Unsteady Poiseuille Flow of an Incompressible Elastico-Viscous Fluid in a Tube of Spherical Cross Section on a Porous Boundary
Exact solution of an unsteady flow of elastico-viscous
fluid through a porous media in a tube of spherical 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suddenly withdrawn and the resulting fluid motion in a tube of
spherical 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,
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.
[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.
[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.
[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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:69074", author = "Sanjay Baburao Kulkarni", title = "Unsteady Poiseuille Flow of an Incompressible Elastico-Viscous Fluid in a Tube of Spherical 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 spherical 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
spherical 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,
porosity parameter of the bounding surface has significant effect on
the velocity parameter.
", keywords = "Elastico-viscous fluid, Porous media, Second order
fluids, Spherical cross-section.", volume = "9", number = "2", pages = "312-7", }
