1. Introduction. Keywords Stability, Suspended Particles, Magnetic Field, Brinkman Porous Medium. Kapil Kumar 1,*, V. Singh 2, Seema Sharma 1

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1 American Journal of Fluid Dynamics, (): DOI:.59/j.ajfd.. Stability of an Oldroydian Viscoelastic Fluid Permeated with Suspended Particles through a Brinkman Porous Medium with Variable Gravity Field in Hydromagnetics Kapil Kumar,*, V. Singh, Seema Sharma Department of Mathematics & Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar, 4944, India Department of Applied Sciences, Moradabad Institute of Technology, Moradabad, 44, India Abstract The effect of suspended particles, variable gravity magnetic field on the thermal stability of an Oldroydian viscoelastic fluid through Brinkman porous medium is considered. Following the linearized stability theory normal mode method, the dispersion relation is obtained. It has been found that for stationary convection, Oldroydian viscoelastic fluid behaves like an ordinary Newtonian fluid medium permeability, medium porosity suspended particles have destabilizing effects on the system whereas the magnetic field Darcy-Brinkman number have stabilizing effects on the system. The sufficient conditions for the non-eistence of overstability are obtained it has also been found that the modes may be oscillatory non-oscillatory. It is also noted that the principle of echange of stabilities is valid under certain conditions. Keywords Stability, Suspended Particles, Magnetic Field, Brinkman Porous Medium. Introduction The theoretical eperimental results of the onset of convection in a horizontal layer of Newtonian fluid heated from below under varying assumptions of hydrodynamics hydromagnetics has been discussed in detail by Chrasekhar[]. The flow through porous medium is important in filtering equipment, chemical reactors, oil gas etraction, recovery of crude oil fro m earth s interior, movement of ground water, etc. The physical properties of comets, meteorites interplanetary dust strongly suggest the importance of porosity in astrophysical situations (McDonnel[]). A porous medium is defined as a material volu me consisting of solid matri with an interconnected void. It is mainly characterized by its porosity which is defined as the ratio of the void space to the total volume of the medium. A comprehensive detailed study of convection through porous medium has been given by Nield Bejan[]. At low velocities a saturated porous medium with high porosity is well described by the Darcy-Brinkman approimation require that classical frictional term be added in the classical Darcy equation. The dynamic viscosity µ associated with the Brinkman[4] term is known as effective viscosity. * Corresponding author: kkchaudhary@gmail.com (Kapil Kumar) Published online at Copyright Scientific & Academic Publishing. All Rights Reserved Tissues can be treated as a porous medium as it is composed of dispersed cells separated by connective voids which allow for flow of nutrients, minerals, etc. There are several evidences, both theoretical eperimental, which suggest that the Darcy s equation gives inadequate results of the hydrodynamic conditions, particularly near the boundaries of a porous medium. The Darcy-Brinkman equation, which takes into account the boundary effects, has been employed in recent years in biomedical hydrodynamic studies[5], modeling a thin fibrous surface layer coating blood vessels (endothelial surface layer) as it is a highly permeable porous medium[6,7]. It is, therefore, of interest to study the characteristics of the Darcy-Brinkman equation which have relevance importance in several biological, geological, chemical reactors medical science applications, thus forming a motivation for the present study. In geophysical contet, the fluid is often not pure but may instead be permeated with dust particles. The effects of suspended particles on the stability of superposed fluids have industrial scientific importance in geophysics, chemical engineering astrophysics. Scanlon Segel[8] have considered the effect of suspended particles on the onset of Bénard convection found that the critical Rayleigh number was reduced solely because the heat capacity of the pure fluid was supplemented by that of the particles. So the effect of suspended particles was thus found to increase the instability of the layer. An eperimental demonstration by Toms Strawbridge[9] reveals that a dilute solution of methyl methacrylate in n-butyl acetate behaves in

2 American Journal of Fluid Dynamics, (): accordance with the theoretical model of Oldroyd fluid[]. The present analysis would be relevant to the stability of some polymer solutions like a dilute solution of methyl methacrylate in n-butyl acetate also find its importance in several geographical situations in chemical technology. Bhatia Steiner[] have studied the problem of thermal instability of a Mawellian visco-elastic flu id in the presence of a magnetic field found that the magnetic field has a stabilizing influence on the overstable mode of convection in a visco-elastic flu id layer. The problem of thermal instability of a fluid layer under variable gravity field has been considered by Pradhan Samal[]. Since the earth s gravity field varies with its distance from its surface so it is of utmost important to take into account the effect of variable gravity field having its importance in several convective flows, atmospheric science, astrophysics, geophysics also in material processing technology. Sharma[] studied the stability of a layer of an electrically conducting Oldroyd fluid in the presence of a magnetic field found that the magnetic field has a stabilizing influence while the effect of suspended particles on the onset of Bénard convection in hydromagnetics have been studied by Sharma et al.[4]. Sharma Sharma[5] have investigated the effect of suspended particles on thermal instability in Mawellian Oldroydian viscoelastic fluids in a porous medium found that for stationary convection both the Mawellian Oldroydian fluids behave like Newtonian fluid the medium permeability suspended particles have destabilizing effects on the system. Sharma Sunil[6] considered the thermal instability o f an Oldroydian viscoelastic fluid with suspended particles in hydromagnetics in porous medium concluded that, for stationary convection, magnetic field has a stabilizing effect whereas medium permeability suspended particles have destabilizing effects on the system. The problem of the onset of electroconvective instability of an Oldroydian viscoelastic liquid layer through Brinkman porous medium under the simultaneous action of an electric field a vertical temperature gradient has been investigated by El-Sayed[7]. The purpose of the present note treated here is to eamine theoretically the problem of the onset of stability of an Oldroydian viscoelastic fluid permeated with suspended particles through a Brinkman porous medium under the simultaneous action of varying gravity field vertical magnetic field.. Formulation of the Problem Perturbation Equations Let Tij, τij, eij, δ ij, qi, p, µλλ,, ( < λ) i denote, respectively, the total stress tensor, shear stress tensor, rate of strain tensor, Kronecker delta, velocity vector, pressure, viscosity, stress relaation time, strain retardation time position vector. Then the Jeffrey (Oldroydian) constitutive relations for performing linear stability of Rayleigh Bénard convection in clear viscoelastic fluids are Tij = pδij + τij, ( + λ ) τ ij = µ ( + λ ) eij, q q i j () eij = + j i Relations of the type () were proposed studied by Oldroyd[]. He also showed that many rheological equations of general validity reduce to () when linearized. λ = yields the Mawellian fluid, whereas λ = λ = gives the Newtonian viscous fluid. Here we consider an infinite horizontal layer of an Oldroydian viscoelastic fluid permeated with suspended particles bounded by the planes z= z=d in a porous medium of porosity medium permeability k. The fluid layer is acted on by a uniform vertical magnetic field H (,, H) a variable gravity force g = h( z) g, where h( z) can be positive or negative according as the gravity increase or decreases upward fro m its value g ( > ). The layer is heated from below so that a uniform temperature gradient β = dt dz is maintained. The governing equations of motion continuity for the Oldroydian viscoelastic fluid based on the Darcy-Brinkman approimation are ρ q + ( q. ) q K N p + ρx + i = + λ () µ e ( qd q) + ( H) H 4π µ µ + + λ q k. q = () ρ, µµµ,,, q, q, N, t X = gλ e d i i denote, respectively, the density of fluid, viscosity, effective viscosity, magnetic permeability, velocity of pure fluid, velocity of suspended particles, number density of the suspended particles the gravitational acceleration. (, y z) =, K = 6πρυδ where δ being particle radius, is the Stokes drag coefficient. The presence of suspended particles adds an etra force term, in equation of motion, proportional to velocity difference between particles fluid. Since the force eerted by the fluid on the particles is equal opposite to

3 6 Kapil Kumar et al.: Stability of an Oldroydian Viscoelastic Fluid Permeated with Suspended Particles through a Brinkman Porous Medium with Variable Gravity Field in Hydromagnetics that eerted by the particles on the fluid, there must be an etra force term, equal in magnitude but opposite in sign, in the equations of motion for the particles. Inter-particle reactions are ignored as the distances between the particles are assumed to be quite large compared with their diameters. The effects of pressure, magnetic field gravity on the particles are very s mall hence ignored. If mn is the mass of particles per unit volume, then the equations of motion continuity for the particles are: mn q + q q K N t = q q N +.( Nq d ) = (. ) d d d d The energy equation is T ρcv+ ρscs( ) + ρcv( q. ) T t + mncpt + qd. T = k T, c, c, c, T k s s v pt (4) (5) (6) ρ denote, respectively, the density of solid material, heat capacity of solid material, the specific heat at constant volume, heat capacity of suspended particles, the temperature the thermal conductivity. The Mawell s equation yields H = ( q H) +η H (7). H =, (8) η denote the electrical resistivity. The equation of state is ρ = ρ + α T T (9) The steady state solution corresponding to the system of equations ()-(9) are: [ ] q = (,,), T = T β z, ρ = ρ + αβ z, p p g αβ z = ρ +, H =,, H [ ] () Let the initial state solution described by equation () be slightly perturbed. We assume that q (u,v,w), q d (l,r,s), N, θ, δ p, δρ h( h, hy, h z) denote, respectively, the perturbation in fluid velocity q(,,), perturbation in particle velocity q d (,,), perturbation in particle number density N, temperature T, pressure p, density ρ magnetic field H. The change in density δρ caused by perturbation θ in temperature, is given by δρ = αθρ () The governing perturbation equations (using Oberbeck- Boussinesq approimation) due to linearization procedure are: where q + λ K N ( δ p) + gαθλi + ρ ρ = + λ () t µ e ( qd q) + {( h) H} 4πρ υ υ + + λ q k. q = () m + qd K = q (4) N +.( Nq d ) = (5) θ ( E + b ) = β( w + bs) + κ θ (6). h = (7) h = ( H) q+η h (8) µ µ k υ =, υ =, κ =, λi = (,,), w s ρ ρ ρ C denote, respectively, the kinematic viscosity, the effective kinematic viscosity, the co-efficient of thermometric conductivity, unit vertical vector, vertical fluid velocity suspended particles velocity ρscs mnc pt E =+ ( ) b =. ρcv ρcv p Eliminating δ between the three component equations of () using equations () (4), we obtain: m + λ + ' ( w) K m = + λ + ' K µ eh gα + θ + ( hz ) y 4πρ z mn v ( w) + λ + + λ ρ m υ υ + ' K k ( w) (9)

4 American Journal of Fluid Dynamics, (): m ζ + λ + ' K m µ eh ξ = + λ + ' K 4 πρ mn ζ + λ + + λ ρ m υ υ + ' ζ K k v u hy h ζ = ξ = y y () st for the z-component of vorticity z-component of current density, respectively. From equation (6), we obtain: m m + ' ( E+ b) κ θ = β + b w ' + K K () The y component equations of (8) along with (7) yields: ξ ζ = H +η ξ z the z- component equation of (8) is h = w η z + + y z z = H + hz () () is the three dimensional Laplacian operator.. Dispersion Relation Discussion Decomposing the disturbances into normal modes by seeking solutions whose dependence on, y t is given by: [ w,,, h, ] = W( z), Θ( z), Z( z), K( z), X ( z) θζ z ξ ep ( ik + ikyy + nt) k, k y (4) are the wave numbers along y directions, respectively k ( + ) = is the k k y resultant wave number n is the frequency of the harmonic disturbance, which is, in general, a comple constant. Using epression (4) making the substitutions of the non-dimensional quantities of the form nd mυ ρ f z = z d a = kd = = N = υ Kd m,, σ, τ,, λυ λυ E = E+ b, B = b +, F =,F =. d d µ k DA = k, is the Darcy-Brinkman number, P µ d, l = is d υ p =, is the κ υ p =, is the magnetic η the dimensionless medium permeability, thermal Prtl number Prtl number. We obtain the non-dimensional form of Equations (9)-() (after dropping the asterisk for convenience) as: σ f ( + F σ ) DA ( D a ) + + τσ ( + Fσ) g αadθ (5) ( D a ) W( z) + υ µ ehd ( D a ) DK = 4πρ υ ( ) σ f ( + F σ ) DA D a + Z + τσ ( + Fσ) µ ehd = DX 4πρυ βd B τσ ( D a ) Ep σ + Θ= W κ + τσ Hd pσ ( D a ) K = DW η Hd pσ ( D a ) X = DZ η Eliminating Θ K (6) (7) (8) (9) from equations (5), (7) (8), we obtain: σ f ( + F σ ) DA ( D a ) + + τσ ( + Fσ) ( D a ) Epσ pσ ( D a ) ( D a ) () W( z) Q( D a ) Epσ ( D a ) D W( z) B + τσ = Ra h( z) pσ ( D a ) W ( z) + τσ 4 gαβd R =, is the thermal Rayleigh number υκ

5 6 Kapil Kumar et al.: Stability of an Oldroydian Viscoelastic Fluid Permeated with Suspended Particles through a Brinkman Porous Medium with Variable Gravity Field in Hydromagnetics Q µ Hd e 4πρυη =, is the Chrasekhar number. From equations (6) (9), we obtain: ( ) σ f ( + F σ ) DA D a + + τσ ( + Fσ) pσ ( D a ) QD ( Z, X ) = () It is apparent from equation () that for the problem under consideration X = Z = () that is the z- components of vorticity current density vanish identically for the problem on h. The boundary conditions appropriate to two free perfectly conducting boundaries are defined as: W = D W = DZ = DK = h z = Θ = on z =. () Equation () together with the boundary conditions () () constitute an eigen-value problem for the problem on h. It is evident that when F =, the system reduces to that for a Mawell fluid whereas for F = F =, the system reduces to that for an ordinary viscous fluid. 4. Solution of the Eigen-value Problem The boundary conditions () () suggest that a proper solution for W belonging to the lowest mode is defined by W = W sinπ z (4) where W is a constant. Substituting solution (4) in equation () letting R a σ Q DA R =, =, iσ =, P= π P, Q =, D =, 4 l A π π π π π we get the eigenvalue relationship of the form: ( + iσπ F) ( + iσπ F ) ( + iσ f DA ` iσπ τ P P ( + + iσep iσp + ( + ( + + Q( + ( + + iσ Ep B+ iσπ τ = Rh ( z) iσ p + ( + iσπ τ + (5) Equation (5) is required dispersion relation accounting the effects of magnetic field, suspended particles medium permeability on thermal instability of an Oldroyd visco-elastic flu id through a Brinkman porous medium. 5. The Stationary Convection When instability sets in as stationary convection, the marginal state will be characterized by putting σ = in equation (5) get the eigenvalue relationship for a stationary instability of the form: R ( + ) ( + ) DA Q = Bh z P P ( (6) Thus it is clear from equation (6) that for stationary convection Oldroydian visco-elastic fluid behaves like an ordinary Newtonian fluid since the stress relaation parameter F strain retardation parameter F vanishes with σ. Minimizing equation (6) with respect to, yields the third order equation in as DA + ( D ) A + ( DA ) ( DA QP ) = (7) Substituting the value of critical wave number c thus obtained from equation (7), in equation (6), gives the value of critical Rayleigh number for stationary instability. In order to investigate the effects of suspended particles, magnetic field, medium permeability, Darcy-Brinkman number medium porosity, we eamine the behaviour of dr dr dr dr dr,,, analytically. db dq dp dda d Equation (6) yields ( + ) ( + ) dr DA Q ( db = + h( z) B P P + + (8) which is negative implying thereby that the suspended particles have a destabilizing effect on the viscoelastic Oldroydian fluid for the case of stationary convection when the gravity field increases upward fro m its value g i.e. ( h( z ) > ). This destabilizing effect of suspended particles agrees with previous published work by Scanlon Segel[8], Sharma Sharma[5] Sharma Sunil[6]. From equation (6), we get dr dq ( + = (9) h z B which shows that magnetic field always has a stabilizing effect on the viscoelastic Oldroydian fluid for the case of stationary convection when the gravity field increases. This stabilizing ( h z > ) upward from its value g i.e. effect of magnetic field is an agreement of the previous results by Sharma[] Sharma Sunil[6].

6 American Journal of Fluid Dynamics, (): From equation (6), we get dr = + dp Bh( z) P P ( + D ( + A (4) which is always negative implying thereby that the medium permeability has a destabilizing effect on the viscoelastic Oldroydian fluid for the case of stationary convection as the gravity field increases upward from its value g i.e.. This destabilizing result agrees with the ( h( z ) > ) earlier work reported by Sharma Sharma[5] Sharma Sunil[6]. From equation (6), we get dr ( + ) = (4) dda BP h( z) which is always positive. Therefore, the modified Darcy-Brinkman number has a stabilizing effect on the system for the case of stationary convection as the gravity field increases upward from its value g i.e. From equation (6), we get ( + h z >. dr DA = + Q ( + d Bh( z) P (4) which is negative implying thereby that the effect of medium porosity is to destabilize the system for the case of stationary convection as the gravity increases upward from its value g. This destabilizing effect of porosity ( h z > ) i.e. agrees with the result of El-Sayed[7]. 6. Results Discussion The dispersion relation (6) is analyzed numerically. The region of stationary instability for different values of suspended particles, magnetic field, medium permeability, Darcy-Brinkman number medium porosity is shown in figures,,, 4, 5, respectively. It is clear from figure that the critical Rayleigh number decreases with an increase in the value of suspended particles. Figure illustrates that critical Rayleigh number increases with an increase in magnetic field. Figure shows a decrease in critical Rayleigh number with permeability. Figure 4 illustrates that an increase in the value of Darcy-Brinkman number increases the critical Rayleigh number, thereby depicting the stabilizing effect of Darcy-Brinkman number for the problem on h. From figure 5, we find that an increase in the medium porosity decreases the critical Rayleigh number, hence decreases the region of stability. R Fi gure. Variation of Rayleigh number R with wave number, for A 5 D =, Q = 5, =.5, P =. B=5,,5,,5 R Figure. Variation of Rayleigh number R with wave number, for D =, B =, =.5, P = R A Q=,,,4,5 B=5 B= B=5 B= B=5 Q= Q= Q= Q=4 Q=5 P=. P=. P=. P=.4 P= Figure. Variation of Rayleigh number R with wave number, for D A =, B =, =.5, Q = 5 P=.,.,.,.4,.5

7 64 Kapil Kumar et al.: Stability of an Oldroydian Viscoelastic Fluid Permeated with Suspended Particles through a Brinkman Porous Medium with Variable Gravity Field in Hydromagnetics R X Figure 4. Variation of Rayleigh number R with wave number, for P=.5, B =, =.5, Q = 5 D A =,,, 4, 5 DA= DA= DA= DA=4 DA=5 R Figure 5. Variation of Rayleigh number R with wave number, for D =, B =, P =.5, 5 A X ϵ=. ϵ=. ϵ=. ϵ=.4 ϵ=.5 Q = =.,.,.,.4,.5 7. Overstability Case Here we eamine the possibility of whether the instability may occur as overstability. Since for overstability we wish to determine the critical Rayleigh number for the onset of instability via a state of pure oscillations, it suffices to find the conditions for which equation (5) will admit solutions with σ real. Equating the real imaginary parts of equation (5), we get ( + fσ π τ + σ π FF D 4 A` ( + σ Epp ( + + σ π τ + σ π F P P ( F F ) DA ( + f σ σπ ` ( + σp + σep σ σ π τ σ π F + + P P ( ) ( )( σ πτ 4 ) σ πτ + Q + = R h z + B+ + p B ( + fσ π τ + σ π FF D f σ σπ D ( ) F F A + ` σ Q σ π τ σ π F σ Ep P P 4 A` ( + σp + σep + ( + σ Epp ( + + σ π τ + σ π F P P 4 ( ) σπτ( ) σ ( σ πτ ) = R h z + B + p B+ Eliminating R between equations (4) (44) assuming y 4 4 (4) (44) σ =, we get a fourth order polynomial in y of the form: a y + a y + a y + a y+ a = (45) { } 4 4 a = + π τ FE pp PfF + π τfd FP B + + π τ PF p p Ep + pπ τ EPF p (46)

8 American Journal of Fluid Dynamics, (): where D D ( a ( + {( + f ) PB + π ( F F) BD ( B) πτd} + ( + {( ) πτ } + { }( + ) + ( + ) 4 = B QP QE pbp pbqp EpBD 4 { A } = + + ` a, a a involving large number of terms are not included to save spaces. Since σ The values of the coefficients is real for overstability to occur therefore all the roots of y should be positive. Fro m equation (45), the product of root a a 4 is negative if is positive if a 4 = a is positive. B > E p > p or B > Eη > κ. (48) (47) B > F > F or B > λ > λ. (49) The inequalities defined by (48) (49) are the sufficient conditions for the non-eistence of overstability. 8. Principal of Echange of Stabilities Oscillatory Modes Here we determine the condition, if any, under which principle of echange of stabilities is satisfied the possibility of oscillatory modes for the Oldroydian viscoelastic fluid under the influences of variable gravity, suspended particle magnetic field through Brinkman porous medium. For this, we multiply equation (5) by W * (the comple conjugate of W), integrating over the range of z making use of equations (7) (8) with the help of boundary conditions () (), we obtain = + 4 ( Fσ) ( Fσ) σ f I + τσ + σ + σ { DI} A F F * gαa κ + τσ βυ B τ σ + 4πρ p * µ e * ( I Ep σ I4) ( pσ I5 I6) = 4 = = Θ + Θ I DW a W dz, I D W a W a DW dz, I D a dz, 5 = + I = Θ dz, I DK a K dz, I = D K + a 4 K + a DK dz All the above integrals I -I 6 are positive definite. Putting σ = iσ, where σ is real, into equation (5), equating imaginary part, we obtain ( ) + ( + τ σ ) ( + f ) + τ σ ( F F) τ B I B EpI 4 gαa κ µ e I 5 σ I+ ( I+ DI) A + = (5) ( + τ σ ) ( + F σ ) P 4 l B + τ βυ πρ σ Equation (5) implies that σ = or σ which mean that modes may be non oscillatory or oscillatory. The oscillatory modes introduced due to presence of magnetic field, suspended particles, visco-elasticity, medium permeability Darcy-Brinkman parameter. In the absence of magnetic field, equation (5) reduces to (5)

9 66 Kapil Kumar et al.: Stability of an Oldroydian Viscoelastic Fluid Permeated with Suspended Particles through a Brinkman Porous Medium with Variable Gravity Field in Hydromagnetics ( ) + ( + τ σ ) ( + f ) + τ σ ( F F) gαa κ τ B I B EpI 4 σ I+ ( I+ DI) A + = (5) ( + τ σ ) ( + F σ ) P βυ l B + τ σ B > F > F then the term inside the bracket is positive which implies It is evident from equation (5) that if that σ =, thus the modes are non-oscillatory the principle of echange of stabilities is satisfied. 9. Conclusions The problem of thermal instability of an Oldroydian viscoelastic fluid permeated with suspended particles in the presence of variable gravity vertical magnetic field through a Brinkman porous medium is considered in the present paper. For stationary convection, Oldroydian fluid behaves like an ordinary Newtonian fluid. The effect of suspended particles, medium porosity medium permeability are found to hasten the onset of thermal instability whereas magnetic field Darcy-Brinkman number have a stabilizing effect on the system, as such their effect is to postpone the onset of thermal instability when the gravity field increases upward from its value g i.e. ( h( z ) > ). We thus find that, in case of Brinkman porous medium, the Darcy-Brinkman number has a stabilizing effect for the problem under consideration whereas both medium porosity medium permeability have destabilizing effects on the system for the case of stationary convection. The oscillatory modes are introduced due to presence of magnetic field, suspended particles, visco-elasticity, mediu m permeability Darcy-Brinkman parameter. The principle of echange of stabilities is found to hold under certain conditions. The sufficient conditions for the non-eistence of overstability are also obtained. The present analysis would be relevant to the stability of some polymer solutions Oldroydian viscoelastic fluids finds its importance in several scientific engineering fields like ground water hydrology, petroleum engineering in geophysics. ACKNOWLEDGEMENTS The authors wish to thank the reviewer for useful comments which have lead to the improvement of the paper in the present form. REFERENCES [] Chrasekhar, S.C.: Hydrodynamic Hydromagnetic Stability, Dover Publication, New York, 98. [] McDonnel J.A.M.: Cosmic Dust, John Wiley & Sons, Toronto, Canada, 978. [] Nield, D.A. Bejan, A.: Convection in porous media. Springer, New-York, 6. [4] Brinkman, H.C.: A calculation of the viscous force eerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., A, pp. 8 86, 947. [5] Khaled, A.R.A. Vafai, K. : The role of porous media in modeling flow heat transfer in biological tissues. Int. J. Heat Mass Transfer, Vol. 46, pp ,. [6] Pries, A.R., Secomb, T.W., Jacobs, H., Sperio, M., Osterloh, K. Gaehtgens, P.: Microvascular blood flow resistance: role of endothelial surface layer. Am. J. Physiol, Vol. 7, H7-H79, 997. [7] Pries, A.R., Secomb, T.W., Gaehtgens, P.: The endothelial surface layer. Pflugers Arch.- Eur. J. Physiol, Vol. 44, pp ,. [8] Scanlon, J.W. Segel, L.A.: Effect of suspended particles on onset of Bénard convection, Physics Fluids, 6, pp , 97. [9] Toms B.A. Strawbridge D.J.: Elastic viscous properties of dilute solutions of polymethyl methacrylate in organic liquids, Trans. Faraday Soc., Vol. 49, pp. 5-, 95. [] Oldroyd J.G. : Non-Newtonian effects in steady motion of some idealized elastic-viscous liquids, Proc. Royal Soc. London, A 45, pp , 958. [] Bhatia, P.K. Steiner, J.M.: Thermal instability of fluid layer in Hydromagnetics, J. Math. Anal. Appl.,4, pp. 7-89, 97. [] Pradhan, G.K. Samal, P.C.: Thermal instability of a fluid layer under variable body forces, J. Math. Anal. Appl.,, pp , 987. [] Sharma R.C.: Thermal instability in a viscoelastic fluid in hydromagnetics, Acta Phys. Hung., Vol. 8, pp. 9-98, 975. [4] Sharma R.C., Prakash K. Dube S.N.: Effect of suspended particles on the onset of Bénard convection in hydromagnetics, Acta Phys. Hung., Vol. 4, pp. -, 976. [5] Sharma R.C. Sharma N.D.: Thermal instability in Mawellian Oldroydian viscoelastic fluids with suspended particles in porous medium, Czechoslovak J. Phys., Vol. 4, no. 5, pp. 5-5, 99. [6] Sharma R.C. Sunil.: Thermal instability of Oldroydian viscoelastic fluid with suspended particles in hydromagnetics in porous medium, Polymer astics Technology Engineering, Vol., Issue, pp. -9, 994. [7] El-Sayed M. F. : Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium, Arch. Appl. Mech. Vol. 78, pp. 4, 8.