J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
A Characterization of the Nonoscilatory Motions in Couple-Stress Fluid with Suspended Particles AJAIB S. BANYAL Department of Mathematics, NSCBM, Govt. College Hamirpur, 177005 (HP) India ABSTRACT The thermal instability of a couple-stress fluid with suspended particles heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper mathematically establishes that the onset of instability at marginal state, cannot manifest itself as stationary convection, if the thermal Rayleigh number R, the couple-stress parameter F and the suspended particles parameter B, satisfy the inequality
{
1 B
}
R ≤ 4π 4 1 + π 2 F , the result which clearly established the stabilizing character of the couple-stress and the destabilizing character of the suspended particles. Keywords: Thermal convection; Couple-Stress Fluid; Fine Dust; Suspended Particles. MSC 2000 No.: 76A05, 76E06, 76E15.
1. INTRODUCTION Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root. 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 etc. 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 Chandrasekhar². 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. Sharma et al.7 has considered the effect of suspended particles on the onset of Bénard convection in hydromagnetics. The fluid has been considered to be Newtonian in all
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
above studies. Chandra3 observed that in an air layer, convection occurred at much lower gradients than predicted if the layer depth was less than 7mm and called this motion “columnar instability”. However for a layer deeper than 10mm, Bénard–type cellular convection was observed. Thus there is a contradiction between the theory and experiment. Scanlon and Segel5 have considered the effect of suspended particles on the onset of Be’nard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure fluid was supplemented by that of the particles. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. Stokes11 proposed and postulated the theory of couple-stress fluid. One of the applications of couple-stress fluid is its use to the study of the mechanism of lubrication of synovial joints, which has become the object of scientific research. A human joint is a dynamically loaded bearing which has articular cartilage as the bearing and synovial fluid as lubricant. When fluid film is generated, squeeze film action is capable of providing considerable protection to the cartilage surface. The shoulder, knee, hip and ankle joints are the loaded-bearing synovial joints of human body and these joints have low-friction coefficient and negligible wear. Normal synovial fluid is clear or yellowish and is a viscous, nonNewtonian fluid. According to the theory of Stokes11, couple-stresses are found to appear in noticeable magnitude in fluids very large molecules. Since the long chain hylauronic acid molecules are found as additives in synovial fluid. Walicki and Walicka13
modeled synovial fluid as couple-stress fluid in human joints. Sharma and Sharma10 have studied the couple-stress fluid heated from below in porous medium. The use of magnetic field is being made for the clinical purposes in detection and cure of certain diseases with the help of magnetic field devices. Sharma and Thakur8 have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma9 have studied the effect of suspended particles on couplestress fluid heated from below in the presence of rotation and magnetic field and found that rotation has a stabilizing effect while dust particles have a destabilizing effect on the system. Sunil et al.12 have studied the effect of suspended particles on couple-stress fluid heated and soluted from below in porous medium and found that suspended particles have stabilizing effect on the system. Kumar and Kumar4 have studied the combined effect of dust particles, magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection, found that dust particles have destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the system, however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions. Keeping in mind the importance of non-Newtonian fluids, the present paper is an attempt to characterize the onset of instability at marginal state as stationary convection analytically, in a layer of incompressible couple-stress fluid heated from below in the presence of suspended particles.
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
2. FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS
Where the suffix zero refer to the values at
Considered an infinite, horizontal, incompressible couple-stress fluid layer in the presence of suspended particles, of thickness d, heated from below so that, the temperature and density at the bottom surface z = 0 are T0 , ρ 0 respectively and at
acceleration due to gravity, x = ( x, y , z ) and K = 6 πµη ′ , η ′ being particle radius, is the Stokes’ drag coefficient. Assuming uniform particles size, spherical shape and small relative velocities between the fluid and particles, the presence of particles adds an extra force term, in the equation of motion (1), proportional to the velocity difference between particles and fluid. The force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. Interparticle reactions are ignored for we assume that the distances between the particles are large compared with their diameters. If mN is the mass of the particles per unit volume, then the equation of motion and continuity for the particles are:
the lower surface z = d are Td , ρ d and that a uniform adverse temperature gradient
β =
dT dz
is maintained. Let ρ , p, T and
→
q (u, v, w) denote respectively the density, pressure, temperature and velocity of the →
_
_
fluid; q d x, t and N x, t denote the velocity and number density of suspended particles respectively. Then the momentum balance, mass balance equations of the couple-stress fluid (Stokes11; Chandrasekhar2 and Scanlon and Segel5) are → ∂ q ∂t
→ →
+ q .∇ q = −
µ → ν − ∇ 2 ∇ 2 q + ρ 0 '
1
ρ
0
KN
ρ
0
→
∇p + g 1 +
+ ρ 0 ,
_
→ ∂q → → → → mN d + q d .∇ q d = KN q − qd , ∂t → ∂N + ∇ N q d = 0 , ∂t
δρ
(1)
→ → q − q d
→
∇. q = 0 ,
→
the reference level z = 0. Here g (0,0,− g ) is
(2)
(5)
Let c v , c pt denote the heat capacity of the fluid at constant volume and the heat capacity of the particles. Assuming that the particles and the fluid are in thermal equilibrium, the equation of heat conduction gives ∂ → + q .∇ T + mNc pt ∂t
ρ0cv
The equation of state ρ = ρ0 [1 − α (T − T0 )] ,
(3)
(4)
→ ∂ → + q d .∇ T = q ∇ 2T ∂t
,
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011) →
Or
∇. q = 0 ,
mNc pt ∂T → + q .∇ T + ρ0cv ∂t →
∂ + q d .∇ T = κ∇ 2T ∂t
,
(6)
The kinematic viscosityν , couplestress viscosity µ ' , thermal diffusivity κ , and coefficient of thermal expansion α are all assumed to be constants. The initial state of the system is taken to be a quiescent layer (no settling) with a uniform particle distribution number. The basic motionless solution is →
q = (0,0,0)
T
→
, q d = (0,0,0) , = T0 − β z , ρ = ρ 0 (1 + αβ z ) ,
N = N 0 , is a constant.
(7)
Assume small perturbations around the basic solution and let δρ , N, δp , θ , →
q (u, v, w)
→
and qd (l , r , s ) denote respectively the perturbations in density, suspended particles number density N 0 , pressure p, temperature T, couple-stress fluid velocity (0,0,0) and particles velocity (0,0,0). The change in density δρ caused mainly by the perturbation θ in temperature is given by (8) δρ = −αρ 0θ . Then the linearized perturbation equations of the couple-stress fluid becomes →
→ ∂q 1 =− ∇δp − g αθ + ρ0 ∂t ' → → → ν − µ ∇ 2 ∇ 2 q + KN 0 q − q ρ0 ρ 0 d
(10)
→
→ → ∂q mN 0 d = KN 0 q − qd , ∂t
(1 + b ) ∂θ ∂t
(11)
= β (w + bs ) + κ∇ 2θ ,
Where b =
mN 0c pt
ρ 0cv
(12)
.
3. NORMAL MODE ANALYSIS Analyzing the disturbances into normal modes, we assume that the Perturbation quantities are of the form
[w,θ ] = [W (z ), Θ(z )] Exp (ik x x + ik y y + nt )
(13)
Where k x , k y are the wave numbers along the x and y-directions respectively
(
k = kx2 + k y2
)
1 2
,
is
the
resultant
wave
number and n is the growth rate which is, in general, a complex constant. Using (13), equations (9) and (12), on using (10) and (11), in non-dimensional form, become
(D
2
=−
)
M − a 2 σ 1 + 1 + τ 1σ
(
+ F D 2 − a 2
) − (D 2
2
)
− a 2 W
gα d 2 a 2 Θ
ν
(14)
,
(9)
(D
2
)
− a 2 − Bp1σ Θ = −
βd 2 (B + τ 1σ ) W, κ (1 + τ 1σ )
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(15)
Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
where a = kd , σ =
nd 2
mN 0 τν m ,τ = ,τ 1 = 2 , M = , ρ0 K d
ν d ν µ' p1 = , B = 1 + b , F = ,D = 2 κ ρ0d ν dz
and D⊕ = dD and dropping (⊕) for convenience. Here is the thermal prandtl number, F is the couple-stress parameter and B is the suspended particles density parameter. βd 2 Θ ⊕ in Substituting W = W⊕ and Θ = κ equations (14) and (15) and dropping (⊕ ) for convenience, in non-dimensional form becomes,
(D
2
M 2 + F(D2 − a2 ) − (D2 − a2 ) −a 2 )σ 1+ , (16) 1+τ1σ
W = −Ra2Θ
(D
2
− a 2 − Bp1σ )Θ = −
(B + τ σ ) W , (1+ τ σ ) 1
(17)
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The constitutive equations of the couplestress fluid are
τ ij = (2µ − 2µ '∇2 )eij , eij =
1 ∂vi ∂v j + . 2 ∂x j ∂xi
The conditions on a free surface are
∂u ∂w + = 0, ∂z ∂x ∂v ∂w = 0 . = µ − µ ' ∇ 2 + ∂z ∂y
τ xz = (µ − µ ' ∇ 2 ) τ yz
(
)
(19)
From the equation of continuity (10) differentiating with respect to z, we conclude that 2 ∂2 ∂ 2 ∂ 2 w ' ∂ µ − µ + + ∂x 2 ∂y 2 ∂z 2 ∂z 2 = 0 , (20)
1
Where R =
gαβ d 4
κν
which implies that , is the thermal Rayleigh
number. Since both the boundaries are maintained at constant temperature, the perturbations in the temperature are zero at the boundaries. The case of two free boundaries is little artificial but it is most appropriate for Stellar atmospheres and enables us to find analytical solutions and to make some qualitative and quantitative conclusions. The appropriate boundary conditions with respect to which equations (16) and (17) must be solved are W = 0, Θ = 0 at z = 0 and z = 1. (18)
∂2w ∂4w , = 0 = 0 , at z = 0 and z = d. ∂z 2 ∂z 4
(21)
Using (13), the boundary conditions (21) in non-dimensional form transform to
D 2W = D 4W = 0 at z = 0 and z = 1. (22) We prove the following theorem:
Theorem: If R 〉 0 , F 〉 0 and B=1+b, b 〉 0 and σ = 0 then the necessary condition for the existence of non-trivial solution (W , Θ ) of equations (16) and (17) together with boundary conditions (18) and (22) is that
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
F ∫ W ∗ (D 2 − a 2 ) Wdz − ∫ W ∗ (D 2 −a 2 )
R 〉 4π 4 {1 + π 2 F } 1 B
1
2
[
],
− a 2 ) F (D 2 − a 2 ) − (D 2 − a 2 ) 2
W = − Ra Θ 2
(D
2
0
Wdz =
Ra B
{ + ∫ {D W
∗
F ∫ W ∗ (D 2 − a 2 ) Wdz − ∫ W ∗ (D 2 − a 2 ) 1
1
3
0 1
,
(26)
2
+ 2a 2 DW + a 4 W dz
2
0
Taking complex conjugate on both sides of equation (24), we get
{
− a )Θ = − BW ,
}
}
}
We first note that since W and Θ satisfy W (0) = 0 = W (1) and Θ(0) = 0 = Θ(1) in addition to satisfying to governing equations and hence we have from the Rayleigh-Ritz inequality [6] ∫ DW dz ≥ π 2 ∫ W dz , 1
1
2
0
2
(30)
0
and
∫ DΘ
2
dz ≥ π
1 2
∫Θ
2
dz ,
(31)
0
Further, for W (0) = 0 = W (1) , Banerjee et al.1 have show that
∫DW 2
(27)
2
2
(29)
1 ∗
2
0
Wdz = − Ra 2 ∫ W ∗ Θdz
(D
2
Ra 2 1 2 2 2 ∫ DΘ + a Θ dz B 0
1
2
0
2
0
=
Multiplying equation (23) by W (the complex conjugate of W) throughout and integrating the resulting equation over the vertical range of z, we get
∗
2
F ∫ D 3W + 3a 2 D 2W + 3a 4 DW + a 6 W dz 1
(24)
(28)
0
0
− a 2 )Θ = − BW ,
2
.
∗ 2 2 ∫ Θ (D − a )Θdz
2 1
Integrating the terms on both sides of equation (28) for an appropriate number of times by making use of the appropriate boundary conditions (25), we get 1
(23)
together with the relevant boundary conditions, W = 0 = Θ = D 2W = D 4W , on both the horizontal boundaries at z = 0 and z = 1. (25)
2
2
0
Proof: When the instability sets in stationary convection and the `exchange principle` is valid, the neutral or marginal state will be characterized by σ = 0 and hence the relevant governing equations (16) and (17) reduces to
(D
1
3
0
2
dz ≥ π
1 2
∫ DW
2
dz ,
(32)
0
∗
∗
Substituting for W in the right hand side of equation (26), we get
Further, multiplying equation (24) by Θ (the complex conjugate of Θ ), integrating by parts each term of the resulting equation
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
on the right hand side for an appropriate number of times and making use of namely boundary condition on Θ Θ(0) = 0 = Θ(1) , it follows that
{
}
11 2 2 2 ∫ DΘ + a Θ dz = B0
Real part of Θ ∗Wdz ,
∫
0
}
(36)
Now, if R 〉 0 and B=1+b, b 〉 0, utilizing the inequalities (32) and (36), the equation (29) gives,
1
{
11 2 2 2 ∫ DΘ + a Θ dz B0 1 2 B ≤ 2 DW dz 2 2 ∫ (π + a )π 0
RBa 2 I1 + I 2 + {a 2 (π 2 + a 2 )F + (π 2 + a 2 )} − 2 (π + a 2 )π 2
1
≤ ∫ Θ ∗Wdz ,
1
2
∫ DW dz 〈0,
0
0
1
(37)
≤ ∫ Θ ∗W dz , Where
0 1
1
{
}
I1 = ∫ D 3W + 2a 2 D 2W + 2a 4 DW + a 6 W dz ,
∗
≤ ∫ Θ W dz ,
0
2
2
2
2
and I = ∫ {a DW + a W }dz 1
0
2
1
2
2
4
2
0
are positive definite. and therefore, we must have
≤ ∫ Θ W dz ,
R 〉 π (π
0
2
1 2
1 2
1 2 1 2 ≤ ∫ Θ dz ∫ W dz , 0 0
(33)
+ a2 ) 1 {1 + a 2 F } , a2 B 2
2
(38)
and thus we necessarily have 1 R 〉 4π {1 + π F } , 4
(39)
2
B
(Utilizing Cauchy-Schwartz-inequality), So that using inequality (31), we obtained from above inequality
Since the minimum value of
(π
and this completes the proof of the theorem.
2
+ a ) 1 2 1 2 ∫ Θ dz ≤ ∫ W dz , 0 0 B 1 2
2
1 2
(34)
And hence inequality (33) and (34), gives
{
}
1 11 2 2 2 B 2 W dz , ∫ DΘ + a Θ dz ≤ 2 2 ∫ B0 (π + a ) 0
Using inequality (30), inequality (35) becomes
(35)
π 2 (π 2 + a 2 )
2
a2
is
4π 4 at a 2 = π 2 〉0 .
The theorem implies from the physical point of view that the onset of instability at marginal state in a couple-stress fluid heated from below in the presence of suspended particles, cannot manifest itself as stationary convection, if the thermal Rayleigh number R, The couple-stress parameter F and suspended particles density
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
544
Ajaib S. Banyal, J. Comp. & Math. Sci. Vol.2 (3), 537-545 (2011)
parameter B, satisfy the inequality R ≤ 1 4π 4 {1 + π 2 F } B
, when the boundaries are
dynamically free.
REFERENCES
4. CONCLUSIONS In this paper, the effect of suspended particles on a couple-stress fluid heated from below is considered and the immediate analytic conclusions of the theorem proved above, are as follows: (a).
The necessary condition for the onset of instability as stationary convection at marginal state for configuration under consideration, is that the inequality (39) is satisfied. Thus the sufficient condition for the non-existence of stationary convection at marginal state is that R ≤
character of the suspended particles, for the configuration under consideration, as derived by Sharma and Sharma9.
1 4π 4 {1 + π 2 F } B
, for the
configuration under consideration. (b). In the inequality (39), the thermal Rayleigh number R 〉 0, is directly proportional to the couple-stress parameter F 〉 0, which clearly mathematically established the stabilizing character of the couplestress, for the configuration under consideration as derived by Sharma and Sharma9. (c). The suspended particles density parameter B is greater than equal to one (B ≥ 1) and in the inequality (39), the thermal Rayleigh number R 〉 0, is inversely proportional to the suspended particles density parameter B ≥ 1, which clearly mathematically established the destabilizing
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10. Sharma, R.C. and Sharma S. On couplestress fluid heated from below in porous medium, Indian J. Phys, Vol. 75B, pp.59-61. (2001). 11. Stokes, V. K. Couple-stress in fluids, Phys. Fluids, Vol. 9, pp.1709-15. (1966). 12. Sunil, Sharma, R.C. and Chandel, R.S. Effect of suspended particles on couple-
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stress fluid heated and soluted from below in porous medium, J. of Porous Media, Vol. 7, No.1 pp. 9-18 (2004). 13. Walicki, E. and Walicka, A. Inertial effect in the squeeze film of couplestress fluids in biological bearings, Int. J. Appl. Mech. Engg., Vol. 4, pp. 363-73 (1999).
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