Unsteady MHD elasticoviscous fluid flow of second order type in a tube of hyperbolic cross section on the porous boundary
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Author(s)
S. B. Kulkarni, Professor and Head of Department of Applied Mathematics, Finolex Academy of Management & Technology, Ratnagiri415 639.
Abstract
Exact solution of an unsteady flow of elasticoviscous fluid through a porous media in a tube of hyperbolic 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 After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of hyperbolic cross section by taking into account of the magnetic parameter and porosity factor of the bounding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 ( ), magnetic parameter(M) and elasticoviscosity parameter ( ), which depends on the NonNewtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as , K and . It is seen that the effect of elasticoviscosity parameter ( ), Magnetic parameter(M) and the porosity parameter ( ) of the bounding surface has significant effect on the velocity parameter.
Keywords
Elastico viscous fluid, second order fluid, Elliptic crosssection, porous media, Separation of variables, Magentic parameter
Cite this paper
S. B. Kulkarni,
Unsteady MHD elasticoviscous fluid flow of second order type in a tube of hyperbolic cross section on the porous boundary, SCIREA Journal of Mechanics. Vol.
1
, No.
1
,
2016
, pp.
48

63
.
References
[ 1 ]  Rajagopal, K. R and Koloni, P. L., (1989) Continuum Mechanics and its Applications, Hemisphere Press, Washington, DC. 
[ 2 ]  Walters, K., (1970) Relation between ColemanNall, RivlinEricksen, GreenRivlin and Oldroyd fluids, ZAMP, 21, pp. 592 – 600 
[ 3 ]  Dunn, J. E and Fosdick, R. L., (1974) Thermodynamics stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ratl. Mech. Anal, 56, pp. 191  252. 
[ 4 ]  Dunn, J. E and Rajagopal, K. R., (1995) Fluids of differential typecritical review and thermodynamic analysis, J. Eng. Sci., 33, pp. 689  729. 
[ 5 ]  Das, U. N and Ahmed, N., (1992) Free convective MHD flow and heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall, J. Pure & App. Math, 23, pp. 295 304. 
[ 6 ]  Pattabhi Ramacharyulu, N. Ch., (1964) Exact solutions of two dimensional flows of second order fluid, App. Sc Res, Sec  A, 15 pp. 41 – 50. 
[ 7 ]  Lekoudis, S.G, Nayef, A.H and Saric., (1976) Compressible boundary layers over wavy walls, Physics of fluids, 19, pp. 514  19. 
[ 8 ]  Lessen, M and Gangwani, S.T., (1976) Effects of small amplitude wall waviness upon the stability of the laminar boundary layer, Physics of the fluids, 19, pp. 510 513. 
[ 9 ]  Shankar, P.N and Shina, U.N., (1976) The Rayeigh problem for wavy wall, J. Fluid Mech, 77, pp. 243 – 256. 
[ 10 ]  Vajravelu, K and Shastri, K.S., (1978) 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, pp.365 – 383. 
[ 11 ]  Rajagopal, K. R., (1992) Flow of viscoelastic fluids between rotating discs, Theor. Comput. Fluid Dyn., 3, pp. 185  206. 
[ 12 ]  Patidar, R. P and Purohit, G. N., (1998) Free convection flow of a viscous incompressible fluid in a porous medium between two long vertical wavy walls, J. Math, 40, pp. 76 86. 
[ 13 ]  Murthy, Ch. V. R and Kulkarni, S. B., (2007) On the class of exact solutions of an incompressible fluid flow of second order type by creating sinusoidal disturbances, J. Def.Sci, 57, 2, pp. 197209. 
[ 14 ]  S.B.Kulkarni, (2014)Unsteady poiseuille flow of second order fluid in a tube of elliptical cross section on the porous boundary 
[ 15 ]  Noll, W., (1958) A mathematical theory of mechanical behaviour of continuous media, Arch. Ratl. Mech. & Anal., 2, pp. 197 – 226 
[ 16 ]  Coleman, B. D and Noll, W., (1960) An approximate theorem for the functionals with application in continuum mechanics, Arch. Ratl. Mech and Anal, 6, pp. 355 – 376 
[ 17 ]  Rivlin, R.S and Ericksen, J. L., (1955) Stress relaxation for isotropic materials. J. Rat. Mech, and Anal, 4, pp.350 – 362. 
[ 18 ]  Reiner, M., (1964) A mathematical theory of diletancy, Amer.J. of Maths, 64, pp. 350  362. 
[ 19 ]  Erdogan, E. M and Imrak, E., (2004) Effects of the side walls on the unsteady flow of a Secondgrade fluid in a duct of uniform crosssection, Int. Journal of NonLinear Mechanics, 39, pp. 13791384. 
[ 20 ]  Islam, S. Bano, Z. Haroon, T and Siddiqui, A. M., (2011) Unsteady poiseuille flow of second grade fluid in a tube of elliptical crosssection, 12, 4/2011. 291295. 
[ 21 ]  Taneja, R and Jain, N. C., (2004) 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., pp.21  29. 
[ 22 ]  H. Darcy, “Les Fontaines Publiques de la Ville de, Dijon, Dalmont, Paris” 1856. 
[ 23 ]  S. B. Kulkarni and P. S. Soman, (2016) Unsteady poiseuille flow of second order fluid in a tube of hyperbolic cross section on the porous boundary. 