 Open Access
 Total Downloads : 436
 Authors : R. Vasanthakumari, R. Sekar, K. Thirumurugan
 Paper ID : IJERTV1IS9419
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Kelvin – Helmholtz Instability Of Superposed Non – Newtonian Viscoelastic Fluid.
Kelvin Helmholtz Instability Of Superposed Non Newtonian Viscoelastic Fluid.
R. Vasanthakumari1, R. Sekar2, K. Thirumurugan3
1Principal, Kasthurba College for Women, Villianur, Puducherry, India 2Professor & Head, Department Of Mathematics, Pondicherry Engineering College, India. 3Research Scholar, Department of Mathematics, Bharathiar University, Coimbatore, India
Abstract
An analysis on the stability of incompressible superposed NonNewtonian fluids is considered. Kelvin Helmholtz instability is selected for this purpose. Rivilin Ericksen elasticoviscous superposed fluid is considered. Normal mode analysis is applied. The dispersion relation is solved analytically. The stability of the system is studied for exponentially varying density, viscosity and viscoelasticity.
Key words – NonNewtonian fluid, Rivilin Ericksen Elasticoviscous superposed fluid, Kelvin Helmholtz instability.

Introduction
A NonNewtonian fluid is a fluid in which viscosity changes with the applied shear force. Therefore, NonNewtonian fluids may not have a well defined viscosity. The role of NonNewtonian fluid dynamics is important as it relates to plastic manufacture, performance of lubricants, clay suspensions, drilling muds, paints, processing of food and moment of biological fluids which contain higher molecular weight components. The convective stability of a general viscoelastic fluid heated from below is analyzed by Sokolov, R. I. Tanner [1], S. Rosenblat [5] and J. Martinez Mardones [6]. The instability of the plane interface separating two uniform superposed streaming fluids, under varying assumption of hydrodynamics, has been discussed in the celebrated monograph by Chandrasekhar [3]. With the growing importance of non newtonian viscoelastic fluids in modern technology and industries and the investigation on such fluids are desirable. RivlinEricksen fluid is one such viscoelastic fluid. Johri [2] has discussed the viscoelastic RivlinEricksen incompressible fluid under time dependent pressure gradient. Sharma and Kumar [7] have studied the thermal instability
of a layer of RivlinEricksen elasticoviscous fluid acted on by a uniform rotation and that rotation has a stabilizing effect and introduces oscillatory modes in the system. The instability of the plane interface between two uniform superposed and streaming fluids through porous medium has been investigated by Sharma and Spanos [4]. Sharma et al. [8] have studied the thermosolutal convection in RivlinEricksen fluid in a porous medium in the presence of uniform vertical magnetic fluid. G.
C. Rana et. al [12] investigated the stability of incompressible RivlinEricksen ElasticoViscous Superposed fluids under rotation in porous medium. Instability of superposed streaming fluids through a porous medium. Aiyub Khan et. al [10] have studied the instability of superposed streaming uids through a porous medium. Neild [11] gave note on the onset of convection in a layer of a porous medium saturated by a NonNewtonian Nanofluid of PowerLaw Type. Pardeep Kumar and Roshan Lal [9] have studied the stability of two superposed viscousviscoelastic fluids. When two superposed fluids flow one over the other with a relative horizontal velocity, the instability of the plane interface between two fluids, when it occurs in this instance, is known as KelvinHelmholtz instability. The present paper focus on the Kelvin
Helmholtz Instability of Superposed Non Newtonian Viscoelastic Fluid.

Mathematical Formulation
An incompressible elasticoviscous RivlinEricksen in fluid which there is a horizontal steaming in the direction with velocity () is considered. The equilibrium of this initial state is analysed by supposing that the system is slightly disturbed and then following its further evolution.
Let , , , , , ( , 0, 0) denote, respectively, the pressure, density, acceleration due to gravity, kinematic viscosity, kinematic viscoelasticity and velocity of RivlinEricksen
Substituting for , Eq.(2.4) with the help of Eqs.(2.5), (2.6) and expression (2.7) simplifies to
+ + + + =
viscoelastic fluid.
+ ( )
+
(2.8)
The governing equations are given by equation of motion, continuity and incompressibility for the RivlinEricksen elastico
where
= .
is unit vector in the direction and
viscous fluid as follows
+ . = +
+ (2.1)
Writing the three component equations of (2.8) and eliminating , and with the help of (2.5), we obtain
+ + +
+ + + =
2 ()
+
(2.9)
. = 0 (2.2)
+ . = 0 (2.3)

Superposed Uniform Fluids
Consider the case when two superposed streaming fluids of uniform densities 1 and 2, uniform viscosities 1 and 2 and uniform
viscoelsticities and are separated by a
1 2
Let , and (, , ) denote the perturbations in pressure , density and velocity
( , 0, 0) respectively.
Then, the linearized perturbation equations of fluid layer become
horizontal boundary layer at = 0. The superscript 1 and 2 distinguish the lower and the upper fluid respectively.
The density 2 of the upper fluid is taken to be less than the density 1 of lower fluid so that, in the absence of the streaming, the configuration is
+ . + .
= +
+ (2.4)
stable. Let the two fluids be streaming with constant velocities 1 and 2. Then in each of the
two regions of constant , , and , Eq. (2.9)
. = 0 (2.5)
becomes
(2 2) = 0 (3.1)
+ ( . ) =
(2.6)
Analyzing the disturbances into normal
here are:
The boundary conditions to be satisfied
modes, we seek solutions whose dependence on
, and is of given by
exp[i( + + )] (2.7)
(A)
+
(3.2)
where is the growth rate, = (2 + 2 )1 2 is the
must be continuous at an interface, since is
resultant wave number and , are horizontal wave numbers.
discontinuous at = 0.

Integrating Eq.(2.9) between 0 and
1,2
0 + and passing to be limit = 0, we obtain, in view of (3.2), the jump condition for = 0 is,
where
= 1,2 ,
= 1,2 , = .
1,2
'
1 +2
1,2
1,2
1,2
1,2
0 + + +
= 2 ()
(3.3)
1 =
1
, = 1 , 2 =
2
and = 2 are
0
+
1
1 1
2
2 2
the kinematic viscosities and kinematic viscoelasti cities of the lower and upper fluid respectively.
while the equation valid everywhere else ( 0) is
+ + +
Equation (3.7) yields
= + 11 + 22 + 2 11+22 +
2 1 1 + 2 2 Â± 11 + 22 2
+ + 1 2
' 2
4 12 1 2 1 2 + 42 1
+ = ()
+
(3.4)
1 2 2
2 ( 2 +
2 1 1 2 1 1 1 1
22 2 + 12 1 1 + 2 2 +
Here 0 = 0 + 0 0 0 is the 2 2 2 1
jump which a quantity experiences at the interface
= 0; and the superscript 0 distinguish the value a
12 1 2 1 2 + 1 1 +
2 1 1
2 + 42 2
2 2 2 1 2 1 2
quantity, known to be continuous at an interface,
4 1 2 [1 + 1 + 2 ]}1 2
(3.8)
1 2
takes at the interface = 0.
The general solution of Eq.(3.1) is a linear combination of the integrals + and . Since
/( + ) must be continuous on the surface
Some cases of interest are now considered.

When = 0, equation (3.8) yields
= 0 and cannot increase exponentially on
= + Â± { + 2
1 1 2 2 1 1 2 2
either side of the interface, the solution appropriate
4 1 2 [1 + 1 + 2 ]}1 2
(3.9)
1 2
for two regions are
Here we assume kinematic viscosities
1, 2 and kinematic viscoelasticities , of
1 2
= + + , ( < 0) (3.5)
1 1
the two fluids to be equal, that is 1 = 2 = ,
= = . However, any of the essential
1 2
2 = + 2 , ( > 0) (3.6)
Applying the boundary condition (3.3) to the solutions (3.5) (3.6), we obtain the dispersion relation
1 + 1 + 2 2 + 2 11 + 22 +
features of the problem are not obscured by this simplifying assumption. Eq.(3.9), then, becomes
= Â± [2 4 1 2 {1 + }]1 2
(3.10)

Unstable case
For the potentially unstable configuration
1 2
1 1 + 2 2 11 + 22 +
(2 > 1), it is evident from Eq.(3.10) that one of
1 2 values of is positive which means that the
2 (12 + 22) 111 + 222
1 2 perturbations growth with time and so the system
1 2 = 0 (3.7)
is unstable.

Stable case
For the potentially stable configuration (2 < 1), Eq.(3.10) yields that both values of in are either real, negative or complex conjugates with negative real parts implying stability of the system.
It is interesting to note from above that for the special case when perturbation in the direction of streaming are ignored ( = 0), the system is unstable for potentially unstable configuration and system is stable for potentially stable configuration and not depending upon kinematic viscoelasticity.


In every other direction, instability occurs when
2

Chandrasekhar S., Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York, 1981.

Sharma R. C. and Spanos T.J.T., Canadian J. Phys., (60)(1982), 1391.

S. Rosenblat, Thermal convection in a viscoelastic liquid, J. NonNewtonian Fluid Mech. 21 (1986) 534 539.

J. MartinezMardones, C. PerzGarcia, Linear instability in viscoelastic fluid convection, J. Phys. Condens. Matter 2 (1990) 12811290.

Sharma R. C. and Kumar P. Z., Naturforsch, 51a(1996), 881.

Sharma R. C., Sunil and Chand S., Appl. Mech.
1 2
2
(1 2)2 > 1 2 (3.11)
Thus for a given difference in velocity
Engg., 3(1) (1998), 1.

Pardeep Kumar and Roshan Lal, J. Fluids Eng.,
1 2 and for a given direction of the wave vector , instability occurs for all wave numbers
2007, Vol. 129, Issue 1, 116.

Aiyub Khan et al., Turkish J. Eng. Env. Sci.
> 1 2
12 1 2 2 cos2
(3.12)
34, 59 68 (2010).

D. A. Nield, A note on the Onset of
where is the angle between the direction of
( , , 0) and (, 0, 0), that is, = cos . Hence, for a given velocity differences 1 2 , instability occurs for the least wave number when
is in the direction of and this minimum wave number; , is given by
Convection in a Layer of a Porous Medium Saturated by NonNewtonian Nanofluid of PowerLaw Type, Transp. Porous Med. (2010) DOI 10.1007/s112420109671z.

Rana G. C., Sharma. V and Sanjeev Kumar, J. Comp & Math Sci. Vol.2(2), 316321, (2011).
= 1 2
12 1 2 2
(3.13)
For > , the system is unstable.



Conclusion
It has been observed that for 1 < 2, the system is found unstable and for 1 > 2 the system is stable.

References

M. Sokolov, R. I. Tanner, convective stability of general viscoelastic fluid heated below, Phys. Fluids 15 (1986) 613623.

Johri A. K., Acta Ciencia Indica, 24(1976), 377.