Unsteady Flow of an Incompressible Elastico-Viscous Fluid of Second order Type in Tube of Ellipsoidal Cross Section on a Porous Boundary
Exact solution of an unsteady flow of elastico-viscous
fluid through a porous media in a tube of ellipsoidal cross section
under the influence of 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suddenly withdrawn and the resulting fluid motion in a tube of
ellipsoidal cross section by taking into account of the 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) 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 tends to zero and porosity tends to
infinity. It is seen that the effect of elastico-viscosity parameter and
the porosity parameter of the bounding surface has significant effect
on the velocity parameter.
[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 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.
[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 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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:69520", author = "Sanjay Baburao Kulkarni", title = "Unsteady Flow of an Incompressible Elastico-Viscous Fluid of Second order Type in Tube of Ellipsoidal Cross Section on a Porous Boundary", abstract = "Exact solution of an unsteady flow of elastico-viscous
fluid through a porous media in a tube of ellipsoidal cross section
under the influence of 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
ellipsoidal cross section by taking into account of the 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) 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 tends to zero and porosity tends to
infinity. It is seen that the effect of elastico-viscosity parameter and
the porosity parameter of the bounding surface has significant effect
on the velocity parameter.
", keywords = "Elastico-viscous fluid, Ellipsoidal cross-section,
Porous media, Second order fluids.", volume = "9", number = "2", pages = "365-6", }
