Validity Of Principle Of Exchange Of Stabilities Of Rivilin- Ericksen Fluid Permeated with Suspended Particles in Porous Medium Under The effect of Rotation with Variable Gravity Field By Using Operator Method.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 01 | Jan -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1198
Validity of Principle of Exchange of Stabilities of Rivilin- Ericksen Fluid
Permeated with Suspended Particles in Porous Medium Under The effect
of Rotation with Variable Gravity Field By Using Operator Method.
Pushap Lata Sharma,Assistant Professor Department of Mathematics, R.GGovt. Degree College,
Chaura Maidan Shimla-4 (H.P.), India pl_maths@yahoo.in
--------------------------------------------------------------------****--------------------------------------------------------------------------------------
Abstract - The Thermosolutal Convection in Rivilin-Ericksen elastico-viscous fluid in porous medium is considered to include the
effect of suspended particles and rotation under variable gravity. In the present, to establish the Principle Of Exchange of
Stabilities (PES) by using a method of a Positive Operator, a generalization of a positive matrix Wherein, the resolvent of the
linearized stability operator is analyzed which is in the form of a composition of certain integral operators. Motivated by the
analysis of Weinberger and the works of Herron , our objective here is to extend this analysis of positive operator to establish the
PES. It is established by the method of positive operator of Weinberger that PES is valid for this problem under sufficient
conditions and g (z) is nonnegative throughout the fluid layer .
Keywords : Rivilin-Ericksen, Positive Operator, Principle of Exchange of Stabilities, linearized Stability
Operator, Suspended Particles.
1. INTRODUCTION
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. With the growing importance of non–
Newtonian fluids in modern technology and industries, the investigations of such fluids are desirable. Rivlin–Ericksen
[1955] proposed a non–linear theory of a class of isotropic incompressible elastico–viscous fluids with the
constitutive relations
ijijij pT   ,
ijij e
t








 2  ,
Here  is the coefficient of viscoelasticity. Such elastico–viscous fluids have relevance and importance in
agriculture, communication appliances, chemical technology and in biomedical applications.
Keeping in mind the importance of non–Newtonian fluids in modern technology, industries, chemical
engineering and owing to the importance of variable gravity field in astrophysics etc.
Our objective here is to extend the analysis of Weinberger & Rabinowitz’s [1969] based on the method of positive
operator to establish the PES for a more general convective problems from the domain of non-Newtonian fluid,
namely,Thermal convection of a Rivlin - Ericksen fluid in porous medium heated from below with variable gravity. Lata
[2010,2012,2013,2015,2016] has exclusively worked for the validity of principle of exchange of stabilities by using Positive
Operator Method.
The present work is partly inspired by the above discussions, and the works of Herron [2000,2001] and the
striking features of convection in non-Newtonian fluids in porous medium and motivated by the desire to study the
above discussed problems. Our objective here is to extend the analysis of Weinberger & Rabinowitz’s [1969] 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 Rivilin- Ericken fluid layer heated
from below in porous medium under the effect of suspended particles with variable gravity g(z) is positive throughout
the fluid layer in porous medium heated from below with variable gravity is analyzed and using the positive operator
method, when g (z)( the gravity field) It is established from the present analysis that PES is valid .
2. Mathematical Formulation of the Physical Problem
Consider an infinite horizontal Rivlin -Ericksen fluid layer of thickness d bounded by the horizontal plane z=0 and
z=d in porous medium permeated with suspended particles. This layer is heated from below so that a uniform
temperature gradient 






dz
dT
is maintained across the layer. This layer is acted upon by a vertical variable
gravity field ))z(g0,0(g 

.
3. Basic hydrodynamical equations governing the physical configuration
The basic hydrodynamic equations that govern the physical configurations (c.f. Rivlin and Ericksen [1955],
Spiegel and Veronis [1960], Stokes [1966] and Scanlon and Segal [1973)] ) under Boussinesq approximation[1903]
are given by;
3.1. Equation of Continuity
0. v

(1)
3.2. Equations of Motion
)()(2
1
1).(
11
0
100
vu
SN
q
v
tk
X
p
vv
t
v































 (2)
3.3. The equations of motion and continuity for the particles
The force exerted by the fluids on the particle is equal and opposite to the force exerted by the particles on fluid,
there must be an extra force term, equal in magnitude but opposite in sign, in the equations of motion for the
particles. The buoyancy force on the particles are neglected. Inter –particle reactions are ignored for we assume that
the distances between particles are quite large as compared with their diameter . If mN is the mass of particles per
unit volume, then the equation of motion and continuity for the particles ,under the above assumptions are :
m N )uv(SNu).u(
1
t
u 










(3)
0)u.N(
t
N





(4)
3.4. The equation of heat conduction
Since the volume fraction of the particles is assumed small, the effective properties of the suspension are taken to be
those of the clean fluid. Assuming that the particles and fluid are in thermal equilibrium, the equation of heat
conduction is given as;
  v0ssv0 c
t
T
)1(cc 


 ( TqTu
t
mNcTv pt
2
.). 










 (5)

3.5. The equation of state
  00 1 TT   (6)
In the above equations, p,  , , ,  , 1k , , )w,v,u(v

,Tand X

denote respectively the pressure, density,
temperature, viscosity, viscoelasticity, medium porosity, medium permeability, thermal coefficient of expansion, the
external force field, gradient operator; and velocity of the fluid; S=6  , (  being particle radius), is the Stokes’
drag coefficient, x (x,y,z), s , sc , , vc denote the density and heat capacity of solid (porous) matrix and fluid
respectively, c pt the heat capacity of the particles and q the “effective” thermal conductivity of the fluid . )t,x(u

and
N(x,t) denote the filter velocity and number density of the suspended particles, respectively.
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 of the form   tykxkiexp yx  ,
(c.f. Chandrasekhar [1961] and Siddheshwar and Krishna [2001]);
  




)()(1
1
1
222
zgkRwkDF
P
B
T
l














(7)
      wHRhEkD T   Pr1 22
(8)
together with following dynamically free and thermally and electrically perfectly conducting boundary conditions
wD0w 2
 at 10  zandz (9)
In the forgoing equations, z is the real independent variable,
dz
dD  is the differentiation with respect to z ,
2
k
is the square of the wave number, Pr


 is the thermal Prandtl number, lP = 2
1
d
k
is the dimensionless medium
permeability, Where 2
Sd
m
 and H=h+1 and B=b+1 where b= S , TR =



4
02 dg
R is the thermal Rayleigh
number, lP = 2
1
d
k
is the dimensionless medium permeability and Pr=


is the Prandtl number and F= 2
d

is the
dimensionless Rivilin-Ericksen parameter,



4
02 dg
R is the thermal Rayleigh number,  ir i  is the
complex growth rate associated with the perturabations and ,w are the perturbations in the vertical velocity,
temperature, respectively.
Hence, the system of equations (7) and (8) together with boundary conditions (9) constitutes an eigen value problem for
 for given values of the parameters
2
k , R, F Pr, H ,B and  for the present problem.
The system of equations (7)-(8) together with the boundary conditions (9) constitutes an eigenvalue
problem for for the given values of the parameters of the fluid and 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 (7)-(8) 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 operator to establish PES.
3. METHOD OF POSITIVE OPERATOR
We seek conditions under which solutions of equations (7)-(8) together with the boundary conditions (9)
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
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 operators. When the Green’s function Kernels for these
operators are all nonnegative, the resulting operator is termed positive. The abstract theory is based on the Krein –
Rutman theorem [1962 ], 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 eigen value  , not less in modulus than every other eigen value, and this number corresponds at least
one eigen vector  of the operator A, and at least one eigen vector

  of the operator

A ”. For the present
problem the cone consists of the set of nonnegative functions.
To apply the method of positive operator, formulate the above equations (7) and (8) together with boundary
conditions (9) in terms of certain operators as;
4. MATHEMATICAL ANALYSIS BY USING THE METHOD OF POSITIVE OPERATORS
In the following analysis, we shall first of all construct an equivalent eigen -value problem to the eigen -value problem
described by equations (7) and (8) together with boundary conditions (9) in terms of certain operators.
Let (-D )k22
 w= mw
and define
 
M
~
domw,mM
~
M
~
M
~
domw,wmwM
~
M
~
domw,mwwM
~
22



We have the following forms of equations (2A.38) and (2A.39)
  















)z(gRkMwF1
p
1
1
H 2
1
(10)
       HRphEaD1 1
22
w (11)
The above define domains are contained in cone  , where
  






 
1
0
22
dz1,0L is a Hilbert space with a finite magnitude, by definition [],
with scalar product
    
1
0
dzzz, , , 

and norm
2
1
,
So, the domain of M
~
is
dom M
~
=     010,m,D:   .
In the present a case, we have },w{  .
Substituting the value of  from equation (11) in equation (12), we get
w=  
1
1
F1
p
1
1
H
















 1
M








1
B 122
)]hEPr(M)[z(gkR 
 (12)
or  wKw  (13)
We know that is a Hilbert space, so, the domain of M is
dom M =     010,Bm,D/B  .
In (12), we have
    1
Pr}{Pr)

  hEMhET and exists for
   











0Im,
Pr)(
Re
2
)Pr(




hE
k
CT
hE
k and
Pr)(
Pr)(
2
1
hE
k
hEPT r





for
Re  
Pr)(
2
hE
k



 .
Now,  PrET is an integral operator such that for ,
      
1
0
Pr}{;,Pr}{  dfhEzgfhET ,
where,   Pr)(,, hEzg  is Green’s function kernel for the operator  Pr}{ hEM   , and is
given as
 
     
rr
zrzr
PhEzg r
sinh2
1cosh1cosh
}{Pr,,




where, Pr)(2
hEkr   .
In particular, taking 0 , we have )0(1
TM 
is also an integral operator.
 K defined in (12), which is a composition of certain integral operators, is termed as linearized stability operator.
K(  ) depends analytically on  in a certain right half of the complex plane. It is clear from the composition of K(  )
that it contains an implicit function of  .

               1
0
1
0
1
0
1 
  KIKKKIIKI (14)
If for all 0 greater than some a,
(1)    1
0

 KI is positive,
(2)  K has a power series about 0 in   0 with positive coefficients; i.e.,
h
is positive for all n, then the
right side of (13) has an expansion in  0 with positive coefficients. Hence, we may apply the methods of
Weinberger [1969] and Rabinowitz [1969], to show that there exists a real eigenvalue 1 such that the spectrum of
 K lies in the set   1Re:  . This is result is equivalent to PES, which was stated earlier as “the first
unstable eigenvalue of the linearized system has imaginary part equal to zero.”
5. THE PRINCIPLE OF EXCHANGE OF STABILITIES (PES)
It is clear that  K is a product of certain operators. Condition (1) can be easily verified by following the analysis of
Herron [2000, 2001] for the present operator  K .The operator  0M 1

is an integral operator whose Green’s
function  0;,zg  is nonnegative so
1
M
= T(0) is a positive operator. It is mentioned above that   Pr}{ hET  is an
integral operator its Green’s function kernel g   Pr}{,, hEz  is the Laplace transform of the Green’s function
Pr)hE(
1

G 







Pr)hE(
t
;,z for the boundary value problem
 t,zG
t
Pr)hE(k
z
2
2
2











 ,
where,  t,z  is Dirac –delta function in two-dimension,
with boundary conditions       00;,zGt;,1Gt;,0G 
Following Herron [2000], by direct calculation of the inverse Laplace transform, we can have Green’s function kernel g
 Pr)hE(;,z  is the Laplace transform of the Green’s function
Pr)hE(
1

G 







Pr)hE(
t
;,z , thus by definition
[] of Laplace transform,
g  Pr)hE(;,z  =
tPr)hE(
0
e 

 Pr)hE(
1

G 







Pr)hE(
t
;,z dt
n
d
d







 g  Pr)hE(;,z  =
tPr)hE(
0
n
et 

 Pr)hE(
1

G 







Pr)hE(
t
;,z dt 0

for al l n and for all real 0
Pr)hE(
k2

 . So,   Pr)hE(T has a power series about 0 in (( 0 ) with
positive coefficients forl all real 0
Pr)hE(
k2


Theorem. The PES holds for (7)- (8) when g(z) is nonnegative throughout the layer
}
B
,
Pr)hE(
k
),
)FP(2
)FHP(
)FP(
}
)FP(2
FHP
{max{
2
1
1
1
2
1
1












 ,for
)FP(4)FHP( 1
2
1  .
Proof: As    KI is a nonnegative compact integral operator for
}
B
,
Pr)hE(
k
),
)FP(2
)FHP(
)FP(
)
FP2
FHP
(max{
2
1
1
1
2
1
1
0












 , for
)FP(4)FHP( 1
2
1  .Thus all the conditions of the Krein-Rutman theorem are satisfied ,therefore
   KI has a positive eigen value 1 , which is an upper bound for the absolute values of all the eigenvalues, and the
corresponding eigen function   is nonnegative. We observe that
        01KI 1  ,
Thus, if    KI is nonnegative, then 1 1 ,so the methods of Weinberger[] and Rabinowitz []apply and showing that
“there exits a real eigenvalue a1  such that the spectrum of  K lies in the set “   1Re  ”. This is
equivalent to the PES.
6.Conclusions:
It is established from the present analysis that PES is valid for Rivilin- Ericken fluid layer heated from below in porous
medium under the effect of suspended particles with variable gravity g(z) is positive throughout the fluid layer and
)FP()FHP( 1
2
1  . The following conclusions are deduced from the above result in the light of Remark 1.
7.REFERENCES
1.Chandrasekhar, S. [1961], ‘Hydrodynamic and Hydromagnetic Stability’, Oxford University Press, London.
2.Drazin, P.G. and Reid W.H. [1981], ‘Hydrodynamic Stability’, Cambridge University Press, Cambridge.
3.Garg, A., Srivastava, R. K. and Singh, K. K., (1994), ‘Proc. Nat. Acad. Sci;’ India, 64A (III), 355.
4.Herron, I.H. [2000], On the principle of exchange of stabilities in Rayleigh-Benard Convection, Siam J. Appl. Math.,
61(4), 1362-1368.
5. Lata, Pushap. [2013], Study on the principle of exchange of stabilities in thermal instability of Walter’s fluid in
porous medium with variable gravity by positive operator, International Journal of Physical and Mathematical
Sciences. 4 (1): 496-511
6.Lata, Pushap. [2013], On the principle of exchange of stabilities in Oldroydian fluid in Porous Medium with Variable
Gravity using Positive Operator method, Advances in Applied Research Sciences ,4 (6) : 68-74

7.Lata, Pushap. [2013]. Thermal Instabilty of Walter’s (MODEL B’) fluid Permeated with suspended particles in
porous medium with variable gravity using Positive Operator , International Journal of pure and applied Mathematical
Sciences, 6(3) :261-2725.
8.Lata, Pushap. [2013], On the principle of exchange of stabilities in Rayleigh- Benard Convection in Porous Medium
with Variable Gravity using Positive Operator method, Journal of Applied Mathematics and Fluid Mechanics , 5 (2) :45-
51
9. Lata, Pushap. [2013], Instability of Rivlin- Ericksen Elastico – Viscous fluid in porous medium with variable gravity
by Positive Operator, International Journal of Physical and Mathematical Sciences, 4(1):512-52
10.Lata, Pushap. (2014). On the principle of exchange of stabilities in Thermohaline problem of Veronis type with
Variable Gravity using Positive Operator method, International Journal of physical & mathematical sciences ,5(1):723-
738.
11.Lata, Pushap. (2016). On the principle of exchange of stabilities in the Magnetohydrodynamic Benard Problem with
Variable Gravity by Positive Operator method, International Journal of Advance Research, Ideas and Invotation and
Technology ,2(6):1-6.
12.Lata, Pushap. (2016). Onset of Convection of Maxwellian fluid in Porous Medium with variable Gravity using
Positive Operator method, International Research Journal of Engineering and Technology ,4(1):79-185.
13.Rivlin, R. S. & Ericksen, J. L. (1955), Stress-deformation relations for isotropic materials. J. Rational Mech. and
Analysis, 4, 323.
14.Weinberger, H.F (1969), ‘Exchange of Stabilities in Couette flow’ in Bifurcation Theory and Nonlinear eigenvalue
problems, J.B Keller and S. Antman, eds., Benjamin, New York.
8. Biography
Dr.Pushap Lata is an Assistant Professor in Mathematics at Rajiv Gandhi Govt. Degree College
Chaura Maidan Shimla -4. She is also a life member of Indian science Congress. She has
presented over 17 research papers in the National and International Conferences. She has
published about 14 research papers in reputed Journals with Impact Factor.

Validity Of Principle Of Exchange Of Stabilities Of Rivilin- Ericksen Fluid Permeated with Suspended Particles in Porous Medium Under The effect of Rotation with Variable Gravity Field By Using Operator Method.

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