Effect of Rotation on Thermosolutal Convection. in a Rivlin-Ericksen Fluid Permeated with. Suspended Particles in Porous Medium

Transcription

1 Av. Theor. Appl. Mech., Vol. 3,, no. 4, Effect of Rotation on Thermosolutal Convection in a Rivlin-Ericksen Flui Permeate with Suspene Particles in Porous Meium A. K. Aggarwal Department of Mathematics Jaypee Institute of Information Technology University A-, Sector-6, Noia-Inia amrish.aggarwal@jiit.ac.in Abstract The thermosolutal convection in Rivlin-Ericksen elastico-viscous flui in porous meium is consiere to inclue the effect of suspene particles an rotation. The sufficient conitions for the valiity of principle of exchange of stabilities are obtaine. For stationary convection, a Rivlin-Ericksen elastico-viscous flui behaves like a Newtonian flui an it is foun that suspene particles an meium permeability have estabilizing effect on the system. The stable solute graient an rotation have stabilizing effect on the system. Keywors: Rivlin-Ericksen flui, suspene particles, rotation, porous meium, permeability, stability. Introuction A etaile account of the onset of Bénar convection, uner varying assumptions of hyroynamics an hyromagnetics, has been given by Chanrasekhar []. The problem of thermohaline convection in a layer of flui heate from below an subjecte to a stable salinity graient was investigate by Veronis []. Bhatia an Steiner [3] have stuie the problem of thermal instability of a viscoelastic (Maxwellian) flui in presence of rotation an have foun that rotation

2 78 A. K. Aggarwal has a estabilizing effect in contrast to the stabilizing effect on orinary viscous (Newtonian) flui. Sharma [4] has stuie the thermal instability of a layer of viscoelastic (Olroyian) flui acte on by a uniform rotation an foun that rotation has estabilizing effect as well as stabilizing effects uner certain conitions in contrast to that of a Maxwellian flui where it has estabilizing effect on the system. There are many elastico-viscous fluis that can not be characterize by Maxwell s constitutive relations or Olroy s constitutive relations. Two such classes of fluis are Rivlin-Ericksen s an Walter s (moel B ) fluis. Rivlin an Ericksen [5] have propose a theoretical moel for such one class of elastico-viscous fluis. Sharma an Aggarwal [6] have stuie the effect of compressibility an suspene particles on thermal convection in a Walters (moel B ) elastico-viscous flui in hyromagnetics. In recent years, the investigation of flow of fluis through porous meium has become an important topic ue to the recovery of crue oil from the pores of reservoir rocks. The flow through porous meia is of consierable interest for petroleum engineers an for geophysical flui ynamicists. When the flui permeates a porous material, the gross effect is represente by the Darcy s law. As a result of this macroscopic law, the visual viscous an viscoelastic terms in the equation of Rivlin- Ericksen flui motion are replace by the resistance term μ + μ q, where k t μ an μ are the viscosity an viscoelasticity of the Rivlin-Ericksen flui, k is the meium permeability an q is the Darcian (filter) velocity of the flui. In the present paper, we have stuie the effect of suspene particles alongwith uniform rotation on thermosolutal convection of a Rivlin-Ericksen elasticoviscous flui in porous meium. The problem fins its application in petroleum technology, pulp an paper technology, geophysics, astrophysics, chemical technology, etc. Formulation of the problem an perturbation equations Consier an infinite horizontal flui particle layer of Rivlin-Ericksen elasticoviscous flui of thickness boune by the planes z = an z = in porous meium, which is acte upon by a uniform rotation Ω (,,Ω). This layer is heate from below an subjecte to a stable solute graient such that a uniform temperature graient β ( = T ) an solute graient ( = C ) z β z are maintaine. Then the equations of motion an continuity governing the flow are ρ q KN + ( q ) q = p ρgλ + ( q q) + μ μ q + ( q Ω) t k t () q =, ()

3 Effect of rotation on thermosolutal convection 79 In writing equation () we have assume uniform size of flui particles, spherical shape an small relative velocities between the flui an particles. Then the net effect of the suspene particles on the flui through porous meium is equivalent KN to an extra boy force term per unit volume ( q q). Since the force exerte by the flui on the particles is equal an opposite to that exerte by the particles on the flui, there must be an extra force term, equal in magnitue but opposite in sign, in the equation of motion of the particles. The istances between particles are assume to be so large compare with their iameter that interparticle reactions nee not be accounte for. The effects of pressure, gravity an Darcian force on the suspene particles (assume large istances apart) are negligibly small an therefore ignore. If mn is the mass of particles per unit volume, then the equations of motion an continuity for the particles, uner the above assumptions, are q mn + ( q ) q = KN( q q ) t (3) N + ( N q ) =. (4) t Since the volume fraction of the particles is assume small, the effective properties of the suspension are taken to be those of the pure (clean) flui. If we assume that the particles an the flui are in thermal an solute equilibrium, then the equation of heat an solute conuction gives T [ ρ c + ρ scs ( )] + ρ c( q ) T + mnc pt + q T = q T (5) t t C [ ρ c + ρ scs ( ) ] + ρ c( q ) C + mnc pt + q C = q T (6) t t Since ensity variations are mainly ue to variations in temperature an solute concentration, the equation of state for the flui is given by ρ = ρ α T T + α C (7) [ ( ) ( )] C The initial state of the system, enote by subscript, is taken to be a quiescent layer (no setting) with a uniform particle istribution N. Across the layer an averse linear temperature graient β an an averse linear solute graient β are maintaine. The initial state q = (,,), q = (,,), T = T β z, C = C β z, ρ = ρ ( + αβ z α β ) an z N = N, a constant. (8) is an exact solution to the governing equations. The linearize perturbation equations of motion, continuity, heat conuction an solute conuction for Rivlin-Ericksen elastico-viscous flui-particle layer are q KN = δp + ( q q) q + ( ν + ν g αθ α γ ) λ + ( q Ω) (9) t ρ ρ k t

4 8 A. K. Aggarwal q =, () m + q = q, K t () N + N ( q ) =, t () θ ( E + h ) = β ( w + hs) + κ θ, t (3) γ ( E + h ) = β ( w + hs) + κ γ, t (4) ρ scs Here E = + ( ) is constant an E is a constant analogous to E ρ cv but corresponing to solute rather than heat; w an s are the vertical components of flui an suspene particle velocities, respectively, where, mn c pt h = ρ c, κ = q q μ μ, κ =, ν = an ν =. The kinematic viscosityν, viscosity μ, ρ c ρ c ρ ρ thermal iffusivityκ, solute iffusivityκ, coefficient of thermal expansion α an coefficient of solute expansion α are all assume to be constant. The change in ensityδρ, cause by the perturbationsθ, γ in temperature an solute concentration, is given by δρ = ρ ( αθ α γ ). (5) Eliminating q between equations (9) an (), we get ν mn ν q k t k ρ τ + t = δ p+ g( αθ αγ ) λ + ( q Ω), ρ Writing the scalar components of equation (6), eliminating δp between them by using equations (), (3) an (4) we obtain (6) u, v an mn ν ν w k t k ρ + τ t Ω ζ = g α + θ α + γ, (7) x y x y z

5 Effect of rotation on thermosolutal convection 8 v u where ζ = is the z -component of vorticity. x y Equations (9) an () also yiels ν mn ν Ω w ζ =. (8) k z t k ρ + τ t 3 Dispersion Relations Analyzing the isturbances in to normal moes, we assume that the perturbation quantities are of the form [ w, θ, γ ] = [ W(), z Θ(), z Γ()exp z ] ( ik x+ ik y+ nt x y ), (9) Expressing the coorinates x, y, z in the new unit of length an time t in the new unit of length κ an letting n τν mn ν ν k a = k, σ =, τ =, M =, p =, p =, P ν ρ κ κ l =, H = + h, ν E = E + h, E = E + h, F =, an D =. z The linearize imensionless perturbation equations are σ M + Fσ + + ( D a ) W + τσ P l, () 3 ga Ω + ( αθ α Γ ) + DZ = ν ν σ M + Fσ Ω + + Z = DW, () + τσ P ν l β H + τσ ( D a E pσ ) Θ= W, κ + τσ () β H + τσ ( D a E p σ ) Γ= W. κ + τσ (3) Eliminating Z between equations () an (), we obtain

6 8 A. K. Aggarwal where σ M + Fσ + + ( ) + + τσ P l D a W T DW A σ M + Fσ ga ( αθ α Γ ) =, + τσ P ν l Ω T = is the Taylor number. A ν (4) Eliminating Θ an Γ between equation (7), (9) an (3), we obtain σ M + Fσ ( D a E pσ )( D a E p σ ) + + ( D a ) W + τσ Pl M F ( D a E p σ ) R σ + σ H + τσ + a + + W (5) τσ P + ( ) l D a E pσ S + τσ + + ( D a E pσ )( D a E p σ ) T DW =, A 4 4 where gαβ R = is the thermal Rayleigh number an gα β S = is the νκ νκ analogous solute Rayleigh number. We consier the case where both bounaries are free an perfect conuctors of both heat an solute, while the ajoining meium is assume to be electrically nonconucting. The appropriate bounary conitions for this case are (Chanrasekhar 96) W D W at z = =, Θ =Γ = = an. (6) Using the bounary conitions (6), it can be shown that, all the even orer erivatives of W must vanish for z = anan hence, the proper solution of W characterizing the lowest moe is W = W sinπ z, (7) where W is a constant. Substituting the proper solution (7) in equation (5), we obtain the ispersion relation

7 Effect of rotation on thermosolutal convection 83 R + x + ie pσ = S x ie p σ x + iτσπ + ( + x + ie pσ ) x H + iτσπ iσ M + ifσ π + iτ σ π P T + iτσπ ( + x + ie pσ ) x H + iτ σ π iσ M + ifσ π + iτ σ π + + P. + + (8) A +. where Ω R S T x i P P νκπ νκ π νπ π π 4 4 gαβ gα β a σ =, =, =, =, σ =, = π. 4 4 A l 4 The Stationary Convection When the instability sets in as stationary convection, the marginal state will be characterize byσ =. Putting σ = in equation (8), we get ( + x) T A + x P R = S + + (9) xph x H Equation (9) yiels R + x TP A + x = +, (3) H P x H R + x P =, (3) T x H A R S =, (3) R + x T A + x =. (33) P P xh It is clear from equations (3)-(33) that, for stationary convection, rotation an stable solute graient postpone the onset of convection whereas the meium

8 84 A. K. Aggarwal + x permeability has estabilizing effect when > an has stabilizing effect P + x T A when <, an suspene particles hasten the onset of convection, on the P Rivlin-Ericksen flui permeate with suspene particles, heate an solute from below in porous meium in the presence of rotation. The stable solute graient an rotation have a stabilizing effect on the system. T A The ispersion relation (9) is also analyze numerically. In fig., Rayleigh number R is plotte against suspene particles parameter H for ifferent wave numbers x =, 4, 6, 8, for fixe values of T A =, S = 5, P =.5 an =. It is observe from the graph that suspene particles have estabilizing effect on the system. In fig., Rayleigh number R is plotte against rotation parameter T A for ifferent wave numbers x =,4,6,8, for fixe values of H = 5, S = 5, P =.5 an =.5. It is clear from the graph that rotation has stabilizing effect on the system. In fig. 3, Rayleigh number R is plotte against solute graient parameter S for ifferent wave numbers x =,4,6,8, for fixe values of H =, TA =, P =.5 an =.. It is observe from the graph that solute graient has stabilizing effect on the system. In fig. 4, Rayleigh number R is plotte against permeability parameter P for ifferent wave numbers x =, 4, 6,8, for fixe values of H =, TA = 5, S = 5, an =.5. It is clear from the graph that permeability has stabilizing/estabilizing effect on the system.

9 Effect of rotation on thermosolutal convection Fig.: Rayleigh Number vs. Suspene Particles Fig. : Rayleigh Number vs. Rotation Parameter 53 Rayleigh Number x= x=4 x=6 x=8 x= Rayleigh Number x= x=4 x=6 x=8 x= 5 5 Suspene Particles Rotation 7 Fig. 3: Rayleigh Number vs. Solute Graient Fig. 4: Rayleigh Number vs. Permeability Rayleigh Number x= x=4 x=6 x=8 x= Rayleigh Number x= x=4 x=6 x=8 x= 5 5 Solute Graient Permeability

10 86 A. K. Aggarwal 5 Stability of the system an oscillatory moes In this section, we etermine uner what conitions the principle of exchange of stabilities is satisfie (i.e. n is real an the marginal states are characterize by n = ) an the oscillations come into play. * Multiplying equation () byw, the complex conjugate ofw, an integrating over the range of z (i.e. z = to) an using equations (), (), (3) together with bounary conition (6), we have σ M + Fσ + + I + τ σ P l ακ ( I + E pσ I 3) ga + τσ β ν H τ σ ακ ( I E p σ I 4 5) + + β σ M + Fσ, + τ σ P l I = 6 (34) where { } { } I = DW + a W z = Θ + Θ,, I D a z I = z, 3 Θ I = { }, 4 DΓ + a Γ z I z, 5 = Γ I Z z 6 =. (35) Integrals I I 6 are all positive efinite. Putting σ = (.. ie σ = iσ ) r i in equation (34) an equating imaginary parts,

11 Effect of rotation on thermosolutal convection 87 M F + + I + τ σ i Pl ga τ H ακ α κ + I I 4 ν H + τ σ β β i σ. i = ga H + τ σ ακ α κ i EpI EpI ν H + τ σ β β i M F + I τ σ P i l (36) In the absence of rotation an stable solute graient, equation (36) reuces σ i = ga ακ H + τσ i + Ep I 3 ν β H + τ σ i M F gaτ ακ H + + I + I + τ σ P ν β H τ σ i l + i. (37) The terms in the bracket are positive efinite. Thus σ = implies that i oscillatory moes are not allowe an the principle of exchange of stabilities is satisfie in the absence of rotation an stable solute graient. It is evient from equation (36) that presence of rotation an stable solute graient brings oscillatory moes (as σ may not be zero) which were non-existent in their absence, for a Rivlini Ericksen flui layer with suspene particles, heate an solute from below in the presence of rotation. 6 Discussion of results an conclusions In this paper, we have stuie the effect of rotation an suspene particles on thermosolutal convection in Rivlin-Ericksen elastico-viscous flui heate an solute from below saturating a porous meium. The principal conclusions from the analysis of this paper are as follows: (i) For stationary convection, a Rivlin-Ericksen elastico-viscous flui behaves like Newtonian flui an it is foun that suspene particles have estabilizing effect on the system.

12 88 A. K. Aggarwal + x (ii) The meium permeability has estabilizing effect when > an has P + x T A stabilizing effect when < i.e. meium permeability hastens the P onset of convection in the absence of rotation an also hastens the onset of + x convection in the presence of rotation if T A < whereas the meium P + x permeability postpones the onset of convection if T A >. P (iii) The stable solute graient an rotation have stabilizing effect on thermal convection of the system. (iv) The principle of exchange of stabilities is foun to hol true in the absence of rotation an stable solute graient. It is evient from equation (36) that presence of rotation an stable solute graient brings oscillatory moes (as σ i may not be zero) which were non-existent in their absence. References [] S. Chanrasekhar, Hyroynamic an Hyromagnetic Stability, Dover Publication, New York, 98. [] G. Veronis, On finite amplitue instability in thermohaline convection, J. Marine Res. 3(965),. [3] P. K. Bhatia an J. M. Steiner, Thermal instability in a visco-elastic flui layer in hyromagnetics, J. Math. Anal. Appl., 4(973), 7. [4] R. C. Sharma, Effect of rotation on thermal instability of a viscoelastic flui Acta. Physica Hung. (976). [5] R. S. Rivlin an J. L. Ericksen, Stress eformation relations for isotropic materials, J. Rat. Mech. Anal. 4(955), 33. [6] R. C. Sharma, an A. K. Aggarwal,, Effect of compressibility an suspene particles on thermal convection in a Walters B elastico-viscous flui in hyromagnetics, Int. J. of Applie Mechanics an Engineering, () (6), 39. Receive: August, 9 T A