Hydromagnetic Stability of Stratified Maxwell Fluid in the Presence of Suspended Particles through a Porous Medium

Available online www.ejaet.com European Journal of Advances in Engineering and Technology, 2021, 8(9):127-132 Research Article ISSN: 2394 - 658X 127 Hydromagnetic Stability of Stratified Maxwell Fluid in the Presence of Suspended Particles through a Porous Medium Ravi Prakash Mathur Department of Mathematics S.G.S.G. Government College, Nasirabad, Ajmer (India) [email protected] _____________________________________________________________________________________________ ABSTRACT In the present paper, the stability of a stratified Maxwell viscoelastic fluid in the presence of suspended particles and a variable magnetic field through a porous medium has been investigated. The analysis considers the case where density, viscosity, magnetic field, and suspended particle concentration vary exponentially. It is observed that the system becomes unstable for disturbances of all wave numbers under conditions of potentially unstable stratification. The influence of a variable horizontal magnetic field on the stability characteristics has been examined in detail. The results reveal that the growth rate of instability either decreases or increases with an increase in particle density and kinematic viscosity, whereas it increases or decreases with a rise in the permeability of the porous medium, depending on the specific parameter range considered. Keywords: Maxwell fluid, porous medium, kinematic viscosity, hydro magnetic stability, permeability _____________________________________________________________________________________________ INTRODUCTION The stability of an incompressible heavy fluid of variable density was first investigated by Lord Rayleigh and is known as the Rayleigh–Taylor instability. Several researchers have examined the Rayleigh–Taylor instability for Newtonian fluids under various hydrodynamic and hydromagnetic conditions, and a comprehensive account of these investigations has been presented in the classic monograph by Chandrasekhar [1]. In most of these studies, the medium has been assumed to be nonporous. However, fluid flow through porous media has attracted considerable attention in recent years, particularly among petroleum engineers and geophysical fluid dynamicists [2]. In many geophysical situations, the fluid is not pure but may contain suspended (or dust) particles. Hence, the study of Rayleigh–Taylor instability in fluids within porous media holds great significance in geophysics, soil science, groundwater hydrology, and astrophysics. It is generally accepted that comets consist of dusty “snowballs” made up of mixtures of frozen gases, and the physical properties of comets, meteorites, and interplanetary dust emphasize the role of porosity in astrophysical contexts (McDonnell [3]). In stellar interiors and atmospheres, variable and nonuniform magnetic fields are present and play an important role in influencing the nature of instability. With the increasing importance of non-Newtonian fluids in industrial processes, chemical technology, and geophysical fluid dynamics, significant attention has been given to the investigation of such fluids. Sunil and Chand [4] analyzed the Rayleigh–Taylor instability of plasma in the presence of a variable magnetic field and suspended particles in a porous medium. Sharma and Kumar [5] examined the Rayleigh–Taylor instability of Oldroydian viscoelastic fluids in a porous medium subjected to a variable magnetic field. The stability of shear flow of a heterogeneous or stratified fluid has been considered by many researchers. The stability of stratified Rivlin–Ericksen viscoelastic fluid in the presence of suspended particles and variable magnetic field in porous medium has been studied by Sunil el al [6], also the stability of stratified Walters (Model B′) viscoelastic fluid in stratified porous medium is studied by Sunil el al [7]. Mathur and Kumar [8] have examined the stability of Maxwell viscoelastic fluid of a variable density in an inhomogeneous magnetic field through porous medium. Sengupta and Basak [9] have studied the instability of the plane interface separating two superposed viscoelastic (Maxwell) conducting fluids in a uniform vertical magnetic field. So keeping in mind the importance of nonNewtonian fluids in modern technology and their various applications mentioned above the present paper is devoted to the consideration of the stability of stratified Maxwellian viscoelastic fluid in the presence of suspended particles and variable magnetic field in porous medium. Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(9):127-132 128 FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS Let Tij, τij, eij, δij, P, qi, xi, μ, and λ denote, respectively, the total stress tensor, shear stress tensor, rate of strain tensor, Kronecker delta, scalar pressure, velocity, position vector, viscosity and stress relaxation time. Then the Maxwellian viscoelastic fluid is described by the constitutive relations ijijij PT  +−= , ijij e dt d  21 =      + (1) , 2 1           +   = x j j i ij x q x q e where +   =.q tdt d is the ‘convective derivative’. Relations of the type (1) were proposed and studied by Maxwell. When a fluid flows through a homogeneous and isotropic porous medium, the gross effect is represented by Dracy’s law. As a result, the usual viscous term is replaced by the resistance term v k          1  where k1 and v  denote, respectively, the medium permeability and the filter velocity. Here we consider a static state in which an incompressible, electrically conducting Maxwell fluid layer of variable density is arranged in horizontal strata in the presence of suspended particles and a variable horizontal magnetic field. The pressure P and density ρ are functions of vertical z-coordinate only. The character of the equilibrium of this initial state is determined by supposing that the system is slightly disturbed and then by following its further evolution. The fluid is under the action of the gravity g  ( 0 , 0,- g ) and the magnetic field H  = (H0(z),0,0). This fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium of the porosity ε and medium permeability k1. Let P, ρ, μ and v  (u, v, w) denote, respectively, the pressure, the density, the viscosity and the velocity of pure fluid, ( ) txu ,  and N( x ,t) denote the filter velocity and the number density of the suspended particles. K = 6π μη, where η is the particle radius, is the Stokes drag coefficient. ( ) srlu ,,=  ) and ( ) zyxx ,,= . Let g, ε and k1 stand for acceleration due to gravity, medium porosity, medium permeability. Let μe denote magnetic permeability. Then the equations of motion and continuity for Maxwellian viscoelastic fluid with suspended particle and horizontal variable magnetic field in porous medium are ( ) ( )  ( )      −−+++−         +=      +           +v k vu KN HH e gP t vv t v t     1 1.1       (2) 0. =v  (3) 0. = H  (4) ( ) ( ) HvvH t H    −=  ..  (5) Since the density of a fluid particle remains unchanged as we follow it with its motion, we have ( ) 0. =+      v t  (6) In the equation of motion (2), assuming uniform particle size, spherical shape and small relative velocities between the fluid and particle, the presence of particle adds an extra force term, in the equation of motion (2) 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 equations of motions and continuity for the particles are Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(9):127-132 129 ( ) ( ) ,. uvKNuu t u mN   −=      +   (7) ( ) 0=+  uN t N (8) Let δρ, δP, v  (u,v,w) u  (l.r.s) and h  (hx, hy, hz) denote, respectively, the perturbation in the fluid density ρ, the fluid pressure P, the fluid velocity (0,0,0), the particle velocity (0,0,0) and the magnetic field ( )( ) 0,0, 0zHH  ; then the linearized hydromagnetic perturbation equations of Oldroydian viscoelastic fluid in porous medium with suspended particles are ( ) ( )  ( ) v k vu KN HHHh e gP tt v t      1 4 11       −    −+=   ++−         +=           + (9) 0. =v  (10) 0. = h  (11) ( ) ( ) HvvH t h    −=  ..  (12) ( ) ,  Dw t−=   (13) vu tK m =      +  1 (14) 0. =+  u t M (15) where: , 0 N N M  = 0 N and N stand for initial uniform number density and perturbation in number density, and dz d D= DISPERSION RELATION Analysing the perturbation into normal modes, we assume that the perturbation quantities have an x, y and t dependence of the form )exp( ntyikxik yx ++ (16) where kx, ky are the wave numbers along x and y directions, respectively, 22 yx kkk += is the resultant wave number of disturbance and n stands for the growth rate which is, in general, a complex constant. For perturbations of the form (16) equations (9)–(14), after eliminating u  , give ( ) ( ) ( ) u k DHhPiknnun n mN z e x 1 0 4 11 1 1        −      +−+=+       + + (17) ( ) ( ) ( ) v k hikhikHPiknnvn n mN xyyx e y 1 0 4 11 1 1        −      −+−+=+       + + (18) ( ) ( ) ( )      −−−+−−+=+       + +w k DHhDhHHhikgPDnnwn n mN xxzx e 1 000 4 11 1 1        (19) 0=++ Dwvikuik yx (20) 0=++ zyyxx Dhhikhik (21) Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(9):127-132 130      = = −= wHiknh vHiknh wDHuHiknh xz xy xx 0 0 00 ,    (22)  wDn−= (23) Where 𝜏 = 𝑚 𝐾 Eliminating P  ,  , u , v , x h , y h , z h between (17) – (19) and using (20)-(23) we obtain ( ) ( )    ( ) ( ) ++−+−+ DwHD n k wkkH n wD n gk wkDwDn xe x e2 0 2 2 222 0 22 2 2 44 1      ( ) ( )   ( )   0 1 12 1 2=+    − +wkDwD nk mNwkmNDwD n    (24) CASE OF EXPONENTIALLY VARYING DENSITY, VISCOSITY, MAGNETIC FIELD AND SUSPENDED PARTICLES Here we consider the stratification in magnetic field, density, viscosity and suspended particles in the fluid of the depth d as zzz eHHee   2 0 2 00 ,, === and z eNN  0 = (25) where  ,,,, 0000 NH are constants. Equations (25) imply that the coefficient of kinametic viscosity v and the Alfven velocity VA are constant everywhere. Substituting the values of ρ, μ, 2 0 H , N in (24) and neglecting the effect of heterogeneity on inertia and solving the equation we arrive at ( ) ( ) ( ) 0 1 / 1 1 100 2 22 1 0 2 2 22 =       + ++ + + +− w n mN n Vk k v nn n kg wkD Ax     (26) Where 0 2 0 2 0 0 04 ,     H Vv e A== and K m =  are constants. Assume that the system is restricted by two planes z = 0 and z = d and that both the boundaries are free. The boundary conditions for the case of two free surfaces are 2 Dw = , w = 0 at z =0 and z = d (27) The proper solution of equation (26) satisfying (27) is d zm ww  sin 0 = (28) where 0 w is a constant and m is an integer. Substituting (28) in (26) and simplifying we obtain           −+           ++++      +++ L kg Vk mN k v n mN nn Ax 2 22 0 0 1 0 2 0 0 34 1         ( ) 0 2 22 2 22 1 0=      −            −+++ L kg Vk L kg Vk k v nAxAx    (29) Where 2 2 22 k d m L+=  For the stable stratifications (β < 0), equation (29) does not have any positive root of n implying thereby that the system is stable for disturbances of all wave numbers. Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(9):127-132 131 For the unstable stratifications (β > 0), the system is stable or unstable depending on whether 22 AxVk is greater than or less than L kg 2  , i.e, L kg Vk Ax 2 22   or L kg Vk Ax 2 22   The system is clearly unstable for β > 0 in the absence of a magnetic field and is also unstable if L kg Vk Ax 2 22   However, the system can be completely stabilized by a large magnetic field as can be seen from equation (29) if Lk kg V x A2 2 2   The magnetic field therefore succeeds in stabilizing wave numbers in the range 2 2 2 2sec       − d m V g k A    (30) which are unstable in the absence of magnetic field. Here θ is the angle between kx and k (i.e., kx = kcos θ). The viscoelasticity, the medium permeability and the suspended particles do not have any qualitative effect on the nature of stability nor instability. Thus, if 0  and L kg Vk Ax 2 22   equation (29) has one positive root and one negative root. Let n0 denote the positive root of (29), then    +           −+++++      ++++ L kg Vk mN k v n mN nn Ax 2 22 0 0 1 0 2 0 0 0 3 0 4 01         ( ) 0 2 22 2 22 1 0 0=            −+            −++ L kg Vk L kg Vk k v nAxAx    (31) To study the behaviour of the growth rate of unstable modes with respect to kinematic viscosity, medium permeability, stress relaxation time and strain retardation time and particle number density, we examine the natures of , ,,, 0 1 0 0 0  d dn dk dn dv dn and 0 0 dN dn analytically. Equation (31) yields. ( )               −+++           −++++      +++ = L kg Vk k v L kg Vk mN k v n mN nn k vn dk dn AxAx 2 22 1 0 2 22 0 0 1 0 0 0 0 2 0 3 0 2 1 00 1 0 1234            (32) ( )               −+++           −++++      +++ − = L kg Vk k v L kg Vk mN k v n mN nn mn dN dn AxAx 2 22 1 0 2 22 0 0 1 0 0 0 0 2 0 3 0 0 2 0 0 0 1234            (33) ( )               −+++           −++++      +++ − = L kg Vk k v L kg Vk mN k v n mN nn k n dv dn AxAx 2 22 1 0 2 22 0 0 1 0 0 0 0 2 0 3 0 1 0 0 0 1234            (34) ( ) ( )               −+++           −++++      +++              −++        ++− = L kg Vk k v L kg Vk mN k v n mN nn L kg Vkn mN nnn d dn AxAx Ax 2 22 1 0 2 22 0 0 1 0 0 0 0 2 0 3 0 2 22 0 0 0 2 0 3 00 0 1234 11                (35) Let Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(9):127-132 132           +++ ++ 0 0 2 0 3 0 0 34 2 1     mN nn n R k vmN k v n=    +        +++ 1 0 0 0 1 0 012     and ( ) S mN n n n=      ++ +0 0 0 0 2 01 1    It is evident from (35) that either (i) R L kg Vk Ax − 2 22  and S or R L kg Vk Ax − 2 22  and S, then  d dn0 is negative. Thus , the growth rates decrease r increase with an increase in stress relaxation time. Also, it is evident from (32), (33) and (34) that if R L kg Vk Ax − 2 22  or R L kg Vk Ax − 2 22  then 0 0 dN dn and 0 0 dv dn are always negative or positive and 1 0 dk dn and 0 0  d dn are always positive or negative. CONCLUSION The growth rates, therefore, decrease or increase with an increase in particle density and kinematic viscosity; similarly, growth rates increase or decrease with an increase in medium permeability. Thus, the growth rates both increase (for certain wave numbers) and decrease (for different wave numbers) with an increase in kinematic viscosity, medium permeability, suspended particles number, density, and stress relaxation time. REFERENCES [1]. S Chadrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York, 1981. [2]. W C Chin, Wave Propagation in the Petroleum Engineering, Gulf Publishing Company, Houston, USA, 1993. [3]. J A M Mcdonnell, Cosmic Dust, John Wiley and Sons, 1978, 330. [4]. Sunil and T Chand, Rayleigh–Taylor Instability of a Plasma in Presence of a Variable Magnetic Field and Suspended Particles in Porous Medium, Indian Journal of Physics, 1997, 71 B(1), 95–105. [5]. R C Sharma and P Kumar, Hydromagnetic Rayleigh–Taylor instability of Oldroydian Viscoelastic Fluids in Porous Medium in Presence of Variable Magnetic Field, Indian Journal Pure Applied Mathematics, 1994, 25(10), 1099-1105. [6]. Sunil, R C Sharma and R S Chande, Stability of Stratified Rivlin–Ericksen Fluid Particle Mixture in Hydromagnetic in Porous Medium, Ganita, 2000, Vol. 51, No. 2, 179–185. [7]. Sunil, Divya, R C Sharma and Veena Sharma, Stability of Stratified Walters’ (Model B') Viscoelastic Fluid in Stratified Porous Medium, Studia Geotechnica et Mechanica, 2004, 26 (1–2). [8]. R P Mathur and Nagendra Kumar, Stability of a Viscoelastic (Maxwell) Fluid of Variable Density in an Inhomogeneous Magnetic Field through Porous Medium, Global Journal of Mathematical Sciences: Theory and Practical, 2011, 3 (5), 427-434. [9]. P R Sengupta and P Basak, Stability of Two Superposed Viscoelastic (Maxwell) Fluids in a Vertical Magnetic Field, Indian Journal Pure Applied Mathematics, 2004, 35(7), 905.