Adv. Theor. Appl. Mech., Vol. 4, 0, no. 3, 3 - Hydrodynamic and Hydromagnetic Stability of Two Superposed Walters B' Viscoelastic Fluids in Porous Medium S. K. Kango Department of Mathematics, Govt. College, Haripur (Manali Himachal Pradesh 7536, India sango7@gmail.com Abstract The stability of two superposed Walters B' viscoelastic fluids is considered in porous medium. In the potentially stable case, the system is stable or unstable if the inematic viscoelasticity is less than or greater than the medium permeability divided by medium porosity and in the potentially unstable case, the system is unstable. The effect of a variable horizontal magnetic field is also considered. In the presence of magnetic field, for stable case, the system is stable or unstable if the inematic viscoelasticity is less than or greater than the medium permeability divided by medium porosity, whereas the system is unstable for all wave numbers in the absence of the magnetic field for the potentially unstable arrangement. Mathematics Subject Classification: 76A0, 76D50, 76E5, 76S05 Keywords: Stability; viscoelasticity; Rivlin-Ericsen fluid; suspended particles; porous medium. Introduction The instability of the plane interface separating two Newtonian fluids when one is superposed over the other, under varying assumptions of hydrodynamics and hydromagnetics, has been studied by several researchers and a comprehensive
4 S. K. Kango account of these investigations has been given by Chandrasehar []. The influence of viscosity on the stability of the plane interface separating two incompressible superposed fluids of uniform densities, when the whole system is immersed in a uniform horizontal magnetic field, has been studied by Bhatia [5]. Sharma [7] has studied the thermal instability of a layer of Oldroydian viscoelastic fluid acted on by a uniform rotation and found that the rotation has a stabilizing effect under certain conditions but has a destabilizing effect also under different conditions. This is in contrast to the thermal instability of Maxwellian viscoelastic fluid in the presence of a uniform rotation, considered by Bhatia and Steiner [6], where rotation is found to have a destabilizing effect. Sharma and Sharma [9] have studied the stability of the plane interface separating two Oldroydian viscoelastic superposed fluids of uniform densities. Chandra [3] observed a contradiction between the theory for the onset of convection in fluids heated from below and his experiment. He performed the experiment in an air layer and found that the instability depended on the depth of layer. A Bénard-type cellular convection with fluid descending at the cell centre was observed when the predicted gradients were imposed for layers deeper than 0 mm. A convection which was different in character from that in deeper layers occurred at much lower gradients than predicted, if the layer depth was less than 7 mm and called this columnar instability. He added an aerosol to mar the flow pattern. Motivated by interest in fluid-particle mixtures and columnar instability, Scanlon and Segel [] studied the effect of suspended particles on the onset of Bénard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure gas was supplemented by that of the particles. Generally the magnetic field has a stabilizing effect on the instability problem but a few exceptions are there. For example, Kent [] studied the effect of a horizontal magnetic field, which varies in the vertical direction, on the stability of parallel flows and showed that the system is unstable under certain conditions, while in the absence of magnetic field the system is nown to be stable. Sharma and Singh [8] have considered the stability of stratified fluid in the presence of suspended particles and variable magnetic field. In all the above studies, the fluids have been considered to be Newtonian or viscoelastic (Maxwellian or Oldroydian. There are many elastico-viscous fluids that cannot be characterized by Maxwell s constitutive relations or Oldroyd s constitutive relations. One such fluid of elastico-viscous fluids is Walters B' fluid [4]. When a fluid flows through a porous medium, the gross effect is represented by the usual Darcy s law. The effect of suspended particles on the stability of stratified fluids in porous medium might be of industrial and chemical engineering importance. Further motivation for this study is the fact that nowledge concerning fluid-particle mixtures is not commensurate with their industrial and scientific importance. A study on the stability of two superposed Walters B' viscoelastic liquids has been made by Sharma and Kumar [0]. Keeping in mind the relevance and importance in chemical technology and geophysics, the present paper considers the stability of the plane interface separating two incompressible superposed Walters B' fluids in porous medium. The stability of
Hydrodynamic and hydromagnetic stability 5 electrically conducting superposed Walters B' fluids in porous medium in presence of uniform horizontal magnetic field has also been considered.. Formulation of the problem and perturbation equations Consider a static state in which an incompressible Walters B' viscoelastic fluid is arranged in horizontal strata in a porous medium. The pressure p and density are functions of the vertical coordinate z only. The character of the equilibrium of this initial static state is determined, as usual, by supposing that the system is slightly disturbed and then by following its further evolution. r r Let δ, δpquvw, (,,, g( 0,0, g, νν,, ε and denote respectively the perturbations in density, pressure p, fluid velocity( 0,0,0, gravitational acceleration, inematic viscosity, inematic viscoelasticity, medium porosity and medium permeability. Then the liberalized perturbation equations of Walters B' viscoelastic fluids through porous medium are r q δ p rδ g ( ν ν r = + q, ( ε t t. q r = 0, wd (. t δ = (3 Analyzing the disturbances into normal modes, we see solutions whose dependence on x, y and t is give by exp( i x + i y + nt, (4 x y where x y in general, a complex constant. Writing equations (-(3 in scalar form, are wave numbers along x and y -directions, ( / x y ( = + and n is, nu = ixδ p ( μ μ n u, nv = i yδ p ( μ μ n v, (5 (6
6 S. K. Kango nw = D δ p g δ ( μ μ n w, (7 i u + i v + Dw = 0, (8 x y nδ = ( D w, (9 d where D =. dz Using equations (8 and (9, eliminating uvδ,, andδ p between equations (5-(7, we obtain n g w( D D ( Dw w n + + ε + D{ ( μ μ n Dw} ( μ μ n w = 0. (0 3. Two superposed viscoelastic Walters B' fluids separated by a horizontal boundary Consider the case when two superposed Walters B' viscoelastic fluids of uniform densities and, uniform viscosities μ and μ and uniform viscoelasticities μ and μ are separated by a horizontal boundary at z = 0. The subscripts and distinguish the lower and the upper fluids respectively. Then in each region of constant, constant μ, constant μ and constant mn, equation (0 becomes ( D w 0. = ( The general solution of equation ( is + z z w= Ae + Be, ( where A and B are arbitrary constants. The boundary conditions to be satisfied in the present problem are (i The velocity w 0when z + (for the upper fluid and z (for the lower fluid. (ii wz ( is continuous at z = 0.
Hydrodynamic and hydromagnetic stability 7 (iii The jump conditions at the interference w between the fluids. This is obtained by integrating equation (0 across the interface at z = 0 and is n Dw Dw + n Dw n Dw + [ ] ( μ μ ( μ μ z= 0 Z = 0 + g ( w0 0, n = (3 where w 0 is the common value of w at z = 0. Applying the boundary conditions (i and (ii to the general solution (, we can write w Ae,( z + z = < 0 (4 z w = Ae,( z > 0 (5 where the same constant A has been chosen to ensure the continuity of w at z = 0. Applying the condition (3 to the solutions (4 and (5, we obtain n ( αν αν n + + [ αν + αν ] g ( α α = 0, (6, μ, μ, where α, =, v, =, v, =. +,, Here we mae the assumption that the inematic viscosities and inematic viscoelasticities of the two fluids are equal i.e. v = v = v and v v = = v. However, this simplifying assumption does not obscure any of the essential features of the problem. And equation (6 becomes n ν + n ν + g( α α = 0. (7 (a Stable Case ( < For the potentially stable arrangement ( < andν <, all the ε coefficients of equation (7 are positive. Therefore, both the three roots of equation (7 are either real and negative or there are complex conjugates with negative real parts. The system is, thus, stable in each case. However, for the potentially stable arrangement ( < andν >, the ε coefficients of n in equation (7 is negative. There is a change of sign in the
8 S. K. Kango coefficients of equation (7 and hence equation (7 allows a positive root. The system is, therefore, unstable. This is in contrast to the stability of Newtonian superposed fluids in porous medium where the system is always stable for the stable configuration. For the Walters B' viscoelastic superposed fluids in porous medium, the system can be stable or unstable if the inematic viscoelasticity is less than or greater than the medium permeability divided by medium porosity (b Unstable Case ( > For the potentially unstable case ( >, the constant term in equation (7 is negative. Equation (7 involves a change of sign and hence allows a positive root. So the system is unstable. Thus for the potentially unstable arrangement, the interface between two Walters B' elastico-viscous fluids in porous medium is unstable. 4. Effect of a horizontal magnetic field Here we consider a static state in which an incompressible, infinitely electrically conducting Walters B' elastico-viscous fluid is arranged in horizontal strata in porous medium in the presence of uniform horizontal magnetic field H r [ H ( z,0,0]. Let r h ( hx, hy, hz denotes the perturbation in magnetic field and μ e stands for magnetic permeability, then the linearized hydromagnetic perturbation equations, relevant to the problem, are r q δ p rδ μe g ν ν r r r = + q+ ( h H, t t 4π r. h = 0, h r r = ( H. q, t together with equations (-(3. Writing equations (8-(9 in scalar form, we have nu = ixδ p ( μ μ n u, μ ( e nv = i yδ p μ μ n v + ( ixhy iyhx, 4π μ ( e nw = Dδp gδ μ μ n w + ( ixhz Dhx, 4π (8 (9 (0 ( ( (3
Hydrodynamic and hydromagnetic stability 9 ixhx + i yhy + Dhz = 0, (4 nhx = ixhu, (5 nhy = ixhv, (6 nhz = ixhw, (7 together with equations (8 and (9. Eliminating uvh,, x, hy, hz, δ andδ p between equations (-(3 and using equations (8, (9 and (4-(7, we obtain + + n D ( Dw w D {( μ μ n Dw } ( μ μ n w μehx g + ( D + ( D w= 0. 4π n n (8 Here also we consider the case of two uniform Walters B' viscoelastic fluids of uniform densities and, uniform viscosities μ and μ and uniform viscoelasticities μ and μ, separated by a horizontal boundary. In each region, Equation (8 reduces to equation ( and the proper solutions satisfying the relevant boundary conditions are given by (4 and (5. Integrating equation (8 across the interface at z = 0, we obtain the jump condition n g μex [ Dw Dw] + ( w0 + [ Dw Dw] + Z= 0 Z= 0 n 4π n + ( μ μ n Dw ( μ μ n Dw = 0. Z = 0 (9 Applying the solutions (4 and (5 to the equation (9 and simplifying, we obtain n v + n v + xva g( α α = 0, (30 / μeh where VA = is the Alfvén velocity. As in section 3, the inematic 4 π ( + viscosities and inematic viscoelasticities of the two fluids are assumed equal i.e. v = v = v and v = v = v. (a Stable Case ( < It is evident from equation (30 that for the potentially stable <, the system is stable or unstable according as arrangement (
0 S. K. Kango v < or >. (3 ε Thus for Walters B' viscoelastic superposed fluids for stable case in porous medium in hydromagnetics, the system is stable or unstable if the inematic viscoelasticity is less than or greater than the medium permeability divided by medium porosity. This is in contrast to the stability of Newtonian superposed fluids in porous medium in hydromagnetics where the system is always stable for stable configuration. (b Unstable Case ( > For potentially unstable arrangement ( > and v >, the magnetic field ε has got a stabilizing effect and the system is stable for all wave-numbers which satisfy the inequality ( α α g where * = sec θ ( V x A > g α α, or > *, (3 V A and θ is the angle between r and magnetic field H r. However, for the potentially unstable arrangement ( > and v >, the system is ε unstable. For the potentially unstable configuration and for two viscous superposed Newtonian fluids in porous medium in hydrodynamics, the system is unstable for all wave numbers whereas in hydromagnetics, the magnetic field has stabilizing effect and completely stabilizes the wave-number band > *, where ( α α g * = sec θ V A This is in contrast to the case of Walters B viscoelastic superposed fluids in porous medium in hydromagnetics for the potentially unstable configuration. Here the system is stable for the wave-number band > *, if the inematic viscoelasticity is less than the medium permeability divided by medium porosity. However, the system is always unstable if the inematic viscoelasticity is greater than the medium permeability divided by medium porosity.
Hydrodynamic and hydromagnetic stability References [] A. Kent: Instability of laminar flow of a Magneto-Fluid, Phys. Fluids. 9, p.p.86, 966. [] J.W. Scanlon and L.A. Segel : Some effects of suspended particles on the onset of Bénard convection, Phys. Fluids 6, p.p.573, 973. [3] K. Chandra: Instability of fluid heated from below, Proc. Roy. Soc. (Lon.. A64, p.p.3, 938. [4] K. Walters: The motion of an elastico-viscous liquid contained between coaxial Cylinders, Quart. J. Mech. Apple. Math., 3, p.p.444, 960. [5] P.K. Bhatia: Rayleigh-Taylor instability of two viscous superposed conducting fluids, Nuovo Cimento, 9B, p.p.6, 974. [6] P.K. Bhatia and J.M. Steiner: Convective instability in a rotating viscoelastic fluid layer, Z.Angew, Math.Mech. 5, p.p.3, 97. [7] R.C. Sharma: Effect of Rotation on Thermal Instability of a Viscoelastic Fluid, Acta Physica Hungarica. 40, p.p., 976. [8] R.C. Sharma and B. Singh: stability of stratified fluid in the presence of suspended particles and variable magnetic field, J.Math.Phys.Sci. 4, p.p.45, 990. [9] R.C. Sharma and K.C. Sharma: Stability of the plane interface separating two Oldroydian viscoelastic superposed fluids of uniform densities, Acta Physcia Hungarica. 45, p.p.3, 978. [0] R.C. Sharma and P.Kumar: On the Stability of Two Superposed Walters B' Viscoelastic Liquids, Czech.J.Phys. 47, p.p.97, 997.
S. K. Kango [] S. Chandrasehar: Hydrodynamic and Hydromagnetic Stability, Dover Publications, New Yor 98. Received: February, 0