SIMILARITY SOLUTIONS OF THE THREE DIMENSIONAL BOUNDARY ... · dimensional unsteady incompressible laminar boundary layer flow ... places a vital role because it is the only class

Int. J. of Appl. Math and Mech. 8 (2): 77-88, 2012.

SIMILARITY SOLUTIONS OF THE THREE DIMENSIONAL

BOUNDARY LAYER EQUATIONS OF A CLASS OF GENERAL NON-

NEWTONIAN FLUIDS

V. Patel and M. G. Timol

Department of Mathematics, Veer Narmad South Gujarat University Surat-395003 India

Email: [email protected]

Received 28 May 2011; accepted 16 October 2011

ABSTRACT

Similarity analysis is made of the three dimensional incompressible laminar boundary layer

flow of general class of non-Newtonian fluids. This work is an extension of previous analysis

by Na and Hansen (1967), where the similarity solution of laminar three dimensional

boundary layer equations of Power-law fluids was investigated. For the present flow

situation, it is observed that the similarly solutions exist only for the case of flow over

wedge. Further, it is also observed that for the more general case of the boundary layer

flow of non-Newtonian fluids over anybody shapes yields non –similar solutions. Present

similarity equations are well agreed with those available in literature.

Keywords: Non-Newtonian fluids, Power-law fluid, Reiner-Philippoff fluid, Non-similarity

solution, Similarity solution,Group of Transformations.

1 INTRODUCTION

The classical theory of Newtonian fluid depends upon the hypothesis of linear relationship

between stress tensor and strain tensor, rate of strain tensor and even rate of stress tensor. The

fluids which do not follow such a linear relationship are called non-Newtonian fluid. Non-

Newtonian fluids are generally divided in to two categories like viscoinelastic fluids and

viscoelastic fluids. The common feature of viscoinelastic fluids is that when at the rest they

are isotropic and homogeneous and when they are subjected to a shear the resultant stress

depends only on the rate of shear. However, such types of fluids show diverse behavior in

response to applied stress. Numbers of rheological models have been proposed to explain

such a diverse behavior. Some of this models are; Power-law fluids, Sisko fluids, Ellis fluids,

Prandtl fluids Williamson fluids, Sutterby fluids, Reiner-Rivlin fluids, Bingham plastic,

Eyring fluids, Powell Eyring fluids, Reiner-Philippoff etc. To investigate the non-Newtonian

effects, the class of solutions known as similarity solutions place an important role. This is

because that is the only class of the exact solutions for the governing equations which are

usually non-linear partial differential equations (PDEs) of the boundary layer type. Further

this also serves as a reference to check approximate solutions.

The subject of boundary layer flows of non-Newtonian fluids has been a topic of an

investigation from a long time as it has an application in various industries and in day to day



78

life. It is well known that similarity solutions for the PDEs governing the flow of Newtonian

and non-Newtonian fluids exist only for limited classes of main stream velocities at the edge

of the boundary layer. For example, for two dimensional laminar boundary layer flow of

Newtonian fluids, similarity solutions are limited to the well known Falkner-Skan solutions

Rajagopal et al (1983). Most of the generalization of the Falkner-Skan solutions and

approximate solutions in the literature are limited to the power law fluids; this is because they

are mathematically the easiest to be treated among most of the non-Newtonian fluids.

In past, some special work on the topic was due to Acrivos et al ((1960) , (1965)),Schowalter

(1960),Bizzell et al (1962) ,Hayasi (1965), Kapur et al (1963), Lee and Ames (1966), Hansen

and Na (1968), Walters et al (1964), Denn (1967), Seth (1974), Rajagopal et al ((1983),

(1984)), Banks et al (1986). Recently, the topic developed much rapidly may be availability of

fast computing devices and software facilities. Patil and Timol (2011) have investigated

Similarity solutions of nonlinear partial differential equations governing the motion of three-

dimensional unsteady incompressible laminar boundary layer flow of non-Newtonian power-

law fluid past a flat plate by two parameter group of transformations method in its most

general form.Anjali Devi and Julie Andrews (2011) have discussed problem of

incompressible, viscous, forced convective laminar boundary layer flow of copper water and

alumina water nanofluid over a flat plate. An exact solution of the problem of oscillatory flow

of a fluid and heat transfer along a porous oscillating channel in presence of an external

magnetic field has been discussed in detail by Adhikary and Misra (2011). They have

considered the flow through a channel in which the fluid is injected on one boundary of the

channel with a constant velocity, while it is sucked off at the other boundary with the same

velocity. The two boundaries are considered to be in close contact with two plates placed

parallel to each other. While Parmar and Timol (2011) have derived similarity transformations

for the system of partial differential equations governing the boundary layer equations for

coupled heat and mass transfer natural convection of a viscous, incompressible and

electrically conducting flow of non-Newtonian Power-law fluids over a vertical permeable

cone surface saturated porous medium in the presence of uniform transverse magnetic field

and the thermal radiation effects using one parameter deductive group-theoretic

transformation technique.

Even though considerable progress has been made in our understanding of the flow

phenomena, more work is still needed to understand the effect of various parameters

involving different non-Newtonian models and the formulation of accurate method of analysis

for any body shape of engineering significance. The theoretical research on the topic however

is hampered because of complex nature of the equations describing the flows. Moreover, the

nature of non-linear shearing stress and rate of a strain relationship of the various models

poses additional difficulties.

In the analysis of boundary layer problem, the class of solution known as similarity solutions

places a vital role because it is the only class of exact solution for the boundary layer

equations. For two-dimensional flow of Newtonian fluids, it is well-known that similarity

solutions exist for the class of bodies known as the Falkner-Skan problem, which includes

many practical geometry. On the other hand, for non-Newtonian fluids, the non-linear relation

between shearing stress and the rate of strain causes further restrictions on the class of

problems this can be solved by similarity transformations.It is interesting to note that this non-

linear relationship can be mathematically expressed as a functional relationship between stress

tensor and rate of deformation tensor and for different non-Newtonian fluids this

Similarity Solutions Of The Three Dimensional Boundary Layer Equations


79

relationship may be implicit or explicit functional relationship. For the two dimensional case,

such problem was investigated by Hansen and Na (1968) and they have drawn the conclusion

that for boundary layer flows of non-Newtonian fluids of any model, similarity solutions exist

only for the flow passed a wedge.

There are certain aspects of three-dimensional boundary layers. In two-dimensional flow,

since the boundary layer is restricting to more in the direction of the outer flow due to any

sufficiently strong opposing pressure gradient. These ultimately results in separation of the

flow from the surface and may trigger important readjustment of the outer flows. In the three-

dimensional boundary layer, on the other hand, the flow remains in freedom of choosing the

minimum difficult path and strong unfavorable pressure gradient do not necessarily lead to

detachment of flow from the surface but demonstrate themselves by drastic changes in flow

directions. Consequently, not only is the boundary layer separation modified in the three-

dimensional flow but also it carries different implications (regarding over all effects) from

those which applied in two-dimensional case.

Even though very useful information can be revealed to the various physical parameters on

the boundary layer characteristics from the similarity solution, it is of limited engineering

value since for practical purposes bodies other than wedge will most likely be

encountered. This required a general formulation and solution technique which can solve any

problem of boundary layer flows of non-Newtonian fluids such as the Reiner – Philippoff

fluid treated in this paper – a topic which seems to have been neglected in the literature. In

this paper, we will therefore look beyond the similarity solution of the problem by

considering shapes other than a wedge. A formulation is given in which the boundary

layer equations are transformed to a form which are suitable for solution by some exact or

numerical solution techniques. The formulation is made into such a general form that

boundary layer flows of any shape can be treated by entering the expression of the main

stream velocity into a general function p(x) and q(x) these are also known as body shape

functions. Similarity equations will be presented in this paper for two examples, namely, the

similarity solution of the flow over a wedge and the non-similar solution of the flow over

a semi-infinite flat plate with mainstream parallel to the plate. The second example is known

as Blasius solution, which for the case of two-dimensional flows of Newtonian fluids is in

well agreement. Deviations from similarity solutions as shown in the present paper where

non-Newtonian fluids are treated therefore show clearly the effects of the various parameters

involved in the model.

There are two reasons for studying this particular non-Newtonian fluid model, namely Reiner-

Philippoff fluid model. First, this model correctly represents a class of non-Newtonian fluids

and yet there seems to be luck of reported literature on the boundary layer flow of such fluids.

Second, the present analysis introduces a method of formulation and solution which can be

applied to the boundary layer flow of any non-Newtonian fluid over any body shape in which

the velocity gradient is expressed explicitly as a function of the shearing stress.

2 PROBLEMFORMULATION

The formulation of empirical relations for different non-Newtonian fluid and its evaluation in

terms of known variables is indeed very difficult task. Recently Patel and Timol (2010) have



80

derived empirical relationship for three-dimensional boundary layer equations of different

non-Newtonian fluids models.

A number of industrially important fluids such as molten plastics, polymers, pulps and foods

exhibit non-Newtonian fluid behavior (Nakayama et al. 1988). Due to the growing use of

these non-Newtonian materials, in various manufacturing and processing industries,

considerable efforts have been directed towards understanding their flow characteristics.

Many of the inelastic non-Newtonian fluids encountered in chemical engineering processes

are assumed to follow the so-called "power-law model". This is because of the mathematical

simplicity of the power-law fluids in which the shear stress varies according to a power

function of the strain rate (Metzner et al. 1965). This model is purely phenomenological;

however, it is useful in that approximately describes a great number of real non-Newtonian

fluids. This model behaves properly under tensor deformation. Use of this model alone

assumes that the fluid is purely viscous. But there are certain limitations of this model. For

example, it is deduced from empirical relationship and it indicates an infinite effective

viscosity for low shear rate, thus limiting its range of applicability. For present study we

consider Reiner-Philippoff non-Newtonian fluid, mainly for two reasons: First, thismodel

correctly represents a class of non-Newtonian fluids and yet there seems to be a lackof

reported literature on the boundary layer flow of such fluids. Second, the present analysis

introduces a method of formulation and solution which can be applied to the boundary layer

flow of any non-Newtonian fluid over anybody shape in which the velocity gradient

isexpressed explicitly as a function of the shearing stress.

The governing differential equations for the three dimensional boundary layer flow of a

Reiner – Philippoff non-Newtonian fluid can be written as[Refer Na and Hansen

(1967),Kalthia and Timol (1986)].

(1)

{

}

(2)

{

}

(3)

Where are shearing stresses parallel to Y-direction and acting along X and Z

direction respectively. Following Schowalter (1960), under the boundary layer assumptions,

the only two non-vanishing components that are related explicitly to the velocity

gradient are given by

(

) (

)

(4)

(

) (

)

(5)

The boundary conditions are:



81

y = 0: u(x) = 0; v (x) = 0;w(x) = 0 (6)

(7)

Let us introduce the following dimensionless quantities

√

√

√

√

(8)

And a stream function, such that

(9)

Equations (1)-(7) become

( )

(10)

( )

(11)

(

)

(

)

(12)

(

)

(

)

(13)

Subject to the boundary conditions:

(14)

(15)

Equations (10)-(15) represent a system of non-linear partial differential equations, the solution

of which is quite difficult. One major simplification can be achieved by using the similarity

transformation where the system of non-linear partial differential equations is reduced to a

system of ordinary differential equations. Such transformations are of the limited to some

special forms of the mainstream velocities. For the case of boundary layer flows of general

non-Newtonian fluids, it was proved by Timol and Kalthia (1986) that similarity solutions

exist only if the mainstream velocities are given by

(16)

(17)



82

This corresponds to the three-dimensional boundary layers flow past a wedge. For other

bodyshape, the flow is not self similar and a transformation will be introduced in this paper to

reduce equations (10)-(15) to a form which can be solved by some suitable numerical

technique.

3GROUP THEORETIC ANALYSIS

Similarity analysis by the group theoretic method is based on concepts derived from the

theory of continuous transformation group. This method was first introduced by Birkhoff

(1950) and Morgan (1952) and later on lot of contribution was made in these techniques by

various research workers [Bluman and Cole (1974), Bluman and Kumai (1989) and Bluman

and Anco (2002).]

For the present problem here we select following one parameter linear group of

transformationsG:

{

Where are constants and A is a parameter of group transformation.

We now seek relations among the such that the basic equation will be invariant under this

group of transformation. This can be achieved by substituting the transformation into equation

(10)-(15). Thus, we obtain

(

)

( )

(18)

And

(

)

( )

(19)

(

) (20)

(

) (21)

(22)

(23)

From equations (10)-(15), it is seen that if the basic equation are to be invariant under theG

group of transformation, the powers of A in each term should be equal. Thus, invariance of

equations (18) to (23) under the group G gives following relations among .



83

(24)

Solving equations (24) we get

(25)

The next step in this method is to find the so-c ll “ b ol t i ri t ” r gro p of

transformation. Absolute invariants are function having the same form before and after the

transformation

(26)

Therefore, these functions are absolute invariants under this group of transformation. We

therefore obtain the transformed independent and dependent variables are:

g g (27)

Substituting for independent and dependent variable in equations (10)-(15) expressions found

from equation (27). We expect to obtain a set of equations which are ordinary differential

equation or very close to ordinary differential equations; specifically we obtain similarity

solution flow over a wedge,

g

( )

(28)

g

(29)

g

(g g

g g

) (30)

g

(g g

g g

) (31)

Subject to the boundary conditions:

(32)

(33)



84

3.1 Deduction

For two-dimensional case, the flow will be independent of Z-direction and hence in this case,

equations (29) and (31) will vanish (as g ). Hence equations (28) and (30) along

with boundary conditions (32)- (33) will be reduced to those derived by Sirohi et al (1984)

and Na (1994) which they have independently solved numerically.

4 NON-SIMILARITY SOLUTIONS

For the general case in which the boundary layer over anybody shape is to be analyzed

general transformations are introduced as follows:

√

√

√

g (√

) g

(√

)

(34)

Under these transformations, equations (10)-(15) become,

g {

p

}

p { ( )

} {

} (35)

g {

}

{ } {

} (36)

Subject to the same boundary conditions given by equations (32)-(33).

Where

g

( g

g

g g

) (37)

g

(

g

√ g

√

√ g

g

) (38)

p

(39)

(40)

Here p are known as body shape functions and it is easy to verify that for



85

p

Now for these values of p above set of equations will reduce to those ordinary

differential equations given by (28) and (29). This also confirms that similarity solutions of

present flow problem exist only for the flow past wedge. On the other hand for the body

other than wedge, where are real constants. For

the above mentioned case, the related bodyshape functions will be:

p

;

( )

And for this case, non-similarity equation (35)-(38) will become

g {

}

{ ( )

} {

} (41)

g {

}

{ } {

} (42)

Where corresponding equations (37) and (38) willreduce

g

[ g

g

g g

] (43)

g

(

g

√ g √

g

g

) (44)

With the same boundary conditions are given by equations (39)-(40).

4.1 Deduction

For two-dimensional case, the flow will be independent of Z-direction and hence in this case,

equations (42) and (44) will vanish (as g ). Hence equations (41) and (43) along

with boundary conditions (32) - (33) will be reduced to those derived by Na (1994) which

they have already solved numerically by Kellar-Box method.

5 CONCLUSION

The similarity analysis of the three dimensional boundary layer flow of non-Newtonian

Reiner – Philippoff fluids past external surface is derived. It is observed that similarity

solutions exist for only the flow past at wedge.For the flow past any otherbody shape, the

same formulation can be used and only change is in two functions namely p and

equations (37) and (38) by substituting into these functions the mainstream velocity for that

particular geometry. The present analysis provides useful in formulation for the boundary

layer flow not only for Reiner – Philippoff fluids but for the other fluids too which are studied

by Hansen and Na (1968), Sirohi et al (1984), Timol and Kalthia (1986), Patel and Timol

(2009) and recently, Surati and Timol (2010).



86

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SIMILARITY SOLUTIONS OF THE THREE DIMENSIONAL BOUNDARY ... · dimensional unsteady incompressible laminar boundary layer flow ... places a vital role because it is the only class