J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
Bounds for Complex Growth Rate of Perturbations in Rotatory Rivlin-Ericksen Viscoelastic Fluid Configuration AJAIB S. BANYAL Department of Mathematics, Govt. College Nadaun, Hamirpur-177033, H.P., INDIA (Received on: June 24, 2012) ABSTRACT The thermal instability of a Rivlin-Ericksen viscoelastic fluid acted upon by uniform vertical rotation and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical rotation, for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle complex
σ r2 + σ i2 〈
σ -plane, where TA
TA , in the right half of a π 2F
is the Taylor number and F is the
viscoelasticity parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory Rivlin-Ericksen viscoelastic fluid heated from below. Further, It is established mathematically that for stationary convection, in the absence on rotation Rivlin-Ericksen fluid behaves like an ordinary Newtonian fluid. Keywords: Thermal convection; Rivlin-Ericksen Fluid; Rotation; PES; Rayleigh number; Taylor number. MSC 2000 No.: 76A05, 76E06, 76E15; 76E07.
1. INTRODUCTION Stability of a dynamical system is closest to real life, in the sense that
realization of a dynamical system depends upon its stability. Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root.
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in Geophysics, interiors of the Earth, Oceanography and Atmospheric Physics, and has been investigated by several authors (e.g., Bénard5, Rayleigh15, Jeffreys11) under different conditions. A detailed account of the theoretical and experimental study of the onset of Bénard Convection in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar8. The use of Boussinesq approximation has been made throughout, which states that the density changes are disregarded in all other terms in the equation of motion except the external force term. Bhatia and Steiner6 have considered the effect of uniform rotation on the thermal instability of a viscoelastic (Maxwell) fluid and found that rotation has a destabilizing influence in contrast to the stabilizing effect on Newtonian fluid. The thermal instability of a Maxwell fluid in hydromagnetics has been studied by Bhatia and Steiner7. They have found that the magnetic field stabilizes a viscoelastic (Maxwell) fluid just as the Newtonian fluid. Sharma17 has studied the thermal instability of a layer of viscoelastic (Oldroydian) fluid acted upon by a uniform rotation and found that rotation has destabilizing as well as stabilizing effects under certain conditions in contrast to that of a Maxwell fluid where it has a destabilizing effect. In another study Sharma18 has studied the stability of a layer of an electrically conducting Oldroyd fluid13 in the presence of magnetic field and has found that the magnetic field has a stabilizing influence.
There are many elastic-viscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s13 constitutive relations. Two such classes of fluids are Rivlin-Ericksen’s and Walter’s (model B’) fluids. Rivlin-Ericksen9 have proposed a theoretical model for such one class of elastic-viscous fluids. Sharma and Kumar19 have studied the effect of rotation on thermal instability in Rivlin-Ericksen elastico-viscous fluid and found that rotation has a stabilizing effect and introduces oscillatory modes in the system. Kumar et al.14 considered effect of rotation and magnetic field, with free boundaries only, on Rivlin-Ericksen elastico-viscous fluid and found that rotation has stabilizing effect, where as magnetic field has both stabilizing and destabilizing effects. A layer of such fluid heated from below or under the action of magnetic field or rotation or both may find applications in geophysics, interior of the Earth, Oceanography, and the atmospheric physics. Pellow and Southwell14 proved the validity of PES for the classical RayleighBénard convection problem. Banerjee et al.1 gave a new scheme for combining the governing equations of thermohaline convection, which is shown to lead to the bounds for the complex growth rate of the arbitrary oscillatory perturbations, neutral or unstable for all combinations of dynamically rigid or free boundaries and, Banerjee and Banerjee2 established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al.10. However no such result existed for non-Newtonian fluid configurations, in general and for RivlinEricksen viscoelastic fluid configurations, in
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
particular. Banyal4 have characterized the non-oscillatory motions in couple-stress fluid. Keeping in mind the importance of non-Newtonian fluids, the present paper is an attempt to prescribe the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude, in a layer of incompressible Rivlin-Ericksen viscoelastic fluid heated from below in the presence of uniform vertical rotation opposite to force field of gravity, when the bounding surfaces are of infinite horizontal extension, at the top and bottom of the fluid are rigid. 2. FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS Considered an infinite, horizontal, incompressible electrically conducting Rivlin-Ericksen viscoelastic fluid layer, of thickness d, heated from below so that, the temperature and density at the bottom surface z = 0 are T0 and ρ 0 and at the upper surface z = d are Td and ρ d respectively, and that a uniform adverse temperature
gradient β =
dT dz
is maintained. The fluid
is acted upon by a uniform vertical rotation →
Ω(0,0, Ω ) , parallel to the force field of →
gravity g (0,0,− g ) . The equation of motion, continuity and heat conduction, governing the flow of Rivlin-Ericksen viscoelastic fluid in the presence of rotation (Rivlin and Ericksen (1955); Chandrasekhar (1981) and Kumar et al (2006)) are
→ p → → ∂q 1 → → + q .∇ q = −∇ − Ω× r ∂t 2 ρo → δρ ' ∂ + g 1 + + ν + ν ∂ t ρ 0
2
→ → 2→ ∇ q + 2 q × Ω
,
(1)
→
∇. q = 0 ,
(2)
→ ∂T + ( q .∇ ) T = κ ∇ 2 T , ∂t
(3)
→
Where ρ , p, T,ν ,ν and q (u , v, w) denote respectively the density, pressure, temperature, kinematic viscosity, kinematic viscoelasticity and velocity of the fluid, '
→
respectively and r ( x, y , z ) . The equation of state for the fluid is
ρ = ρ 0 [1 − α (T − T0 )],
(4)
Where the suffix zero refer to the values at →
the reference level z = 0. Here g (0,0,− g ) is acceleration due to gravity and α is the coefficient of thermal expansion. In writing the equation (1), we made use of the Boussinesq approximation, which states that the density variations are ignored in all terms in the equation of motion except the external force term. The thermal diffusivity κ is assumed to be constant. The initial state is one in which the velocity, density, pressure, and temperature at any point in the fluid are, respectively, given by →
q = (0,0,0 ) , ρ = ρ ( z ) ,p=p(z),T=T(z), (5)
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
Assume small perturbations around the basic solution and let δρ , δp , θ , and →
q (u , v, w)
denote respectively the perturbations in density ρ , pressure p, →
temperature T and velocity q (0,0,0) . The change in density δρ , caused mainly by the perturbation θ in temperature, is given by ρ + δρ = ρ 0 [1 − α (T + θ − T0 )] , = ρ − αρ 0θ
i.e. δρ = −αρ 0θ .
(6)
∂2 ∂2 ∂2 + + Where ∇ = ∂x 2 ∂y 2 ∂z 2 ∂v ∂u − and; ς = ∂x ∂y 2
3. NORMAL MODE ANALYSIS Analyzing the disturbances into normal modes, we assume that the Perturbation quantities are of the form
[w,θ , ς ] = [W (z ), Θ(z ), Z (z )] exp
→
→ 1 ∂q = − ∇ δ p − g αθ ∂t ρ0
→ → 2→ ∇ q + 2 q × Ω
,
(7)
(ik
x
the
∇. q = 0 ,
(8)
∂θ = β w + κ∇ 2θ , ∂t
x + ik y y + nt ) ,
(9)
Within the framework of Boussinesq approximation, equations (6) - (9) , become
(
x-
and
k = kx + k y 2
y-directions,
),
1 2 2
(D
2
)[
) ]
(
∂ θ ∂ θ + ∂x 2 ∂y 2
− 2Ω ∂ς ∂z
(10)
∂ς ∂ ∂w = ν + ν ' ∇ 2ς + 2 Ω , (11) ∂t ∂t ∂z
,
[(1 + Fσ )(D − a ) − σ ]Z = − DW , (D − a − p σ )Θ = −W , 2
,
is the resultant wave
− a 2 (1 + F σ ) D 2 − a 2 − σ
W = Ra Θ + T A DZ
∂ 2 ∂ ∇ w = ν + ν ' ∇ 4 w + gα ∂t ∂t
respectively,
number, and n is the growth rate which is, in general, a complex constant. Using (13), equations (10) – (12), in nondimensional form transform to 2
2
(13)
Where k x , k y are the wave numbers along
→
2
(12)
denote the z-component of vorticity.
Then the linearized perturbation equations are
∂ + ν + ν ' ∂t
∂θ = β w + κ∇ 2θ ∂t
2
2
2
1
(14) (15) (16)
Where we have introduced new coordinates (x' , y' , z ') = (x/d , y/d, z/d) in new units of
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
length d and D = d / dz ' . For convenience, the dashes are dropped hereafter. Also we have 2 substituted a = kd ,σ = nd , p1 = ν , is the thermal κ
ν
Prandtl number;
p2 =
Prandtl number;
F=
Ericksen parameter;
ν η
2
, is the Rilvin-
κν
TA =
4Ω 2 d 4
ν
2
2Ωd
ν
Z⊕
, and
D⊕ = dD ,
W = W⊕
,
Θ=
βd 2 Θ⊕ κ
,
and dropped (⊕ ) for
convenience. We now consider the case where both the boundaries are rigid and perfectly conducting and are maintained at constant temperature, then the perturbations in the temperature are zero at the boundaries. The appropriate boundary conditions with respect to which equations (14)-(16), must possess a solution are W = DW = 0, Θ = 0 and Z=0 at z = 0 and z = 1.
(17)
We first note that since W and Z satisfy W (0) = 0 = W (1) and Z (0) = 0 = Z (1) in addition to satisfying to governing equations and hence we have from the Rayleigh-Ritz inequality24.
∫ DW 0
1
2
dz ≥ π 2 ∫ W dz , 2
0
2
(19)
0
(18)
W (0) = 0 = W (1)
for
Z (0) = 0 = Z (1) , Banerjee et al.
and have
25
shown that
∫
1
2
D 2W dz ≥ π 2 ∫ DW dz And 2
0
0
1
1
∫D
2
2
Z dz ≥ π 2 ∫ DZ dz , 2
0
(20)
0
4. MATHEMATICAL ANALYSIS We prove the following lemma: Lemma 1: For any arbitrary oscillatory perturbation, neutral or unstable 1
∫
2
Z dz 〈
0
Equations (14)-(16), along with boundary conditions (17), poses an eigenvalue problem for σ and we wish to Characterize σ i when σ r ≥ 0 .
1
2
Further,
1
, is the Taylor number.
Also we have Substituted Z=
∫
1
DZ dz ≥ π 2 ∫ Z dz ,
0
kinematic viscoelasticity gαβ d 4 , is the thermal Rayleigh R=
number and
1
, is the magnetic
ν' d
and
∫ {DZ
1
1
σ
2
∫ DW
dz and
0
1
2
2
}
+ a 2 Z dz〈 2
0
1
1
π 2F 2 σ
2
∫ DW
2
dz .
0
Proof: Further, multiplying equation (15) with its complex conjugate, and integrating by parts each term on both sides of the resulting equation for an appropriate number of times and making use of boundary condition on Z namely Z (0) = 0 = Z (1) along with (17), we get 2 1 (1 + 2 Fσ r + F 2 σ ) ∫ D 2 Z 0
2
2 1 dz + ( 2σ r + 2 F σ ) ∫ DZ 0
2 + a2 Z dz
2
+ 2a 2 DZ
2
2 + a4 Z
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
419 +σ
Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012) 1 2
∫
2
0
(21)
0
Since σ r ≥ 0 and on utilizing the inequalities (19) and (20), equation (21) gives 1
∫Z
2
dz 〈
0
∫ {DZ
σ
∫ DW
2
2
2
dz and
}
+ a 2 Z dz 〈 2
0
1
1
π F σ 2
2
2
∫ DW
2
dz ,
(22)
0
Therefore, using (26) and appropriate boundary condition (17), we get 1
1
1
0
0
0
(
1
)
(
1
2
0
)
(
)
{
1 0
(
) }
{
2
2
2
}
1
0
{
2
2
}dz
0
1 1 1 2 2 2 2 2 2 = Ra 2 ∫ DΘ + a 2 Θ + p1σ ∗ Θ dz − TA (1 + Fσ * ) ∫ DZ + a 2 Z dz − TAσ * ∫ Z dz 0 0 0
∗
Proof: Multiplying equation (14) by W (the complex conjugate of W) throughout and integrating the resulting equation over the vertical range of z, we get
)
1
2
(
)
1
1
0
0
(1 + Fσ ) ∫ W ∗ D 2 − a 2 Wdz − σ ∫ W ∗ D 2 −a 2 Wdz = Ra 2 ∫ W ∗Θdz + TA ∫ W ∗DZdz 0
(23) Taking complex conjugate on both sides of equation (16), we get
)
− a 2 − p1σ ∗ Θ∗ = −W ∗ , Therefore, using (24), we get 1
1
0
0
(
(24)
)
∗ 2 2 ∗ ∗ ∫ W Θdz = − ∫ Θ D − a − p1σ Θ dz , (25)
Also taking complex conjugate on both sides of equation (15), we get
,
(28) Integrating the terms on both sides of equation (25) for an appropriate number of times by making use of the appropriate boundary conditions (19), along with (17), we get 1
2
,
0
(1 + Fσ )∫ D 2W + 2a 2 DW + a 4 W dz + σ ∫ DW + a 2 W
T σ 〈 2A . π F 2
(D
) }
(1 + Fσ ) ∫ W ∗ D 2 − a 2 Wdz − σ ∫ W ∗ D 2 − a 2 Wdz 1
and σ i ≠ 0 then the necessary condition for the existence of non-trivial solution (W , Θ, Z ) of equations (16), (17) and (18) together with boundary conditions (19) is that
(
(
= − Ra 2 ∫ Θ D 2 − a 2 − p1σ ∗ Θ ∗dz + TA ∫ Z (1 + Fσ * ) D 2 − a 2 − σ ∗ Z ∗dz
Theorem 1: If R 〉 0 , F 〉 0, T A 〉 0, σ r ≥ 0
0
{
∗ ∗ * 2 2 ∗ ∗ ∫ W DZdz = − ∫ DW Zdz = ∫ Z (1 + Fσ ) D − a − σ Z dz
0
We prove the following theorems:
1
]
)
− a 2 − σ ∗ Z ∗ = − DW ∗ , (26)
Substituting (25) and (27) in the right hand side of equation (23), we get
0
1
2
(27)
1
1
[(1 + Fσ )(D *
1 2
Z dz = ∫ DW dz ,
,
(29) And equating imaginary parts on both sides of equation (29), and cancelling σ i (≠ 0) throughout, we get 1
{
} { 1
2
F ∫ D 2W + 2a 2 DW + a 4 W dz + ∫ DW + a 2 W 2
2
0
1
1
{
2
0
}
2
}dz
1
= − Ra 2 p1 ∫ Θ dz +T A F ∫ DZ + a 2 Z dz + T A ∫ Z dz , 2
0
2
2
0
2
0
(30) Now R 〉 0, F 〉 0 and T A 〉 0, utilizing the inequalities (19, (20) and (22), the equation (30) gives, TA (1 + π 2 F )1 − 2 π F σ
2
1 2 〈 0, (31) ∫ DW dz + I 1 0
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
Where I1 = F ∫ 2a 2 DW 0 1
2
+ a 4 W dz + a 2 ∫ W dz + Ra 2 p1 ∫ Θ dz 0 0 2
1
2
1
2
,
Is positive definite. And therefore, we must have
σ 〈 2
TA π 2F
Hence, if then σ 〈 2
(32)
σ r ≥ 0 and σ i ≠ 0 , TA . π 2F
And this completes the proof of the theorem. Theorem 2: For stationary convection the Rivlin-Ericksen viscoelastic fluid behaves like an ordinary Newtonian fluid i. e. for T A = 0 implies that σ r = 0 and σ i = 0 , when both the bounding surfaces are rigid. Proof: The inequality (32), can be written as
σ r2 + σ i2 〈
TA , π 2F
If T A = 0 , then we necessarily have,
σ r = 0 And σ i = 0 ,
Thus, For stationary convection the Rivlin-Ericksen viscielastic fluid behaves like an ordinary Newtonian fluid, when both the bounding surfaces are rigid and it mathematically establishes the result of Kumar et al. (2006). This completes the proof. 5. CONCLUSIONS The inequality (32) for σ r ≥ 0 and
σ i ≠ 0 , can be written as σ r 2 + σ i 2 〈
TA , π 2F
420
The essential content of the theorem, from the point of view of linear stability theory is that for the configuration of Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension heated form below, having top and bottom bounding surfaces rigid, in the presence of uniform vertical rotation parallel to the force field of gravity, the complex growth rate of an arbitrary oscillatory motions of growing amplitude, must lie inside a semi-circle in the right half of the σ r σ i - plane whose centre is at the origin and radius is
TA , where TA is the π 2F
Taylor number and F is the viscoelasticity parameter. 6. REFERENCES 1. Banerjee, M.B., Katoch, D.C., Dube, G.S. and Banerjee, K., Bounds for growth rate of perturbation in thermohaline convection. Proc. R. Soc. A, 378, 301-304 (1981). 2. Banerjee, M. B., and Banerjee, B., A characterization of non-oscillatory motions in magnetohydronamics. Ind. J. Pure & Appl Maths., 15(4): 377-382 (1984). 3. Banerjee, M.B., Gupta, J.R. and Prakash, J., On thermohaline convection of Veronis type, J. Math. Anal. Appl., Vol.179, pp. 327-334 (1992). 4. Banyal, A.S, The necessary condition for the onset of stationary convection in couple-stress fluid, Int. J. of Fluid Mech. Research, Vol. 38, No.5, pp. 450-457 (2011). 5. Bénard, H., Les tourbillions cellulaires dans une nappe liquid, Revue Genérale
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
421
6.
7.
8.
9.
10.
11.
12.
Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.3 (3), 414-421 (2012)
des Sciences Pureset Appliquees 11, 1261-1271, 1309-1328 (1900). Bhatia, P.K. and Steiner, J.M., Convective instability in a rotating viscoelastic fluid layer, Zeitschrift fur Angewandte Mathematik and Mechanik 52, 321-327 (1972). Bhatia, P.K. and Steiner, J.M., Thermal Instability in a viscoelastic fluid layer in hydromagnetics, Journal of Mathematical Analysis and Applications 41, No. 2,271-283 (1973). Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York (1981). Rivlin, R.S. and Ericksen, J.L., Stress deformation relations for isotropic materials, J. Rat. Mech. Anal. 4, 323429 (1955). Gupta, J.R., Sood, S.K., and Bhardwaj, U.D., On the characterization of nonoscillatory motions in rotatory hydromagnetic thermohaline convection, Indian J. pure appl.Math.,17(1), pp 100107 (1986). Jeffreys, H., The stability of a fluid layer heated from below, Philosophical Magazine 2, 833-844 (1926). Kumar, P., Mohan, H. and Lal, R., Effect of magnetic field on thermal instability of a rotating Rivlin-Ericksen
13.
14.
15.
16.
17.
18.
19.
viscoelastic fluid, Int. J. of Maths. Math. Scs., Vol., 1-10 (2006). Oldroyd, J.G., Non-Newtonian effects in steady motion of some idealized elastic-viscous liquids, Proceedings of the Royal Society of London A245, 278297 (1958). Pellow, A., and Southwell, R.V., On the maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London A, 176, 312-43 (1940. Rayleigh, L., On convective currents in a horizontal layer of fluid when the higher temperature is on the underside, Philosophical Magazine 32,529-546 (1916). Schultz, M.H.. Spline Analysis, Prentice Hall, Englewood Cliffs, New Jersy (1973). Sharma, R.C., Effect of rotation on thermal instability of a viscoelastic fluid, Acta Physica Hungarica 40, 11-17 (1976). Sharma, R.C., Thermal instability in a viscoelastic fluid in hydromagnetics, Acta Physica Hungarica 38, 293-298 (1975). Sharma, R.C. and Kumar, P., Effect of rotation on thermal instability in RivlinEricksen elastic-viscous fluid, Zeitschrift fur Naturforschung 51a, 821-824 (1996).
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