Numerical Investigation of Unsteady MHD Flow of Second Order Fluid in a Tube of Elliptical Cross-Section on the Porous Boundary

Exact solution of an unsteady MHD flow of elasticoviscous fluid through a porous media in a tube of elliptic cross section under the influence of magnetic field and constant pressure gradient has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of elliptical cross section by taking into account of the porosity factor and magnetic parameter of the bounding surface is investigated. The problem is solved in two-stages the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a non-dimensional porosity parameter, magnetic parameter and elastico-viscosity parameter, which depends on the Non-Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter, magnetic parameter tends to zero, and porosity tends to infinity. The numerical results were simulated in MATLAB software to analyze the effect of Elastico-viscous parameter, porosity parameter, and magnetic parameter on velocity profile. Boundary conditions were satisfied. It is seen that the effect of elastico-viscosity parameter, porosity parameter and magnetic parameter of the bounding surface has significant effect on the velocity parameter.




References:
[1] K. R. Rajagopal, P. L. Koloni, “Continuum Mechanics and its
Applications”, Hemisphere Press, Washington, DC, 1989.
[2] K. Walters, “Relation between Coleman-Nall, Rivlin-Ericksen, Green-
Rivlin and Oldroyd fluids”, ZAMP, 21, 1970 pp. 592 - 600.
[3] J. E. Dunn, R. L. Fosdick, “Thermodynamics stability and boundedness
of fluids of complexity 2 and fluids of second grade”, Arch. Ratl. Mech.
Anal, 56, 1974, pp. 191 - 252.
[4] J. E. Dunn, K. R. Rajagopal, “Fluids of differential type-critical review
and thermodynamic analysis”, J. Eng. Sci., 33, 1995, pp. 689 - 729.
[5] K. R. Rajagopal, “Flow of visco-elastic fluids between rotating discs”,
Theor. Comput. Fluid Dyn., 3, 1992, pp. 185 - 206.
[6] N. Ch. PattabhiRamacharyulu, “Exact solutions of two dimensional
flows of second order fluid”, App. Sc Res, Sec - A, 15. 1964, pp. 41 – 50.
[7] S. G. Lekoudis, A. H. Nayef and Saric., “Compressible boundary layers
over wavy walls”, Physics of fluids, 19, 1976, pp. 514 - 19.
[8] P. N. Shankar, U. N. Shina, “The Rayeigh problem for wavy wall”, J.
Fluid Mech, 77, 1976, pp. 243 – 256.
[9] M. Lessen, S. T. Gangwani, “Effects of small amplitude wall waviness
upon the stability of the laminar boundary layer”, Physics of the fluids,
19, 1976, pp. 510 -513.
[10] K. Vajravelu, K. S. Shastri, “Free convective heat transfer in a viscous
incompressible fluid confined between a long vertical wavy wall and a
parallel flat plate”, J. Fluid Mech, 86, 1978, pp.365 – 383.
[11] U. N. Das, N. Ahmed, “Free convective MHD flow and heat transfer in
a viscous incompressible fluid confined between a long vertical wavy
wall and a parallel flat wall”, I.J. Pure & App. Math,23, 1992, pp. 295 -
304.
[12] R.P Patidar, G. N. Purohit, “Free convection flow of a viscous
incompressible fluid in a porous medium between two long vertical
wavy walls”, I. J. Math, 40, 1998, pp. 76 -86.
[13] R. Taneja, N. C. Jain, “MHD flow with slip effects and temperature
dependent heat source in a viscous in compressible fluid confined
between a long vertical wavy wall and a parallel flat wall”, J. Def. Sci.,
2004, pp.21 - 29.
[14] Ch. V. R. Murthy, S.B. Kulkarni, “On the class of exact solutions of an
incompressible fluid flow of second order type by creating sinusoidal
disturbances”, J. Def.Sci, 57, 2, 2007, pp. 197-209.
[15] S. B. Kulkarni, “Unsteady poiseuille flow of second order fluid in a tube
of elliptical cross section on the porous boundary”, Special Topics &
Reviews in Porous Media., 5, 2014, pp. 269 – 276.
[16] W. Noll, “A mathematical theory of mechanical behaviour of continuous
media”, Arch. Ratl. Mech. & Anal., 2, 1958, pp. 197 – 226.
[17] B. D. Coleman, W. Noll, “An approximate theorem for the functionals
with application in continuum mechanics”, Arch.Ratl.Mech and Anal, 6,
1960, pp. 355 – 376.
[18] R. S. Rivlin, J. L. Ericksen, “Stress relaxation for isotropic materials”, J.
Rat. Mech, and Anal, 4, 1955, pp.350 – 362.
[19] M.Reiner, “A mathematical theory of diletancy”,Amer.J. ofMaths, 64,
1964,pp.350 - 362.
[20] H. Darcy, “Les FontainesPubliques de la Ville de, Dijon, Dalmont,
Paris” 1856.
[21] E. M. Erdogan, E. Imrak, “Effects of the side walls on the unsteady flow
of a Second-grade fluid in a duct of uniform cross-section”, Int. Journal
of Non-Linear Mechanics, 39, 2004, pp. 1379-1384.
[22] S. Islam, Z. Bano, T. Haroon and A.M. Siddiqui, “Unsteady poiseuille
flow of second grade fluid in a tube of elliptical cross-section”, 12, 4,
2011. 291-295.
[23] S. B. Kulkarni, “Unsteady flow of an incompressible viscous electrically
conducting fluid in tub of elliptical cross section under the influence of
magnetic field”, International Journal of Mathematical, Computational,
Physical and Quantum Engineering, 8(10), 2014, pp. 1311 – 1317.
[24] S. B. Kulkarni, “Unsteady poiseuille flow of an incompressible viscous
fluid in a tube of spherical cross section on a porous boundary”,
International Journal of Mechanical Aero Space Industrial and
Mechatronics Engineering, 9(2), 2015, pp. 240 – 246.
[25] S.B. Kulkarni, “Unsteady MHD flow of elastico–viscous incompressible
fluid through a porous media between two parallel plates under the influence of magnetic field”, Defence Science Journal, 65(2), 2015, pp.
119 – 125.
[26] S. B. Kulkarni, “Unsteady MHD flow of elastic-viscous fluid in tube of
spherical cross section on porous boundary”, International Journal of
Mechanical Aero Space Industrial and Mechatronics Engineering, 9(4),
2015, pp. 590 – 596.
[27] Amos Gilat, “MATLAB An Introduction with applications”, John Wiley
& Sons Inc., 2010.