EFFECT OF MAGNETIC FIELD ON THERMAL INSTABILITY OF A ROTATING RIVLIN-ERICKSEN VISCOELASTIC FLUID

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1 EFFECT OF MAGNETIC FIELD ON THERMAL INSTABILITY OF A ROTATING RIVLIN-ERICKSEN VISCOELASTIC FLUID PARDEEP KUMAR, HARI MOHAN, AND ROSHAN LAL Received 22 February 25; Revised 6 September 25; Accepted 23 October 25 The thermal instability of a rotating Rivlin-Ericksen viscoelastic fluid in the presence of uniform vertical magnetic field is considered. For the case of stationary convection, Rivlin-Ericksen viscoelastic fluid behaves like a Newtonian fluid. It is found that rotation has a stabilizing effect, whereas the magnetic field has both stabilizing and destabilizing effects. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. The rotation and magnetic field are found to introduce oscillatory modes in the system which were nonexistent in their absence. Copyright 26. All rights reserved. 1. Introduction The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in geophysics, interior of the Earth, oceanography, and the atmospheric physics, and so forth, and has been investigated by several authors e.g., Bénard [1],Rayleigh [8],Jeffreys [6]) under different conditions. A detailed account of the theoretical and experimental study of thermal instability Bénard convection) in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar [4]. The use of Boussinesq approximation has been made throughout, which states that the density may be treated as a constant in all the terms in the equations of motion except the external force term. Bhatia and Steiner [2]have considered the effect of a uniform rotation on the thermal instability of a viscoelastic Maxwell) fluid and have 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 Steiner [3]. They have found that the magnetic field stabilizes a viscoelastic Maxwell) fluid just as the Newtonian fluid. Sharma [1] has studied the thermal instability of a layer of viscoelastic Oldroydian) fluid acted on by a uniform rotation and found that rotation has destabilizing as well as stabilizing effects under certainconditionsin contrastto that of a Maxwell fluid where it has a destabilizing effect. In another study, Sharma [9] has studied the stability International Mathematics and Mathematical Sciences Volume 26, Article ID 2842, Pages 1 1 DOI /IJMMS/26/2842

2 2 Thermal instability of viscoelastic fluid of a layer of an electrically conducting Oldroyd fluid [7] in the presence of a magnetic field and has found that the magnetic field has a stabilizing influence. There are many elastico-viscous fluids that cannot be characterized by Maxwell s constitutive relations or Oldroyd s [7] constitutive relations. One such class of elasticoviscous fluids is Rivlin-Ericksen fluid. Srivastava and Singh [13] have studied the unsteady flow of a dusty elastico-viscous Rivlin-Ericksen fluid through channels of different crosssections in the presence of a time-dependent pressure gradient. In another study, Garg et al. [5] have studied the rectilinear oscillations of a sphere along its diameter in a conducting dusty Rivlin-Ericksen fluid in the presence of a uniform magnetic field. Sharma and Kumar [11] 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. A layer of such fluid heated from below under the action of magnetic field and rotation may find applications in geophysics, interior of the Earth, oceanography, and the atmospheric physics. Keeping in mind the importance of non-newtonian fluids, convection in fluid layer heated from below, magnetic field, and rotation, the present paper attempts to study the effect of uniform vertical magnetic field on Rivlin-Ericksen viscoelastic fluid heated from below in the presence of a uniform rotation. 2. Formulation of the problem and perturbation equations Consider an infinite, horizontal, incompressible electrically conducting Rivlin-Ericksen viscoelastic fluid layer of thickness d, heated from below so that the temperatures and densities at the bottom surface z = aret and ρ and at the upper surface z = d are T d and ρ d, respectively, and that a uniform temperature gradient β= dt/dz ) ismaintained. The gravity field g,, g), a uniform vertical magnetic field H,,H), and a uniform vertical rotation Ω,,Ω)actonthesystem. The equations of motion, continuity, heat conduction, and Maxwell s equations governing the flow of Rivlin-Ericksen viscoelastic fluid in the presence of magnetic field and rotation are v t + v ) p v = 1 Ω r 2) + g ρ 2 + μ e H ) H +2 v Ω ), 4πρ 1+ δρ ρ ) + υ + υ ) 2 v t 2.1) v =, 2.2) T t + v ) T = χ 2 T, 2.3) H =, 2.4) H t = H ) v + 2 H, 2.5) where vu,v,w), p, ρ, T, υ, andυ denote the velocity, pressure, density, temperature, kinematic viscosity, and kinematic viscoelasticity, respectively, and r x, y,z).

3 Pardeep Kumar et al. 3 The equation of state for the fluid is ρ = ρ [ 1 α T T )], 2.6) where ρ, T are, respectively, the density and temperature of the fluid at the reference level z = and α is the coefficient of thermal expansion. In writing 2.1), we made use of the Boussinesq approximation, which states that the density variations are ignored in all terms in the equations of motion except the external force term. The magnetic permeability μ e,thermaldiffusivity χ, and electrical resistivity are all 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 v =,,), ρ = ρz), p = pz), T = Tz). 2.7) Let vu,v,w), δp, δρ, θ,and hh x,h y,h z ) denote, respectively, the perturbations in velocity v initially zero), pressure p, density ρ, temperature T, and the magnetic field H,,H). Thechangeindensityδρ, caused by the perturbation θ in temperature, is given by ρ + δρ = ρ [ 1 α T + θ T )] = ρ αρ θ, i.e., δρ = αρ θ. 2.8) Then the linearized perturbation equations are v t = 1 δp) gαθ+ υ + υ ) 2 v + μ e ) h H +2 v Ω ), ρ t 4πρ v =, θ t = βw + χ 2 θ, h =, h t = H ) v + 2 h. Within the framework of Boussinesq approximation, 2.9) become t 2 w = υ + υ ) 4 w + μ eh 2 t 4πρ ζ t = υ + υ t ) hz + gα 2 ) θ z x θ y 2 2Ω ζ z, ) 2 ζ +2Ω w [ t χ 2 ] θ = βw, [ t 2 ] h z = H w z, [ t 2 ] ξ = H ζ z, z μ eh ξ 4πρ z, 2.9) 2.1)

4 4 Thermal instability of viscoelastic fluid where 2 = 2 / x / y / z 2 and ζ = v/ x u/ y; ξ = h y / x h x / y stand for the z-components of vorticity and current density, respectively. 3. Dispersion relation We now analyze the disturbances into normal modes, assuming that the perturbation quantities are of the form [ w,θ,hz,ζ,ξ ] = [ Wz),Θz),Kz),Zz),Xz) ] exp ik x x + ik y y + nt ), 3.1) where k x, k y are the wave numbers along x-andy-directions, respectively, k = k 2 x + k 2 y) 1/2 is the resultant wave number, and n is the growth rate which is, in general, a complex constant. Using expression 3.1), 2.1) in nondimensional form transform to [ σ D 2 a 2) gαd 2 W + )a 2 Θ + 2Ωd3 υ υ DZ μ ehd D 2 a 2) ] DK 4πρ υ = [1 + Fσ] 3.2) D 2 a 2) 2 W, [ {1+Fσ} D 2 a 2) σ ] Z = 2Ωd υ [ D 2 a 2 p 1 σ ] Θ = βd 2 [ D 2 a 2 p 2 σ ] K = Hd [ D 2 a 2 p 2 σ ] X = Hd ) μe Hd DW 4πρ υ χ ) DX, 3.3) ) W, 3.4) ) DW, 3.5) ) DZ, 3.6) where we have introduced new coordinates x, y,z ) = x/d, y/d,z/d) innewunitsof length d and D = d/dz. For convenience, the dashes are dropped hereafter. Also we have put a = kd, σ = nd 2 /υ, F = υ /d 2 ; p 1 = υ/χ is the Prandtl number and p 2 = υ/ is the magnetic Prandtl number. We now consider the case where both the boundaries are free as well as perfect conductors of heat, while the adjoining medium is also perfectly conducting. The case of two free boundaries is slightly artificial, except in stellar atmospheres see Spiegel [12]) and in certain geophysical situations where it is most appropriate. However, the case of two free boundaries allows us to obtain analytical solution without affecting the essential features of the problem. The appropriate boundary conditions, with respect to which 3.2) 3.6) must be solved, are W = D 2 W =, DZ =, Θ = atz =, z = 1, DX =, K =, 3.7) on a perfectly conducting boundary.

5 Pardeep Kumar et al. 5 Using the above boundary conditions, it can be shown that all the even-order derivatives of W must vanish for z = andz = 1, and hence the proper solution of W characterizing the lowest mode is W = W sinπz, 3.8) where W is a constant. Eliminating Θ, K, Z,andX between 3.2) 3.6)and substituting 3.8)intheresultant equation, we obtain the dispersion relation R 1 = ) [{ 1+x 1+iF1 σ 1 π 2) }{ } ][ ] 1 + x)+iσ 1 1+x + iσ1 p 2 + Q1 1+x + iσ1 p ) 1 x 1+x + iσ1 p 2 ) ) T 1 1+x + iσ1 p 2 1+x + iσ1 p 1 + { { x[ 1+iF1 σ 1 π 2) }{ } ]}, 1 + x)+iσ 1 1+x + iσ1 p 2 + Q1 3.9) where R=gαβd 4 /υχ, Q =μ e H 2 d 2 /4πρ υ, T A =4Ω 2 d 4 /υ 2 stand for the Rayleigh-number, the Chandrasekhar number, the Taylor number, respectively, and we have also put x = a2 π 2, R 1 = R π 4, iσ 1 = σ π 2, F 1 = π 2 F, T 1 = T A π 4, Q 1 = Q π 2, i = ) 4. The stationary convection When the instability sets in as stationary convection, the marginal state will be characterized by σ =. Putting σ =, the dispersion relation 3.9)reducesto R 1 = ) 1+x [1 + x) 2 ] T + Q x) 2 x x [ ], 1 + x) 2 4.1) + Q 1 a result given by Chandrasekhar [4, equation 59), page 22]. We thus find that for the stationary convection, the viscoelasticity parameter F vanishes with σ and Rivlin-Ericksen viscoelastic fluid behaves like an ordinary Newtonian fluid. To study the effects of rotation and magnetic field, we examine the natures of dr 1 /dt 1 and dr 1 /dq 1 analytically. Equation 4.1)yields dr x) 2 = dt 1 x [ ], 1 + x) 2 + Q 1 4.2) dr x) T x) 2 = dq 1 x x [ 1 + x) 2 ] 2. + Q 1 4.3) It is evident from 4.2) that for a stationary convection, dr 1 /dt 1 is always positive, thus, the rotation has a stabilizing effect on the system. It is also clear from 4.3)thatfor

6 6 Thermal instability of viscoelastic fluid 8 R T 1 x = 1 x =.2 Figure 4.1. The variation of R 1 with T 1 for fixed values of Q 1 = 1 and x =.2, R T 1 x = 3 x = 4 Figure 4.2. The variation of R 1 with Q 1 for fixed values of T 1 = 1 and x = 3,4. a stationary convection, dr 1 /dq 1 may be positive as well as negative, thus, the magnetic field has both stabilizing and destabilizing effects on the system. The dispersion relation 4.1) is also analyzed numerically. In Figure 4.1, R 1 is plotted against T 1,forfixedvalueofQ 1 = 1 and wave numbers x =.2,1. The Rayleigh number R 1 increases with increase in rotation parameter T 1 showing its stabilizing effect on the system. Figure 4.2 shows the variation of R 1 with respect to Q 1,forfixedvalueofT 1 = 1 and wave numbers x = 3,4. It clearly depicts both the stabilizing and destabilizing effects of the magnetic field on the system. 5. Stability of the system and oscillatory modes Multiplying 3.2)byW, the complex conjugate of W, integrating the resulting equation over the range of z and using 3.3) 3.6), together with the boundary conditions 3.7),

7 Pardeep Kumar et al. 7 we obtain σi 1 + gαχa2 ) I2 + p 1 σ I 3 d 2 1+Fσ ) I 4 d 2 σ I 5 υβ μ ed 2 ) μ e ) I6 + p 2 σi 7 I8 + p 2 σ I 9 = 1 + Fσ)I1, 4πρ υ 4πρ υ 5.1) where I 1 = I 3 = I 5 = I 7 = I 9 = DW 2 + a 2 W 2) dz, I 2 = DΘ 2 + a 2 Θ 2) dz, Θ 2 ) dz, I 4 = DZ 2 + a 2 Z 2) dz, Z 2 ) dz, I 6 = DX 2 + a 2 X 2) dz, X 2 ) dz, I 8 = D 2 K 2 +2a 2 DK 2 + a 4 K 2) dz, DK 2 + a 2 K 2) dz, I 1 = D 2 W 2 +2a 2 DW 2 + a 4 W 2) dz, 5.2) and σ is the complex conjugate of σ. The integrals I 1,...,I 1 are all positive definite. Putting σ = σ r + iσ i,whereσ r, σ i are real and equating the real and imaginary parts of 5.1), we obtain [ σ r I 1 + gαχa2 υβ p 1I 3 d 2 FI 4 d 2 I 5 μ ed 2 4πρ υ p 2I 7 = gαχa2 υβ I 2 + d 2 I 4 + μ ed 2 4πρ υ I 6 + [ σ i I 1 + gαχa2 υβ 6. Discussion μ e 4πρ υ I 8 + I 1, p 1I 3 d 2 FI 4 d 2 I 5 + μ ed 2 4πρ υ p 2I 7 μ ] e 4πρ υ p 2I 9 FI 1 5.3) μ ] e 4πρ υ p 2I 9 + FI 1 =. 5.4) From 5.4), it is clear that σ i is zero when the quantity multiplying it is not zero and arbitrary when this quantity is zero. If σ i, then 5.4)gives gαχa 2 υβ p 1I 3 d 2 FI 4 d 2 I 5 μ e 4πρ υ p 2I 9 = I 1 μ ed 2 4πρ υ p 2I 7 FI ) Substituting in 5.3), we have I 1 + μ e 4πρ υ I 8 + μ ed 2 4πρ υ I 6 + d 2 I 4 +2σ r [ I 1 + μ ed 2 4πρ υ p 2I 7 + FI 1 ] = gαχa2 υβ I )

8 8 Thermal instability of viscoelastic fluid Equation 6.2) on using Rayleigh-Ritz inequality gives π 2 + a 2) 3 a 2 π W 2 dz a 2) a 2 { μe 4πρ υ I 8 + μ ed 2 [ 4πρ υ I 6 + d 2 I 4 +2σ r I 1 + μ ed 2 ]} 4πρ ν p 2I 7 + FI 1 gαχ νβ W 2 dz. 6.3) Therefore, it follows from 6.3) that [ 27π 4 gαχ ] π W 2 dz a 2) 4 νβ a 2 { μe 4πρ υ I 8 + μ ed 2 [ 4πρ υ I 6 + d 2 I 4 +2σ r I 1 + μ ed 2 ]} 4πρ ν p 2I 7 + FI 1, 6.4) since minimum value of π 2 + a 2 ) 3 /a 2 with respect to a 2 is 27π 4 /4. Now, let σ r, we necessarily have from 6.4)that Hence, if gαχ νβ > 27π ) gαχ νβ 27π4 4, 6.6) then σ r <. Therefore, the system is stable. Therefore, under condition 6.6), the system is stable and under condition 6.5) the system becomes unstable. In the absence of rotation and magnetic field, 5.4)reducesto [ ] σ i I 1 + gαχa2 υβ p 1I 3 + FI 1 =, 6.7) and the terms in brackets are positive definite. Thus, σ i =, which means that oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied for Rivlin- Ericksen viscoelastic fluid heated from below. Acknowledgment The authors are thankful to the referee for useful technical comments and valuable suggestions, which led to a significant improvement of the paper.

9 Pardeep Kumar et al. 9 Nomenclature d T g g H,,H) Ω,,Ω) v p rx, y,z) δp hhx,h y,h z ) k x, k y k n p 1 p 2 R Q a F Greek letters μ μ ρ β ν ν μ e α χ δρ θ σ ξ Depthoflayer Temperature Acceleration due to gravity Gravity field Uniform vertical magnetic field Uniform vertical rotation field Filter velocity Fluid pressure Space coordinates Perturbation in pressure Perturbation in magnetic filed Wave numbers in x-andy-directions Resultant wave number Growth rate Prandtl number Magnetic Prandtl number Rayleigh number Chandrasekhar number Dimensionless wave number Dimensionless kinematic viscoelasticity Fluid viscosity Fluid viscoelasticity Density Uniform temperature gradient Kinematic viscosity Kinematic viscoelasticity Magnetic permeability Coefficient of thermal expansion Thermal diffusivity Electrical resistivity Perturbation in density Perturbation in temperature z-component of vorticity z-component of current density References [1] H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Revue Genérale des Sciences Pures etappliquées 11 19), , [2] P. K. Bhatia and J. M. Steiner, Convective instability in a rotating viscoelastic fluid layer, Zeitschrift für Angewandte Mathematik und Mechanik ),

10 1 Thermal instability of viscoelastic fluid [3], Thermal instability in a viscoelastic fluid layer in hydromagnetics, Mathematical Analysis and Applications ), no. 2, [4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, [5] A. Garg, R. K. Srivastava, and K. K. Singh, Drag on sphere oscillating in conducting dusty Rivlin- Erickson elastico-viscous liquid, Proceedings of the National Academy of Sciences. India A ), no. 3, [6] H. Jeffreys, The stability of a fluid layer heated from below, Philosophical Magazine ), [7] J.G.Oldroyd,Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proceedings of the Royal Society of London A ), [8] L. Rayleigh, On convective currents in a horizontal layer of fluid when the higher temparture is on the under side, Philosophical Magazine ), [9] R. C. Sharma, Thermal instability in a viscoelastic fluid in hydromagnetics, Acta Physica Hungarica ), [1], Effect of rotation on thermal instability of a viscoelastic fluid, Acta Physica Hungarica ), [11] R. C. Sharma and P. Kumar, Effect of rotation on thermal instability in Rivlin-Ericksen elasticoviscous fluid, Zeitschrift für Naturforschung 51a 1996), [12] E. A. Spiegel, Convective instability in a compressible atmosphere, TheAstrophysicalJournal ), 168. [13] R. K. Srivastava and K. K. Singh, Drag on a sphere oscillating in a conducting dusty viscous fluid in presence of uniform magnetic filed, Bulletin of the Calcutta Mathematical Society ), Pardeep Kumar: Department of Mathematics, International Centre for Distance Education and Open Learning ICDEOL), Himachal Pradesh University, Shimla-1715, India address: drpardeep@sancharnet.in Hari Mohan: Department of Mathematics, International Centre for Distance Education and Open Learning ICDEOL), Himachal Pradesh University, Shimla-1715, India address: hm math hpu@redffmail.com Roshan Lal: Department of Mathematics, International Centre for Distance Education and Open Learning ICDEOL), Himachal Pradesh University, Shimla-1715, India

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