the production of a conducting thin

film in the presence of transverse magnetic field. Specifically, Dandapat et al.

[2000] extended the pioneering work of Wang [1990] and analyzed the

accompanying heat transfer in the liquid film driven by an unsteady stretching

surface and discussed the physical mechanisms that govern the observed thermal

characteristics for several values of the Prandtl number and the unsteady

parameter. Andersson et al. [2000] examine the effect of heat transfer on a liquid

film on an unsteady stretching surface. Dandapat et al. [2003] considered the

thermocapilarity in a liquid film on an unsteady stretching surface. Usha et al.

[2005] explores the dynamics and stability of a thin liquid film on a heated rotated

disk film with variable viscosity. Dandapat and Maity [2006] worked on the flow

of a thin film on an unsteady stretching sheet. Wang and Pop [2006] were the first

to analyse the flow of a power law fluid film on an unsteady stretching surface by

means of the homotopy analysis method (HAM). Chen [2006] extended the work

of Andersson et al. [1996] and examined numerically the behavior of a liquid film

of an incompressible non-Newtonian fluid flow and heat transfer obeying a power-

law model induced by an accelerating surface in the presence of viscous

dissipation. Liu and Andersson [2008] generalized the above problem for

prescribed surface temperature for the thermal characteristics of a liquid film

driven by an unsteady stretching surface. Nadeem and Awais [2008] analyzed the

Introduction

27

effect of a thin film over an unsteady shrinking sheet with variable viscosity.

Abbas et al. [2008] presented on an unsteady flow of a second grade fluid film

over an unsteady stretching sheet. Subhas Abel et al. [2009a] examine the heat

transfer problem in the presence of an external magnetic field and viscous

dissipation. Santra and Dandapat [2009] considered the effect of thin film flow

over a non-linear stretching sheet. Mostafa et al. [2009] carried out the effect of

MHD flow and heat transfer in a non-Newtonian liquid film over an unsteady

stretching sheet with variable fluid properties. Aziz et al. [2011] explored

analytically the influence of internal heat generation/absorption on the flow and

heat transfer characteristics by means of Homotopy Analysis Method. Muhammad

Hussan et al. [2012] considered the Mass transfer analysis for unsteady thin film

flow over stretched heated plate. Recently, Vajravelu et al. [2012] examine the

effects of viscous dissipation and the temperature-dependent thermal conductivity

on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian

Ostwald–de Waele fluid over a horizontal porous stretching surface. Chung Liu et

al. [2013] worked on the effect of heat transfer in a liquid film due to an unsteady

stretching surface with variable heat flux. Aziz et al. [2013] analysed the flow and

heat transfer in a liquid film over a permeable stretching sheet.