Abstract
The problem of the creeping flow through a spherical droplet with a non-homogenous porous layer in a spherical container has been studied analytically. Darcy’s model for the flow inside the porous annular region and the Stokes equation for the flow inside the spherical cavity and container are used to analyze the flow. The drag force is exerted on the porous spherical particles enclosing a cavity, and the hydrodynamic permeability of the spherical droplet with a non-homogeneous porous layer is calculated. Emphasis is placed on the spatially varying permeability of a porous medium, which is not covered in all the previous works related to spherical containers. The variation of hydrodynamic permeability and the wall effect with respect to various flow parameters are presented and discussed graphically. The streamlines are presented to discuss the kinematics of the flow. Some previous results for hydrodynamic permeability and drag forces have been verified as special limiting cases.
Similar content being viewed by others
Abbreviations
- a, b, c, :
-
radii of spherical particles
- (r, θ, φ):
-
spherical polar coordinates
- vr,vθ :
-
velocity components of fluid
- p, :
-
fluid pressure
- σ̃rr :
-
normal stress
- σ̃rθ :
-
tangential stress
- U :
-
uniform velocity of fluid
- k 0 :
-
permeability parameter
- γ:
-
particle volume fraction
- λ:
-
viscosity ratio
References
DARCY, H. P. G. Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris (1856)
JONES, I. P. Low Reynolds number flow past a porous spherical shell. Proceedings of the Cambridge Philosophical Society, 73, 231–238 (1973)
RAJA-SEKHAR, G. P. and AMARANATH, T. Stokes flow inside a porous spherical shell. Journal of Applied Mathematics and Physics, 51(3), 481–490 (2000)
PADMAVATHI, B. S., AMARANATH, T., and PALANIAPPAN, D. Stokes flow about a porous spherical particle. Archives of Mechanics, 46(1-2), 191–199 (1994)
RAJA-SEKHAR, G. P. and AMARANATH, T. Stokes flow past a porous sphere with an impermeable core. Mechanics Research Communications, 23(5), 449–460 (1996)
HAMDAN, M. H. and KAMEL, M. T. Flow through variable permeability porous layers. Advances in Theoretical and Applied Mechanics, 4(3), 135–145 (2011)
BEAVERS, G. S. and JOSEPH, D. D. Boundary condition at naturally permeable wall. Journal of Fluid Mechanics, 30, 197–207 (1967)
OCHOA-TAPIA, J. A. and WHITAKER, S. Momentum transfer at the boundary between a porous medium and a homogenous fluid I: theoretical development. International Journal of Heat and Mass Transfer, 38, 2635–2646 (1995)
OCHOA-TAPIA, J. A. and WHITAKER, S. Momentum transfer at the boundary between a porous medium and a homogenous fluid II: comparison with experiment. International Journal of Heat and Mass Transfer, 38, 2647–2655 (1995)
BRINKMAN, H. C. A calculation of viscous force exerted by a flowing fluid on a dense swarm of particles. Journal of Applied Sciences Research, 1, 27–34 (1947)
YADAV, P. K., TIWARI, A., DEO, S., FILIPPOV, A. N., and VASIN, S. I. Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition. Acta Mechanica, 215, 193–209 (2010)
YADAV, P. K., DEO, S., YADAV, M. K., and FILIPPOV, A. N. On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles. Colloid Journal, 75(5), 611–622 (2013)
SRINIVASACHARYA, D. and PRASAD, M. K. On the motion of a porous spherical shell in a bounded medium. Advances in Theoretical and Applied Mechanics, 5(6), 247–256 (2012)
SRINIVASACHARYA, D. and PRASAD, M. K. Axi-symmetric motion of a porous approximate sphere in an approximate spherical container. Archives of Mechanics, 65(6), 485–509 (2013)
SAFFMAN, P. G. On the boundary condition at the surface of a porous medium. Studies in Applied Mathematics, 50(2), 93–101 (1971)
EHLERS, W. and WAGNER, A. Modelling and simulation methods applied to coupled problems in porous-media mechanics. Archive of Applied Mechanics, 89, 609–628 (2019)
RAMKISSOON, H. and RAHAMAN, K. Non-Newtonian fluid sphere in a spherical container. Acta Mechanica, 149, 239–245 (2001)
RAMKISSOON, H. and RAHAMAN, K. Wall effect on a spherical particle. International Journal of Engineering Science, 41, 283–290 (2003)
SAAD, E. I. Translation and rotation of a porous spheroid in a spheroidal container. Canadian Journal of Physics, 88, 689–700 (2010)
SRINIVASACHARYA, D. Motion of a porous sphere in a spherical container. Comptes Rendus Mecanique, 333, 612–616 (2005)
CHERNYSHEV, I. V. The Stokes problem for a porous particle with radially nonuniform porosity. Fluid Dynamics, 35(1), 147–152 (2000)
VASIN, S. I. and KHARITONOVA, T. V. Uniform liquid flow around porous spherical capsule. Colloid Journal, 73(1), 18–23 (2011)
UMAVATHI, J. C., LIU, I. C., PRATHAP-KUMAR, J., and SHAIK-MEERA, D. Unsteady flow and heat transfer of porous media sandwiched between viscous fluids. Applied Mathematics and Mechanics (English Edition), 31(12), 1497–1516 (2010) https://doi.org/10.1007/sl0483-010-1379-6
YADAV, P. K., JAISWAL, S., and SHARMA, B. D. Mathematical model of micropo-lar fluid in two-phase immiscible fluid flow through porous channel. Applied Mathematics and Mechanics (English Edition), 39(7), 993–1006 (2018) https://doi.org/10.1007/s10483-018-2351-8
LI, Q. X., PAN, M., ZHOU, Q., and DONG, Y. H. Drag reduction of turbulent channel flows over an anisotropic porous wall with reduced spanwise permeability. Applied Mathematics and Mechanics (English Edition), 40(7), 1041–1052 (2019) https://doi.org/10.1007/sl0483-019-2500-8
YADAV, P. K., SINGH, A., TIWARI, A., and DEO, S. Stokes flow through a membrane built up by nonhomogeneous porous cylindrical particles. Journal of Applied Mechanics and Technical Physics, 60(5), 816–826 (2019)
HAPPEL, J. and BRENNER, H. Low Reynolds Number Hydrodynamics, Springer, Dordrecht (1983)
VASIN, S. I., FILIPPOV, A. N., and STAROV, V. M. Hydrodynamic permeability of membranes built up by particles covered by porous shells: cell model. Advances in Colloid and Interface Science, 139, 83–96 (2008)
DATTA, S. and DEO, S. Stokes flow with slip and Kuwabara boundary conditions. Proceeding of the Indian Academy of Sciences (Mathematical Sciences), 112(3), 463–475 (2002)
BEAVERS, G. S. and JOSEPH, D. D. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30, 197–207 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Science and Engineering Research Board, New Delhi (No. SR/FTP/MS-47/2012)
Rights and permissions
About this article
Cite this article
Yadav, P.K., Tiwari, A. & Singh, P. Motion through spherical droplet with non-homogenous porous layer in spherical container. Appl. Math. Mech.-Engl. Ed. 41, 1069–1082 (2020). https://doi.org/10.1007/s10483-020-2628-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-020-2628-8