The research paper published by IJSER journal is about Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid 1

ISSN 2229-5518

Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid

Joginder S. Dhiman* and Pushap Lata

Abstract— The theoretical treatments of convective stability problems usually invoked the so-called principle of exchange of stabilities (PES), which is demonstrated physically as convection occurring initially as a stationary convection. Weinberger [1969] used a method of a positive operator, a generalization of a positive matrix operator, to establish the PES, wherein, the resolvent of the linearized stability operator is analyzed, which is a composition of certain integral operators. Motivated by method of positive operator of Weinberger, we aim to extend this analysis of Herron[2000] to establish the PES to more general convective problems from the domain of non-Newtonian fluid. In the present paper, the problem of heated from below with variable gravity is analyzed by the method of positive and it is established that PES is valid for this general problem, when the gravity is a nonnegative throughout the fluid layer and the elastic number of the medium is less than the ratio of permeability to porosity.

Index Terms— Principle of exchange of stabilities; Stationary convection; Positive Operator; Stability Operator; Resolvent; Integral

Operator; Viscoelastic fluid.

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1 INTRODUCTION

Rayleigh–Bénard convection is a fundamental phenomenon found in many atmospheric and industrial applications. The problem has been studied extensively experimentally and theoretically because of its frequent occurrence in various fields of science and engineering. This importance leads the authors to explore different methods to study the flow of these fluids. Many analytical and numerical methods have been applied to analyze this problem in the domain of Newtonian fluids, including the linearized perturbation method, the lattice Boltzmann method (LBM), which has emerged as one of the most powerful computational fluid dynamics (CFD) methods in recent years.

A problem in fluid mechanics involving the onset of convection has been of great interest for some time. The theoretical treatments of convective problems usually invoked the so-called principle of exchange of stabilities (PES), which is demonstrated physically as convection occurring initially as a stationary convection. This has been stated as ―all non decaying disturbances are non oscillatory in time‖. Alternatively, it can be stated as ―the first unstable
eigenvalues of the linearized system has imaginary part
establishment of this principle results in the elimination of unsteady terms in a certain class of stability problems from the governing linerized perturbation equations. Further, we know that PES also plays an important role in the bifurcation theory of instability.
Pellew and Southwell [1] took the first decisive step in the direction of the establishment of PES in Rayleigh-Bénard convection problems in a comprehensive manner. S. H.
Davis [2] proved an important theorem concerning this problem. He proved that the eigenvalues of the linearized stability equations will continue to be real when considered as a suitably small perturbation of a self-adjoint problem, such as was considered by Pellew and Southwell. This was one of the first instances in which Operator Theory was employed in hydrodynamic stability theory. As one of several applications of this theorem, he studied Rayleigh- Bénard convection with a constant gravity and established PES for the problem. Since then several authors have studied this problem under the varying assumptions of hydromagnetics and hydrodynamics.
Convection in porous medium has been studied
with great interest for more than a century and has found
many applications in underground coal gasification, solar
energy conversion, oil reservoir simulation, ground water
contaminant transport, geothermal energy extraction and in many other areas. Also, the importance of non-Newtonian

equal to zero‖. Mathematically, if r 0

i 0 (or

fluids in modern technology and industries is ever
increasing and currently the stability investigations of such
equivalently, i 0

r 0 ), then for neutral stability

fluids are a subject matter of intense research. A non–

( r 0), 0 , where r and i are respectively the real and imaginary parts of the complex growth rate . This is called the ‗principle of exchange of stabilities‘ (PES). The

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Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H. P.), Pin-171 005, INDIA.

Newtonian fluid is a fluid in which viscosity changes with the applied strain rate and as a result of which the non- Newtonian fluid may not have a well-defined viscosity. Viscoelastic fluids are such fluids whose behaviour at sufficiently small variable shear stresses can be characterized by three constants viz. a co-efficient of viscosity, a relaxation time and a retardation time, and
*Corresponding author’s E-mail: jsdhiman66e@gmail.com

The research paper published by IJSER journal is about Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid 2

ISSN 2229-5518

whose invariant differential equations of state for general motion are linear in stresses and include terms of no higher degree than the second in the stresses and velocity gradients together. The problem of the onset of thermal instability in a horizontal layer of viscous fluid heated from below has its origin in the experimental observation of Bénard [3]. Oldroyd [4] proposed and studied the

variable gravity, g(0, 0 g(z)) . Let be the temperature difference between the lower and upper plates. The fluid is assumed to be viscoelastic and described by the Oldroydian constitutive equations. Thus, the governing equations for the Rayleigh-Bénard situation in a viscoelastic fluid– saturated porous medium under Boussinesq approximation and under the effect of variable gravity are (see [6] & [9]);



constitutive relations for viscoelastic fluids in an attempt to
explain the rheological behavior of some non-Newtonian
fluids. Since then numerous research papers pertaining to
the stability investigations of non-Newtonian fluids under
the effects of different external force fields and in presence

1 1 q t t

1 p t

1 X 1

0 t k

(1)
of porous medium have been reported. Shenoy [5] had
reviewed studies of flow in non-Newtonian fluids in
porous medium, with attention concentrated on power-law

.q 0

T 

0 0 1

(2)
fluids. For further reviews of the fundamental ideas, methods and results concerning the convective problems

E ( q. )T

=K 2 T (3)
from the domain of Newtonian/ non-Newtonian fluids,
one may be referred to Chandrasekhar [6], Lin [7], Drazin

0 1 T T0

In the above equations, q , T , K ,
(4)

, 0 and

and Reid [8] and Nield and Bejan [9].
It is clear from the above discussion that the Pellew
and Southwell method is a useful and simple tool for the
establishment of PES in convective problems when the
stand for filter velocity, temperature, density, thermal diffusivity, coefficient of thermal expansion, the relaxation time and the retardation time and the kinematic viscosity,
resulting eigenvalue problem, in terms of differential equations and boundary conditions, is having constant

respectively. Here, E= (1 ) s cs

0cv

is a constant, where
coefficients. Thus, the method is not always useful to

s, cs

stand for density and heat capacity of solid (porous
determine the PES for those convective problems, which
are either permeated with some external force fields, such

matrix) material and 0, ,cv
for fluid, respectively. Here,
as variable gravity, magnetic field, rotation etc., are imposed on the basic Thermal Convection problems and the resulting eigenvalue problems contain variable coefficient/s or an implicit function of growth rate as in the case of non-Newtonian fluids. The present work is partly
inspired by the above discussions, and the works of Herron
the suffix zero refers to the value at the reference level z = 0.
This is to mention here that, when the fluid slowly percolates through the pores of the rock, the gross effect is represented by the usual Darcy`s law. As a consequence, the usual viscous terms has been replaced by the resistance

 in the above equations of motion. Here,

[10], [11] and the striking features of convection in non- Newtonian fluids in porous medium and motivated by the desire to study the above discussed problems.
term
and

q k1

k1 are the viscosity and the permeability of the
Our objective here is to extend the analysis of Weinberger [12] and Rabinowitz [13] based on the method of positive operator to establish the PES to these more general convective problems from the domain of non- Newtonian fluid. In the present paper, the problem of Thermal convection of a viscoelastic fluid in porous
medium.

Following the usual steps of the linearized stability theory, it is easily seen that the nondimensional linearized perturbation equations governing the physical problem described by equations (1)-(4) can be put into the following forms, upon ascribing the dependence of the perturbations
medium heated from below with variable gravity is
of the form
exp i k x x k y y t , (c.f.
analyzed; and using the positive operator method, it is

Chandrasekhar [6] and Siddheshwar and Krishna [14]);
established that PES is valid for this problem, when g (z)
1 1 D2
k 2 w g(z)Rk 2

(5)

(the gravity field) is nonnegative throughout the fluid layer
and the elastic number of the medium is less than the ratio
Pl 1

2 2

of permeability to porosity, i.e.

Pl or

k1 .

D k EPr

w (6)

2 Mathematical Formulation of the Physical

Problem

together with following dynamically free, thermally and
electrically perfectly conducting boundary conditions;

w 0 D2 w at z 0 and z 1 (7)

In the forgoing equations; z is the real

Consider an infinite horizontal porous layer of viscoelastic fluid of depth ‗d‘ confined between two horizontal planes and under the effect of
independent variable, D

d is the differentiation with

The research paper published by IJSER journal is about Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid 3

ISSN 2229-5518

respect to z ,

k 2 is the square of the wave number, Pr

1 1 Mw g(z)R k 2

(8)

pl 1

is the thermal Prandtl number,

Pl = 1

is the
M E Pr Rw (9)
where,
dimensionless medium permeability,

is elastic

Mw mw, w domM; M2 w m2 w, w dom MM ;

and M m , w domM

number, E= (1 ) s cs

0cv

is constant, R 2 g0

is the

The domains are contained in B, where

thermal Rayleigh number, r i i
is the complex
B L2

0,1

2 dz ,

growth rate associated with the perturbations and w, are the perturbations in the vertical velocity, temperature, respectively.
The system of equations (5)-(6) together with the

with scalar product

, z z dz , , B ; and

boundary conditions (7) constitutes an eigenvalue problem for for the given values of the parameters of the fluid and
norm

, 2 .

a given state of the system is stable, neutral or unstable according to whether r is negative, zero or positive.

It is remarkable to note here that equations (5)-(6) contain a variable coefficient and an implicit function of , hence as discussed earlier the usual method of Pellew and Southwell is not useful here to establish PES for this general problem. Thus, we shall use the method of positive

We know that 2 is a Hilbert space, so, the domain of

M is

dom M = B / D , m B, 0 1 0 .

We now formulate the homogeneous problem corresponding to equations (5)-(6) by eliminating from (8) and (9) as;

operator to establish PES.

w k2 R2 M 1 1 1

g z M E Pr

w (10)

3 ABSTRACT FORMULATION

The Method of Positive Operator

We seek conditions under which solutions of

Pl 1

or w K w , (11)

where,

equations (5)-(6) together with the boundary conditions (7)

K( ) k2R 2T(0) 1 1

g z T E Pr w (12)

grow. The idea of the method of the solution is based on the notion of a ‗positive operator‘, a generalization of a positive matrix, that is, one with all its entries positive. Such matrices have the property that they possess a single greatest positive eigenvalue, identical to the spectral radius.
The natural generalization of a matrix operator is an

Pl 1

In equation (12), we have defined

T E Pr M E Pr 1

and this exists for

integral operator with non-negative kernel. To apply the method, the resolvent of the linearized stability operator is
analyzed. This resolvent is in the form of certain integral

T k

Pr E

C Re

1 k 2

, Im 0

E Pr

operators. When the Green‘s function Kernels for these
operators are all nonnegative, the resulting operator is

and
T E Pr for Re
E Pr
.

E Pr

termed positive. The abstract theory is based on the Krein –
Rutman theorem [15], which states that;

―If a linear, compact operator A, leaving invariant a cone  , has a point of the spectrum different from zero, then it has a positive eigenvalue , not less in modulus than every other eigenvalue, and this number corresponds at least one eigenvector  of the operator A, and at
least one eigenvector  of the operator A ‖. For the present problem the cone consists of the set of nonnegative functions.
To apply the method of positive operator,
Here, ‗Re‘ and ‗Im‘ respectively stand for real and
imaginary part of the quantity.

Now, since T E Pr is an integral operator such that for

T E Pr f g z, ; E Pr f d ,

where, g z, , E Pr is the Green‘s function kernel for the operator M E Pr , which can be readily computed following Herron [2000] as;
formulate equations (5) and (6) together with boundary conditions (7) in terms of certain operators as;
g z, , Pr
cosh r 1 z cosh r 1 z

2r sinh r

The research paper published by IJSER journal is about Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid 4

ISSN 2229-5518

where, r k 2

E Pr .

real 0

k 2

E Pr

, and that T E Pr has a power series

In particular, taking 0 , we have M 1
integral operator.
T(0) is also an
about 0 in ( 0 ) with positive coefficients. In other words;

K defined in (12), which is a linear combination of certain integral operators, is termed as linearized stability
for all real 0

k 2

E Pr

, we see that
operator. K( ) depends analytically on in a certain

right half of the complex plane. It is clear from the form of
K( ) that it also contains an implicit function of .
d g z, , E Pr t n e d

E Pr t G z, , t dt 0

We shall examine the resolvent of the K( ) defined as

I K ;

I K

i.e. positive.

In particular, from the above result, taking 0 , we
deduce that T 0 M is also a positive operator.

Since, K is a linear operator, Condition (1) can be easily
I I K 0

K K 0

1 1

I K 0
(13)

verified by following the analysis of Herron [10] for K , i.e. K is a linear, compact integral operator, and has a

If for all 0 greater than some a,

power series about

0 in ( 0 ) with positive coefficients.

(1)

I K 0

is positive,

Thus, K is a positive operator leaving invariant a cone

(2) K has a power series about 0 in 0 with

(set of non-negative functions).
positive coefficients; i.e.,

d n
d K o
is positive for all
Moreover, for real and sufficiently large, the norms of

the operators T 0 and T E Pr become arbitrarily small.
So,

n, then the right side of (13) has an expansion in 0
with positive coefficients. Hence, we may apply the

K 1 .

methods of Rabinowitz [12] and Weinberger [13], to show
that there exists a real eigenvalue 1 such that the spectrum
Hence,

I K has a convergent Neumann series,

of K lies in the set

: Re 1 .
which implies that

I K is a positive operator. This

This is result is equivalent to PES, which was stated earlier

as ―the first unstable eigenvalue of the linearized system
is the content of Condition (1).
To verify Condition (2), we see that
has imaginary part equal to zero.‖ 1 1 1

0 0 > 0 for all 0 real, and

4. THE PRINCIPLE OF EXCHANGE OF STABILITIES (PES)

pl 1 0

pl pl

and

l or < k1 .

As operator

T E Pr (M E Pr ) 1
is an integral

2 p l

pl 2 pl

operator whose kernel g z, , E Pr is the Laplace

Therefore, for all real

transform of the Green‘s function G z, ; t for the initial- boundary value problem

0 max(

k 2

E Pr

2 p l

pl ),

pl 2 pl

k 2 E Pr G z , t z2 t
, (14)
g(z) 0 for all z [0,1] , by the product rule for

differentiation one concludes that K in (12) satisfies
Condition (2).
where, z , t is Dirac–delta function in two-dimension,
with boundary conditions

G 0, ; t G 1, ; t G z, ; 0 0 , (15) Following Herron [10], by direct calculation of the inverse
Laplace transform, we can have
Hence, by the Krein–Rutman theorem, we have the following result;

Theorem. PES holds for (5) - (6) together with boundary conditions (7) when g (z) is nonnegative throughout the layer and

Pl or k1

T E Pr M E Pr

is a positive operator for all

Conclusions

Thus, we see from the present analysis that PES can be established for this general convective problem from the

The research paper published by IJSER journal is about Positive Operator Method to Establish Principle of Exchange of Stabilities in Thermal Convection of a Viscoelastic Fluid 5

ISSN 2229-5518

domain of the non-Newtonian fluid using the method of positive operator, and thus extend the analysis of Weinberger [13] to the domain of non-Newtonian fluids. Theorem proves that, when g(z) (the gravity field) is nonnegative throughout the fluid layer and the elastic constant of the medium is less than the ratio of
permeability to porosity, i.e.

Pl or equivalently

k1 , PES is valid.

In particular, letting 0 , one recover the result of Herron[10] for Bénard convection problem with variable gravity field.

References

[1] A. Pellew and R.V. Southwell, 1940, ―On Maintained Convective Motion

In A Fluid Heated From Below‖, Proc. R. Soc. A, 176, pp.312-343.

[2] S. H. Davis, 1969, ―On The Principle Of Exchange Of Stabilities‖, Proc. Roy.

Soc. A., 310, pp. 341-358.

[3] H. Bénard, 1900, ―Les Tourbillions Cellularies Dans Une Nappe Liquide‖,

Revue Generale Des Sciences Pures et Appliquées, 11, pp. 1261-1309.

[4] J. G. Oldroyd, 1958, ―Non-Newtonian Effects in Steady Motions of Some

Idealized Elastico-viscous liquids‖, Proc. R. Soc. A., 245, pp.278-297.

[5] A.V. Shenoy, 1994, ―Non-Newtonian fluid heat transfer in porous media‖,

Adv. Heat Transfer, 24, pp. 101-190.

[6] S. Chandrasekhar, 1961, Hydrodynamic and Hydromagnetic Stability. Oxford

University Press, London.

[7] C.C Lin, 1955, The Theory of Hydrodynamic and Stability’, Cambridge

University Press. London.

[8] P.G. Drazin, and W.H. Reid, 1981, Hydrodynamic Stability, Cambridge

University Press, Cambridge.

[9] D. A. Nield, and A. Bejan, 1998, Convection in Porous Media, Springer: New

York.

[10] I.H. Herron, 2000, ―On the principle of exchange of stabilities in Rayleigh- Bénard Convection‖, Siam J. Appl. Math., 61(4), pp. 1362-1368.

[11] I.H. Herron, 2001, ―Onset of convection in a porous medium with internal heat source and variable gravity‖, Int. J. Engrg. Sc., 39(2), pp. 201-208.

[12] P. H. Rabinowitz, 1969, ―Non-Uniqueness of Rectangular Solutions of the Bénard Problem‖, ( Eds J. B. Keller and S. Antman,.), Theory and Nonlinear Eigenvalue Problems, Benjamin, New York .

[13] H.. F. Weinberger, 1969, ―Exchange of Stabilities in Couette flow‖, ( Eds J.

B. Keller and S. Antman,.), Theory and Nonlinear Eigenvalue

Problems, Benjamin, New York .

[14] P.G. Siddheshwar and C.V. Sri Krishna, 2001, Rayleigh-Bénard Convection in a Viscoelastic Fluid-Filled High-Porosity Medium With Nonuniform Basic Temperature Gradient, Int. J. Math. Math. Sci., 25(9),

609-619.

[15] M.G. Krein, and M.A. Rutman,1962, ―Linear Operators Leaving Invariant

A Cone in a Banach Space‖, Trans. Amer. Math. Soc., 10, pp. 199-325.