Gy2512651271

Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1265-1271
On The Region For Linear Growth Rate Of Perturbation In
Magneto-Thermal Convection In A Couple-Stress Fluid In A
Porous Medium

Ajaib S. Banyal1* and Daleep K. Sharma2
1*
Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP) INDIA
2
Department of Mathematics, Rajiv Gandhi G. C. Kotshera, Shimla (HP), INDIA

ABSTRACT
A layer of couple-stress fluid heated thermal instability (Bénard Convection) in
from below in a porous medium is considered in Newtonian fluids, under varying assumptions of
the presence of uniform vertical magnetic field. hydrodynamics and hydromagnetics, has been given
Following the linearized stability theory and by Chandrasekhar 1  and the Boussinesq
normal mode analysis, the paper through approximation has been used throughout, which
mathematical analysis of the governing equations states that the density changes are disregarded in all
of couple-stress fluid convection with a uniform other terms in the equation of motion, except in the
vertical magnetic field in porous medium, for any external force term. The formation and derivation of
combination of perfectly conducting free and the basic equations of a layer of fluid heated from
rigid boundaries of infinite horizontal extension below in a porous medium, using the Boussinesq
at the top and bottom of the fluid, established approximation, has been given in a treatise by
that the complex growth rate  of oscillatory
perturbations, neutral or unstable, must lie

Joseph 2 . When a fluid permeates through an
inside a semi-circle isotropic and homogeneous porous medium, the
gross effect is represented by Darcy’s law. The
2 study of layer of fluid heated from below in porous
 R  Pl p 2  media is motivated both theoretically and by its
 
2
  ,

 Ep1 p 2 (1  4 F )   Pl 
2 2

practical applications in engineering. Among the
applications in engineering disciplines one can name
in the right half of a complex  -plane, Where R
the food processing industry, the chemical
is the thermal Rayleigh number, F is the couple- processing industry, solidification, and the
stress parameter of the fluid, Pl is the medium centrifugal casting of metals. The development of
permeability,  is the porosity of the porous geothermal power resources has increased general
interest in the properties of convection in a porous
medium, p1 is the thermal Prantl number and p 2
is the magnetic Prandtl number, which 
medium. Stommel and Fedorov 3 and Linden 4 
prescribes the upper limits to the complex have remarked that the length scales characteristic
growth rate of arbitrary oscillatory motions of of double-diffusive convecting layers in the ocean
growing amplitude in the couple-stress fluid may be sufficiently large so that the Earth’s rotation
heated from below in the presence of uniform might be important in their formation. Moreover, the
vertical magnetic field in a porous medium. The rotation of the Earth distorts the boundaries of a
result is important since the exact solutions of the hexagonal convection cell in a fluid through porous
problem investigated in closed form, are not medium, and this distortion plays an important role
obtainable for any arbitrary combinations of in the extraction of energy in geothermal regions.
perfectly conducting dynamically free and rigid The forced convection in a fluid saturated porous
boundaries. 
medium channel has been studied by Nield et al 5 .
An extensive and updated account of convection in
Key Words: Thermal convection; Couple-Stress porous media has been given by Nield and
Fluid; Magnetic field; PES; Chandrasekhar number.
MSC 2000 No.: 76A05, 76E06, 76E15; 76E07.

Bejan 6 .
The effect of a magnetic field on the
1. INTRODUCTION stability of such a flow is of interest in geophysics,
Right from the conceptualizations of particularly in the study of the earth’s core, where
the earth’s mantle, which consist of conducting
turbulence, instability of fluid flows is being
fluid, behaves like a porous medium that can
regarded at its root. A detailed account of the
theoretical and experimental study of the onset of become conductively unstable as result of
differential diffusion. Another application of the

1265 | P a g e

Vol. 2, Issue 5, September- October 2012, pp.
results of flow through a porous medium in the diseases of joints are caused by or connected with a
presence of magnetic field is in the study of the malfunction of the lubrication. The external
stability of convective geothermal flow. A good efficiency of the physiological joint lubrication is
account of the effect of rotation and magnetic field caused by more mechanisms. The synovial fluid is
on the layer of fluid heated from below has been caused by the content of the hyaluronic acid, a fluid

given in a treatise by Chandrasekhar 1 . of high viscosity, near to a gel. A layer of such fluid
heated from below in a porous medium under the
MHD finds vital applications in MHD
action of magnetic field and rotation may find
generators, MHD flow-meters and pumps for
applications in physiological processes. MHD finds
pumping liquid metals in metallurgy, geophysics,
applications in physiological processes such as
MHD couplers and bearings, and physiological
magnetic therapy; rotation and heating may find
processes such magnetic therapy. With the growing
applications in physiotherapy. The use of magnetic
importance of non-Newtonian fluids in modern
field is being made for the clinical purposes in
technology and industries, investigations of such
detection and cure of certain diseases with the help
fluids are desirable. The presence of small amounts
of magnetic field devices.
 
of additives in a lubricant can improve bearing
performance by increasing the lubricant viscosity Sharma and Thakur 15 have studied the
and thus producing an increase in the load capacity. thermal convection in couple-stress fluid in porous
These additives in a lubricant also reduce the medium in hydromagnetics. Sharma and
coefficient of friction and increase the temperature  
Sharma 16 have studied the couple-stress fluid
range in which the bearing can operate.
heated from below in porous medium. Kumar and
Darcy’s law governs the flow of a
Newtonian fluid through an isotropic and  
Kumar 17 have studied the combined effect of
homogeneous porous medium. However, to be dust particles, magnetic field and rotation on couple-
mathematically compatible and physically stress fluid heated from below and for the case of
consistent with the Navier-Stokes equations, stationary convection, found that dust particles have
Brinkman 7 heuristically proposed the destabilizing effect on the system, where as the
rotation is found to have stabilizing effect on the
 2
introduction of the term  q, (now known as system, however couple-stress and magnetic field
 are found to have both stabilizing and destabilizing
Brinkman term) in addition to the Darcian term effects under certain conditions. Sunil et al. 18  
  have studied the global stability for thermal
   q . But the main effect is through the
k  convection in a couple-stress fluid heated from
 1 below and found couple-stress fluids are thermally
Darcian term; Brinkman term contributes very little more stable than the ordinary viscous fluids.
effect for flow through a porous medium. Therefore,  
Pellow and Southwell 19 proved the
Darcy’s law is proposed heuristically to govern the validity of PES for the classical Rayleigh-Bénard
flow of this non-Newtonian couple-stress fluid
through porous medium. A number of theories of  
convection problem. Banerjee et al 20 gave a new
the micro continuum have been postulated and scheme for combining the governing equations of
 
applied (Stokes 8 ; Lai et al 9 ; Walicka 10 ).   thermohaline convection, which is shown to lead to


the bounds for the complex growth rate of the
The theory due to Stokes 8 allows for polar effects arbitrary oscillatory perturbations, neutral or
such as the presence of couple stresses and body unstable for all combinations of dynamically rigid or

couples. Stokes’s 8 theory has been applied to the free boundaries and, Banerjee and Banerjee 21  
study of some simple lubrication problems (see e.g. established a criterion on characterization of non-
 
Sinha et al 11 ; Bujurke and Jayaraman 12 ;   oscillatory motions in hydrodynamics which was

 
Lin 13 ). According to the theory of Stokes 8 ,
 
further extended by Gupta et al. 22 . However no
such result existed for non-Newtonian fluid
couple-stresses are found to appear in noticeable
configurations, in general and for couple-stress fluid
 
magnitudes in fluids with very large molecules.
Since the long chain hyaluronic acid molecules are configurations, in particular. Banyal 23 have
found as additives in synovial fluid, Walicki and characterized the non-oscillatory motions in couple-
 
Walicka 14 modeled synovial fluid as couple stress fluid.
Keeping in mind the importance of couple-
stress fluid in human joints. The study is motivated
stress fluids and magnetic field in porous media, as
by a model of synovial fluid. The synovial fluid is
stated above,, the present paper is an attempt to
natural lubricant of joints of the vertebrates. The
prescribe the upper limits to the complex growth
detailed description of the joints lubrication has very
rate of arbitrary oscillatory motions of growing
important practical implications; practically all
amplitude, in a layer of incompressible couple-stress

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fluid in a porous medium heated from below, in the 
presence of uniform vertical magnetic field, . H  0 , (4)
opposite to force field of gravity, when the bounding 
surfaces are of infinite horizontal extension, at the
dH   
top and bottom of the fluid and are perfectly   ( H .) q   H ,
2
(5)
conducting with any combination of dynamically dt
free and rigid boundaries.
d  
The result is important since the exact
Where    1 q . , stands for the
solutions of the problem investigated in closed form,
are not obtainable, for any arbitrary combination of
dt t
perfectly conducting dynamically free and rigid convective derivatives. Here
boundaries.  c 
E    (1   ) s s  , is a constant,
 c 
2. FORMULATION OF THE PROBLEM  0 v
AND PERTURBATION EQUATIONS
Here we consider an infinite, horizontal, while  s , c s and  0 , c v , stands for the density
incompressible electrically conducting couple-stress and heat capacity of the solid (porous matrix)
fluid layer, of thickness d, heated from below so material and the fluid, respectively,  is the
that, the temperature and density at the bottom

surface z = 0 are T 0 and  0 , at the upper surface z
medium porosity and r ( x, y, z ) .
= d are T d and  d respectively, and that a uniform The equation of state is
 dT     0 1   T  T0  , (6)
adverse temperature gradient   
 
 is Where the suffix zero refer to the values at the
 dz  
maintained. The fluid is acted upon by a uniform reference level z = 0. Here g 0,0, g  is

acceleration due to gravity and  is the coefficient
vertical magnetic field H 0,0, H  . This fluid layer
of thermal expansion. In writing the equation (1),
is flowing through an isotropic and homogeneous we made use of the Boussinesq approximation,
porous medium of porosity  and of medium which states that the density variations are ignored
permeability k 1 . in all terms in the equation of motion except the
 external force term. The kinematic viscosity ,
Let  , p, T,  ,  e and q u , v, w denote
respectively the fluid density, pressure,
couple-stress viscosity  ' , magnetic permeability
temperature, resistivity, magnetic permeability and  e , thermal diffusivity  , and electrical resistivity
filter velocity of the fluid, respectively Then the  , and the coefficient of thermal expansion  are
momentum balance, mass balance, and energy
balance equation of couple-stress fluid and all assumed to be constants.
Maxwell’s equations through porous medium, The basic motionless solution is

governing the flow of couple-stress fluid in the
q  0,0,0 ,    0 (1   z ) , p=p(z),
presence of uniform vertical magnetic field
(Stokes  
8 ; Joseph 2 ; Chandrasekhar 1 ) are  T   z  T0 , (7)
given by Here we use the linearized stability theory and
    p     1 
the normal mode analysis method. Assume small
1   q 1     '  perturbations around the basic solution, and let  ,
  q .  q     g 1       2  q
     k 
  t      o  0  1  0  
 
 
  p ,  , q u, v, w and h  hx , h y , hz denote
  
respectively the perturbations in density  , pressure
 e (  H )  H , (1)
4 o 

 p, temperature T, velocity q (0,0,0) and the
. q  0 , 
magnetic field H  0,0, H  . The change in
(2)

 density  , caused mainly by the perturbation  in
dT
E  ( q .)T   2T , (3)
temperature, is given by
dt

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     0 1   T    T0      0 , i.e. Where we have introduced new coordinates

   0 . (8)
x' , y' , z ' = (x/d, y/d, z/d) in new units of length d
and D  d / dz' . For convenience, the dashes are
Then the linearized perturbation equations
dropped hereafter. Also we have substituted
of the couple-sress fluid reduces to
nd 2 
 a  kd,   , p1  , is the thermal
1 q 1  1   '        
  p  g      2  q  e    h   H 
 t 0 k1   0  4 0 
   Prandtl number; p 2  , is the magnetic Prandtl
, (9)

 k1
number; Pl 
. q  0 , (10) d2
is the dimensionless medium

  ' /(  0 d 2 )
E  w   2 , (11)
permeability, F

, is the
t dimensionless couple-stress viscosity parameter;

. h  0 , g d 4
(12) R , is the thermal Rayleigh number and
 
 h    
e H 2 d 2
   H .  q   2 h . (13) Q , is the Chandrasekhar number.
t   4 0
Also we have Substituted W  W ,
3. NORMAL MODE ANALYSIS
Analyzing the disturbances into two-  d 2   Hd 
dimensional waves, and considering disturbances
      , K     K  ,

characterized by a particular wave number, we    
assume that the Perturbation quantities are of the and D  dD , and dropped   for convenience.
form Now consider the case for any combination of

w, , h   W z , z , K ( z ) exp
the horizontal boundaries as, rigid-rigid or rigid-free
z, or free-rigid or free-free at z=0 and z=1, as the case
ik x  ik y  nt  ,
x y (14)
may be, and are perfectly conducting. The
boundaries are maintained at constant temperature,
Where k x , k y are the wave numbers along thus the perturbations in the temperature are zero at
the boundaries. The appropriate boundary
the x- and y-directions, respectively, conditions with respect to which equations (15)-
  , is the resultant wave number, n
1
(17), must possess a solution are
k  kx  k y
2 2 2
W = 0=  , on both the horizontal boundaries, (18)
is the growth rate which is, in general, a complex DW=0, on a rigid boundary, (19)
constant and, W ( z ), ( z ) and K (z ) are the D 2W  0 ,on a dynamically free boundary, 20)
functions of z only. K = 0,on both the boundaries as the regions
Using (14), equations (9)-(13), Within the outside the fluid are perfectly conducting, (21)
framework of Boussinesq approximations, in the
non-dimensional form transform to Equations (15)-(17) and appropriately
adequate boundary conditions from (18)-(21), pose
  1 an eigenvalue problem for  and we wish to
D 2

a 2  

 F 2
  D  a2
 P  W  Ra   QDD

2 2
 a2 K  Characterize  i , when  r  0 .
  Pl
  l 

, (15)
D 2 2

 a  p 2 K   DW , (16)
4. MATHEMATICAL ANALYSIS
We prove the following theorems:
and Theorem 1: If R  0 , F  0, Q 0, r  0
D 
and
2
 a 2  Ep1   W , (17)  i  0 then the necessary condition for the
existence of non-trivial solution W , , K  of
equations (15) - (17) and the boundary conditions
(18), (21) and any combination of (19) and (20) is
that

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And equating the real and imaginary parts
 R  Pl p 2  on both sides of equation (28), and cancelling
    .  i ( 0) throughout from imaginary part, we get
 Ep1
  p 2 (1  4 F )   Pl 
2 2

  r 1 1
   
1 1
   DW  a 2 W dz    D 2W  2a 2 DW  a 4 W dz  Ra 2  D  a 2  dz
 2 2 F 2 2 2 2
Proof: Multiplying equation (15) by W (the
 P  
complex conjugate of W) throughout and integrating  l 0 Pl 0   0
the resulting equation over the vertical range of z,
 
 
we get 1 1 1
 Q   D 2 K  2a 2 DK  a 4 K dz   r  Ra 2 Ep1   dz  Qp2  DK  a 2 K dz 
2 2 2 2 2 2
 1   
 W  D 2  a 2 Wdz   W  D 2  a 2  Wdz
1 1
 
F 2    
 P  0 0 0
 l 0 Pl 0 (29)
and
  Ra 2  W  dz  Q  W  DD 2  a 2 Kdz ,
1 1

   
(22) 1
1 1 1

 DW  a W dz  Ra Ep1   dz  Qp2  DK  a K dz
2 2 2 2 2 2 2 2

0
0 0
Taking complex conjugate on both sides of equation 0 0

(17), we get , (30)
D 2

 a 2  Ep1     W  , (23)
Equation (30) implies that,

 dz , (31)
1 1
Ra 2 Ep1   dz  Qp 2  DK  a 2 K
2 2 2
Therefore, using (23), we get

 W dz   D  a  Ep1  dz , (24)
1 1 0 0
 2 2  
is negative definite and also,

 dz  ap
1 2 1
0 0
Q  DK  a 2 K W
2 2 2
Also taking complex conjugate on both sides of dz , (32)
0 2 0
equation (16), we get
 
D 2  a 2  p 2  K    DW  , (25) We first note that since W ,  and K satisfy
Therefore, using (25) and using boundary condition W (0)  0  W (1) , (0)  0  (1) and
(18), we get K (0)  0  K (1) in addition to satisfying to

W DD  a 2 Kdz   DW  D 2  a 2 Kdz   K D 2  a 2 D 2  a 2  p2  K  dz ,
1 1 1 governing equations and hence we have from the
 
 2
Rayleigh-Ritz inequality 24
0 0 0
1 1

 DW dz   2  W dz ;
(26) 2 2
Substituting (24) and (26) in the right hand side of
0 0
equation (22), we get
1 1
 1 
 W  D 2  a 2 Wdz   W  D 2  a 2  Wdz
1 1

 DK dz   2  K dz ,
F 2 2
 
2
 P 
 l 0 Pl 0 0 0

 
 Ra 2  D 2  a 2  Ep1    dz  Q  K D 2  a 2 D 2  a 2  p 2  K  dz
1 1
and Banerjee et al. 25 have proved that,
1 1

DW dz   2  DW dz
0 0 2 2 2
, (27)
0 0
Integrating the terms on both sides of
1 1
equation (27) for an appropriate number of times by
D K dz   2  DK dz
2 2 2
making use of the appropriate boundary conditions (33)
(18) - (21), we get 0 0

  1 1
 
1 Further, multiplying equation (17) and its
DW  a 2 W dz    D 2W  2a 2 DW  a 4 W dz
F
  
2 2 2 2 2
 P    complex conjugate (23), and integrating by parts
 l 0 Pl 0   each term on right hand side of the resulting
equation for an appropriate number of times and
making use of boundary conditions on  namely
(0)  0  (1) along with (22), we get
 
1
 Ra 2  D  a 2   Ep1   dz
2 2 2

 D 
 a  dz  2Ep1 r  D  a  dz  E p1  
1 1 1 1

 dz   W dz
0 2 2 2 2 2 2 2 2 2 2 2

 dz
1 1
 Q   D 2 K  2a 2 DK  a 4 K dz  Qp 2   DK  a 2 K
2 2 2 2 2
  0 0 0 0
0
  0 , (34)
. (28) since  r  0 , i  0 therefore the equation (34)
gives,

1269 | P a g e

 
 
2
 a2
1 1 2


2
D  a  dz W dz ,
2
is 4 at
2 2 2
(35) Since the minimum value of
0 0 a2
And a 2   2 0 .
1 1
1 Hence, if
  dz  W dz ,
2 2
(36) r  0 and i  0,
E 2 p1 
2 2
0 0
  Pl p 2 
It is easily seen upon using the boundary conditions then    R   .
 Ep1  p 2 (1  4 F )   Pl 
2 2
(18) that  
  D 
1
And this completes the proof of the theorem.
 a 2  dz  Real
2 2
part of
0

  D 
1
 1  2 
   5. CONCLUSIONS
   D  a dz    a 2 dz ,
2 2

 0  0
The inequality (40) for r  0 and  i  0 , can
be written as
 
1
    D 2  a 2  dz
2
 R  Pl p 2 
 r   i 
2 2
  ,
 Ep1 p 2 (1  4 F )   Pl 
0 2 2
 
    D 2  a 2  dz ,
1
The essential content of the theorem, from
the point of view of linear stability theory is that for
0
the configuration of couple-stress fluid of infinite
horizontal extension heated form below, having top
   D 2  a 2  dz
1
and bottom bounding surfaces are of infinite
horizontal extension, at the top and bottom of the
0 fluid and are perfectly conducting with any arbitrary
1 1 combination of dynamically free and rigid
1 2  2 1 2
   dz   D 2  a 2  dz  , (37)
2
  boundaries, in the presence of uniform vertical
magnetic field parallel to the force field of gravity,
0  0  the complex growth rate of an arbitrary oscillatory
motions of growing amplitude, lies inside a semi-
(Utilizing Cauchy-Schwartz-inequality) circle in the right half of the  r  i - plane whose
Upon utilizing the inequality (35) and (36),
Centre is at the origin and radius is equal
inequality (37) gives
  Pl p 2 
  D  to  R 
1 1
1  , Where R is the
 a 2  dz  W
2 2 2
dz , (38)
 Ep1  p 2 (1  4 F )   Pl 
2 2
0
Ep1  0
 
thermal Rayleigh number, F is the couple-stress
Now R  0, Pl  0 ,   0 , F  0 and  r  0 , thus
parameter of the fluid, Pl is the medium
upon utilizing (31) and the inequalities (32),( (33)
and (38), the equation (29) gives, permeability,  is the porosity of the porous
 F  1 2 R 
1 medium, p1 is the thermal Prandtl number and p 2 is
 a2   W 2 dz 0 , (39)
Ep1   
I 1  a 2 ( 2  a 2 ) 2  P p  the magnetic Prandtl number. The result is

 Pl  l 2  0 important since the exact solutions of the problem
Where investigated in closed form, are not obtainable, for
 1
 
 a2 1 2
  any arbitrary combinations of perfectly conducting
1 1
I 1   r   DW dz  r  W dz Qa 2  DK  a 2 K dz
2 2 2
  P  0
dynamically free and rigid boundaries and it
 l 0 0 provides an important improvement to Banyal and
Khanna 26 .
,
is positive definite.
Therefore , we must have
REFRENCES
 2 1 ( 2  a 2 ) 2  F  ,
1 .
R
    
P  Chandrasekhar, S., Hydrodynamic and
Ep1  p 2 Pl a2  l Hydromagnetic Stability, Dover
which implies that Publications, New York 1981.
R

4 2 F  2 1
  (40) 
2 . Joseph, D.D., Stability of fluid motions,
Ep1  Pl p 2 Pl vol.II, Springer-Verlag, berlin, 1976.

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