Numerical /Analytical Investigations of Stability of Stratified Rotatory Elastico –Viscous Walter’s (Model ) Fluid in the Presence of Variable Magnetic Field, Suspended Particles Saturating Porous Media

Main Article Content

Sumit Gupta
Virender Sharma


The influence of viscosity, viscoelasticity, medium permeability, medium porosity and suspended particles on the stability of a stratified elastic-viscous fluid is examined for viscoelastic polymeric solutions in the simultaneous presence of a variable horizontal magnetic field  and uniform horizontal rotation in porous medium. These solutions are known as Walters’ (model ) fluid and their rheology is approximated by the Walters’ (model ) fluid constitutive relations, proposed by Walters’. The effects of coriolis force on the stability are chosen along the direction of the magnetic field and transverse to that of the gravitational field . Assuming the exponential stratifications in density, viscosity and viscoelasticity, the appropriate solution for the case of free boundaries is obtained using a linearized stability theory and normal mode analysis method. The dispersion relation is obtained and the behaviour of growth rates with respect to kinematic viscosity, kinematic viscoelasticity, medium permeability, dust particles and medium porosity is examined numerically using Newton-Raphson method through the software Fortran-90 and Mathcad. In contrast to the Newtonian fluids, the system is found to be unstable, for stable stratifications, for small wavelength perturbations. It is found that the magnetic field stabilizes the certain wave number band, for unstable stratification in the presence of rotation, suspended particles and this wave number range increases with the increase in magnetic field and decreases with the increase in kinematic viscoelasticity implying thereby the stabilizing effect of magnetic field,kinematic viscoelasticity , suspended particles and the kinematic viscosity has a stabilizing effect on the system for the low wave number range.  The medium permeability has enhancing effect on the growth rates with its increase for a fixed wave number. These results are shown graphically.

Walters’ (model ) fluid, magnetic field, rotation, viscosity, viscoelasticity, medium permeability, medium porosity, suspended particles.

Article Details

How to Cite
Gupta, S., & Sharma, V. (2020). Numerical /Analytical Investigations of Stability of Stratified Rotatory Elastico –Viscous Walter’s (Model ) Fluid in the Presence of Variable Magnetic Field, Suspended Particles Saturating Porous Media. Asian Journal of Pure and Applied Mathematics, 2(3), 9-20. Retrieved from
Original Research Article


Lapwood ER. Convection of fluid in a porous medium. Proc. Camb.Phil. Soc. 1948;45:508.

Wooding RA. Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 1960;9:183.

McDonnel JAM. Cosmic Dust. John Wiley and Sons. 1978;330.

Lord Rayleigh. Proc. London Math. Soc. 1883;14:170.

Taylor GI. The instability of liquid surfaces when accelerated in direction perpendicular to their planes. Proc. Roy. Soc. (Lon.). 1950;A 201:192.

Hide R. Waves in a heavy, viscous, incompressible, electrically conducting fluid of a variable density in the presence of a magnetic field. Proc. Roy. Soc. (London,). 1955;A233:376.

Reid WH. The effect of surface tension and viscosity on the stability of two superposed fluids Proc. Camb. Phil. Soc. 1960;415.

Chandrasekhar S. Hydrodynamic and hydromagnetic stability. Dover Publication, New York; 1961.

Bellman R, Pennington, RH. Effect of surface tension and viscosity on Taylor Instability. Quart. Appl. Math. 1954;12:151–162.

Gupta AS. Rayleigh–Taylor instability of a viscous electrically conducting fluid in the presence of a horizontal magnetic field. J. Phys Soc. Japan. 1963;18:1073.

Kent A. Instability of laminar flow of a magnetofluid. Phys. Fluids. 1966;9:1286.

Fredricksen AG. Principles and applications of Rheology, Prentice–Hall Inc., New Jersey; 1964.

Joseph DD. Stability of fluid motion II .Springer–Verlag, New York; 1976.

Walters K. The motion of an elastico–viscous liquid contained between concentric .Spheres.Quart. J. Mech. Applied Math. 1960;13:325.

Walters K. Non–Newtonian effects in some elastico–viscous liquids whose behaviour at small rates of shear is characterized by a general linear equation of state.Quart. J. Mech. Applied. Math. 1962;15:63.

Sharma PR, Kumar H. Proc. Nat. Acad. Sci. India. 1995;65(A):175.

Sharma RC, Kumar P. Study of the stability of two superposed Walters’ (model) viscoelastic liquids. Czechoslovak Journal of Physics. 1997;47:197.

SharmaV, Dutt Suneel and Gupta U. Stability of stratified elastico– viscous Walters’ (model ) fluid in presence of uniform horizontal magnetic field and rotation. Arch. Mech. 2006;58(2):187–197.

Yadav RS, Sharma PR. Effects of porous medium on MHD fluid flow along a stretching cylinder. Annals of Pure and Appl. Math. 2014;6(1):104-113.

Gupta S, Sharma V. Stability of stratified Elastico-viscous Walters’ (Model ) fluid in the presence of variable mgnetic field and rotation. Annals of Pure and Applied Mathematics. 2016;12(1):31-40.

Gupta S, Sharma V. Stability of stratified Elastico-viscous Rivlin-Ericksen fluid in the presence of variable magnetic field and rotation saturating porous medium. International Journal of Statistics and Applied Mathematics. 2019;4(5):85-92.

Robey MJ, Puckett GE. Implementation of a volume-of-fluid method in a finite element code wth applications to thermochemical convection in a density .Jour. of Computers & Fluids. 2019;217-253.

Robinet J-C, Gloerfelt X. Instabilities in non-ideal fluids. Journal of Fluid Mechanics. 2019;880:1-4.

Camassa R, Harris MD, Hunt R, KIlic Z and McLaughlin MR. First–principle mechanism for particulate aggregation and self assembly in stratified fluids. Nature Communications. 2019;10(1):1-8.

Ganesh M, Kim S and Dabiri S. Induced mixing in stratified fluids by rising bubbles in a thin gap. Phys. Rev. Fluids. 2020;5(4).

Magnaudet J, Mercier JM. Particles, drops and bubbles moving across sharp interfaces and stratified layers. Annual Review of Fluid Mechanics. 2020;52:61-91.