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##### Unsteady Flow of an Incompressible Viscous Electrically Conducting Fluid in Tube of Elliptical Cross Section under the Influence of Magnetic Field

**Authors:**
Sanjay Baburao Kulkarni

**Abstract:**

Exact solution of an unsteady flow of elastico-viscous electrically conducting fluid through a porous media in a tube of elliptical cross section under the influence of constant pressure gradient and magnetic field 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 After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of elliptical cross section by taking into account of the transverse magnetic field and porosity factor of the bounding surface is investigated. The problem is solved in twostages 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 nondimensional porosity parameter (K), magnetic parameter (m) and elastico-viscosity parameter (β), which depends on the Non- Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter and magnetic parameter tends to zero and porosity tends to infinity. It is seen that the effect of elastico-viscosity parameter, magnetic parameter and the porosity parameter of the bounding surface has significant effect on the velocity parameter.

**Keywords:**
Elastico-viscous fluid,
Elliptic cross-section,
Porous media,
Second order fluids.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1337861

**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”, Acceptance letter dated 21/08/2014 in Special Topics & Reviews in Porous Media

[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 Fontaines Publiques 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.