Existence and uniqueness result for the problem of viscous flow in a granular material with a void
Authors:
Mirela Kohr and G. P. Raja Sekhar
Journal:
Quart. Appl. Math. 65 (2007), 683-704
MSC (2000):
Primary 76D; Secondary 76M
DOI:
https://doi.org/10.1090/S0033-569X-07-01071-1
Published electronically:
August 28, 2007
MathSciNet review:
2370356
Full-text PDF Free Access
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Abstract: The purpose of this paper is to obtain an indirect boundary integral formulation for the three-dimensional viscous flow problem in a granular material with a void. The corresponding existence and uniqueness result of the classical solution to this problem is proved by using the theory of hydrodynamic potentials.
References
- G. S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197–207.
- V. Botte, H. Power, A second kind integral equation formulation for three-dimensional interior flows at low Reynolds number, Bound. Elem. Commun., 6 (1995), 163–166.
- F. J. Briceno, H. Power, The completed integral equation approach for the numerical solution of the motion of $N$ solid particles in the interior of a deformable viscous drop, Eng. Anal. Bound. Elem., 28 (2004), 367–413.
- A. D. H. Cheng, D. Ouazar, Ground Water Flow, Elsevier, 1993.
- W. D. Collins, Note on a sphere theorem for the axisymmetric Stokes flow of a viscous fluid, Mathematika 5 (1958), 118–121. MR 104424, DOI https://doi.org/10.1112/S0025579300001431
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- T. M. Fischer, G. C. Hsiao, and W. L. Wendland, On two-dimensional slow viscous flows past obstacles in a half-plane, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 3-4, 205–215. MR 877902, DOI https://doi.org/10.1017/S0308210500019181
- J. J. L. Higdon, M. Kojima, On the calculation of Stokes flow past porous particles, Multiphase Flow, 7 (1981), 719–727.
- I. P. Jones, Low Reynolds number flow past spherical shell, Proc. Camb. Phil. Soc., 73 (1973), 231–238.
- S. J. Karrila, S. Kim, Integral equations of the second kind for Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations, Chem. Engng. Commun., 82 (1989), 123–161.
- Mirela Kohr, An indirect boundary integral method for a Stokes flow problem, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 5-7, 487–497. MR 1800567, DOI https://doi.org/10.1016/S0045-7825%2800%2900240-1
- Mirela Kohr, An indirect boundary integral method for an oscillatory Stokes flow problem, Int. J. Math. Math. Sci. 47 (2003), 2961-2976. MR 2010743, DOI https://doi.org/10.1155/S016117120321231X
- M. Kohr, A mixed boundary value problem for the unsteady Stokes system in a bounded domain in $\mathbb {R}^n$, Engineering Analysis with Boundary Elements, 29 (2005), 936–943.
- M. Kohr, The Dirichlet problems for the unsteady Stokes system in bounded and exterior domains in $\mathbb {R}^n$, Mathematische Nachrichten, 280 (2007), 534–559.
- M. Kohr and I. Pop, Viscous incompressible flow for low Reynolds numbers, Advances in Boundary Elements, vol. 16, WIT Press, Southampton, 2004. MR 2091145
- Rainer Kress, Linear integral equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989. MR 1007594
- V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, Translated from the second Russian edition, North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1979. Edited by V. D. Kupradze. MR 530377
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
- James A. Liggett and Philip L.-F. Liu, The boundary integral equation method for porous media flow, George Allen & Unwin, London-Boston, Mass., 1983. MR 732790
- Paolo Maremonti, Remigio Russo, and Giulio Starita, On the Stokes equations: the boundary value problem, Advances in fluid dynamics, Quad. Mat., vol. 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999, pp. 69–140. MR 1770189
- G. Neale, N. Epstein, W. Nader, Creeping flow relative to permeable spheres, Chem. Eng. Sci., 28 (1973), 1865-1874. (Erratum: 1974, 29, 1352).
- J. A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid - I, Theoretical development, Int. J. Heat and Mass Transfer, 38 (1995), 2635–2646.
- J. A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid - II, Comparison with experiment, Int. J. Heat and Mass Transfer, 38 (1995), 2647–2655.
- Yu Qin and P. N. Kaloni, Creeping flow past a porous spherical shell, Z. Angew. Math. Mech. 73 (1993), no. 2, 77–84 (English, with English and German summaries). MR 1211619, DOI https://doi.org/10.1002/zamm.19930730207
- B. S. Padmavathi, T. Amaranath, and S. D. Nigam, Stokes flow past a porous sphere using Brinkman’s model, Z. Angew. Math. Phys. 44 (1993), no. 5, 929–939. MR 1241641, DOI https://doi.org/10.1007/BF00942818
- D. Palanippan, Arbitrary Stokes flow past a porous sphere, Mech. Res. Comm., 20 (1993), 309–317.
- Henry Power, The completed double layer boundary integral equation method for two-dimensional Stokes flow, IMA J. Appl. Math. 51 (1993), no. 2, 123–145. MR 1244192, DOI https://doi.org/10.1093/imamat/51.2.123
- Henry Power and Guillermo Miranda, Second kind integral equation formulation of Stokes’ flows past a particle of arbitrary shape, SIAM J. Appl. Math. 47 (1987), no. 4, 689–698. MR 898827, DOI https://doi.org/10.1137/0147047
- H. Power, L. C. Wrobel, Boundary Integral Methods in Fluid Mechanics, WIT Press, Computational Mechanics Publications, Southampton, 1995.
- O. Sano, Viscous flow past a cylidrical hole bored inside porous media - with application to measurement of the velocity of subterranean water by the single boring method, Nagare, 2 (1983), 252–259.
- G. P. Raja Sekhar, B. S. Padmavathi, and T. Amaranath, Complete general solution of the Brinkman equations, Z. Angew. Math. Mech. 77 (1997), no. 7, 555–556. MR 1466445, DOI https://doi.org/10.1002/zamm.19970770716
- G. P. Raja Sekhar and T. Amaranath, Stokes flow inside a porous spherical shell, Z. Angew. Math. Phys. 51 (2000), no. 3, 481–490. MR 1762703, DOI https://doi.org/10.1007/s000330050009
- G. P. Raja Sekhar, O. Sano, Viscous flow past a circular/spherical void in porous medium-an application to measurement of the velocity of ground water by single boring method, J. Phys. Soc. Jpn., 69 (2000), 37–42.
- G. P. Raja Sekhar, O. Sano, Two-dimensional viscous flow past a slightly deformed circular cavity in a porous medium, Fluid. Dyn. Res., 28 (2001), 281–293.
- G. P. Raja Sekhar, O. Sano, Two-dimensional viscous flow in a granular material with a void of arbitrary shape, Physics of Fluids, 15 (2003), 554–567.
- G. P. Raja Sekhar, M. K. Partha, P. V. S. N. Murthy, Viscous flow past a spherical void in porous media - Effect of stress jump boundary condition, J. Porous Media, 9 (2006), 745–767.
- K. B. Ranger, The Stokes drag for asymmetric flow past a spherical cap, Z. Angew. Math. Phys., 24 (1973), 801–809.
- R. Shail, A note on some asymmetric Stokes flows within a sphere, Quart. J. Mech. Appl. Math. 40 (1987), no. 2, 223–233. MR 894200, DOI https://doi.org/10.1093/qjmam/40.2.223
- Werner Varnhorn, An explicit potential theory for the Stokes resolvent boundary value problems in three dimensions, Manuscripta Math. 70 (1991), no. 4, 339–361. MR 1092141, DOI https://doi.org/10.1007/BF02568383
- Werner Varnhorn, The Stokes equations, Mathematical Research, vol. 76, Akademie-Verlag, Berlin, 1994. MR 1282728
- Werner Varnhorn, The boundary value problems of the Stokes resolvent equations in $n$ dimensions, Math. Nachr. 269/270 (2004), 210–230. MR 2074782, DOI https://doi.org/10.1002/mana.200310173
- Josef Wloka, Funktionalanalysis und Anwendungen, Walter de Gruyter, Berlin-New York, 1971 (German). de Gruyter Lehrbuch. MR 0467224
- G. K. Youngren and A. Acrivos, Stokes flow past a particle of arbitrary shape: a numerical method of solution, J. Fluid Mech. 69 (1975), 377–403. MR 398289, DOI https://doi.org/10.1017/S0022112075001486
References
- G. S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197–207.
- V. Botte, H. Power, A second kind integral equation formulation for three-dimensional interior flows at low Reynolds number, Bound. Elem. Commun., 6 (1995), 163–166.
- F. J. Briceno, H. Power, The completed integral equation approach for the numerical solution of the motion of $N$ solid particles in the interior of a deformable viscous drop, Eng. Anal. Bound. Elem., 28 (2004), 367–413.
- A. D. H. Cheng, D. Ouazar, Ground Water Flow, Elsevier, 1993.
- W. D. Collins, Note on a sphere theorem for the axisymmetric Stokes flow of a viscous fluid, Mathematika, 5 (1958), 118–121. MR 0104424 (21:3179)
- L. C. Evans, Partial Differential Equations, American Mathematical Society, vol. 19, Providence, 2002. MR 1625845 (99e:35001)
- T. M. Fischer, G. C. Hsiao, W. L. Wendland, On two-dimensional slow viscous flows past obstacles in a half-plane, Proc. Royal Society of Edinburgh, 104A (1986) 205–215. MR 877902 (88c:76022)
- J. J. L. Higdon, M. Kojima, On the calculation of Stokes flow past porous particles, Multiphase Flow, 7 (1981), 719–727.
- I. P. Jones, Low Reynolds number flow past spherical shell, Proc. Camb. Phil. Soc., 73 (1973), 231–238.
- S. J. Karrila, S. Kim, Integral equations of the second kind for Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations, Chem. Engng. Commun., 82 (1989), 123–161.
- M. Kohr, An indirect boundary integral method for a Stokes flow problem, Comput. Methods Appl. Mech. Engrg., 190 (2000), 487–497. MR 1800567 (2001j:76077)
- M. Kohr, An indirect boundary integral method for an oscillatory Stokes flow problem, Int. J. Math. Math. Sci., 47 (2003), 2961–2976. MR 2010743 (2004h:76059)
- M. Kohr, A mixed boundary value problem for the unsteady Stokes system in a bounded domain in $\mathbb {R}^n$, Engineering Analysis with Boundary Elements, 29 (2005), 936–943.
- M. Kohr, The Dirichlet problems for the unsteady Stokes system in bounded and exterior domains in $\mathbb {R}^n$, Mathematische Nachrichten, 280 (2007), 534–559.
- M. Kohr, I. Pop, Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press, Southampton, UK, 2004. MR 2091145 (2005k:76032)
- R. Kress, Linear Integral Equations, Springer, Berlin, 1989. MR 1007594 (90j:45001)
- V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North Holland, Amsterdam, 1979. MR 530377 (80h:73002)
- O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1969. MR 0254401 (40:7610)
- J. A. Liggett, P. L. F. Liu, The Boundary Integral Equation Method for Porous Media Flow, George Allen and Unwinn, London 1983. MR 732790 (84m:76086)
- P. Maremonti, R. Russo, G. Starita, On the Stokes equations: the boundary value problem. In: Advances in Fluid Dynamics, Quad. Mat. (Aracne, Rome), 1999, 69–140. MR 1770189 (2001g:35212)
- G. Neale, N. Epstein, W. Nader, Creeping flow relative to permeable spheres, Chem. Eng. Sci., 28 (1973), 1865-1874. (Erratum: 1974, 29, 1352).
- J. A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid - I, Theoretical development, Int. J. Heat and Mass Transfer, 38 (1995), 2635–2646.
- J. A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid - II, Comparison with experiment, Int. J. Heat and Mass Transfer, 38 (1995), 2647–2655.
- Y. U. Qin, P. N. Kaloni, Creeping flow past a porous spherical shell, Z. Angew. Math. Mech., 73 (1993), 77–84. MR 1211619 (93m:76024)
- B. S. Padmavathi, T. Amaranath, S. D. Nigam, Stokes flow past a porous sphere using Brinkman’s model, Z. Angew. Math. Phys., 44 (1993), 929–939. MR 1241641 (94h:76022)
- D. Palanippan, Arbitrary Stokes flow past a porous sphere, Mech. Res. Comm., 20 (1993), 309–317.
- H. Power, The completed double layer boundary integral equation method for two-dimensional Stokes flow, IMA J. Appl. Math., 51 (1993), 123–145. MR 1244192 (94i:76020)
- H. Power, G. Miranda, Second kind integral equation formulation of Stokes flows past a particle of arbitrary shape, SIAM J. Appl. Math., 47 (1987), 689-698. MR 898827 (88h:76021)
- H. Power, L. C. Wrobel, Boundary Integral Methods in Fluid Mechanics, WIT Press, Computational Mechanics Publications, Southampton, 1995.
- O. Sano, Viscous flow past a cylidrical hole bored inside porous media - with application to measurement of the velocity of subterranean water by the single boring method, Nagare, 2 (1983), 252–259.
- G. P. Raja Sekhar, B. S. Padmavathi, T. Amaranath, Complete general solution of the Brinkman Equations, Z. Angew. Math. Mech., 77 (1997), 555–556. MR 1466445
- G. P. Raja Sekhar, T. Amaranath, A Stokes flow inside a porous spherical shell, Z. Angew. Math. Phys., 51 (2000), 481–490. MR 1762703 (2001a:76047)
- G. P. Raja Sekhar, O. Sano, Viscous flow past a circular/spherical void in porous medium-an application to measurement of the velocity of ground water by single boring method, J. Phys. Soc. Jpn., 69 (2000), 37–42.
- G. P. Raja Sekhar, O. Sano, Two-dimensional viscous flow past a slightly deformed circular cavity in a porous medium, Fluid. Dyn. Res., 28 (2001), 281–293.
- G. P. Raja Sekhar, O. Sano, Two-dimensional viscous flow in a granular material with a void of arbitrary shape, Physics of Fluids, 15 (2003), 554–567.
- G. P. Raja Sekhar, M. K. Partha, P. V. S. N. Murthy, Viscous flow past a spherical void in porous media - Effect of stress jump boundary condition, J. Porous Media, 9 (2006), 745–767.
- K. B. Ranger, The Stokes drag for asymmetric flow past a spherical cap, Z. Angew. Math. Phys., 24 (1973), 801–809.
- R. Shail, A note on some axisymmetric Stokes flow within a sphere, QJMAM, 40 (1987), 223–233. MR 894200 (88c:76025)
- W. Varnhorn, An explicit potential theory for the Stokes resolvent boundary value problems in three dimensions, Manuscripta Math., 70 (1991), 339–361. MR 1092141 (91k:31011)
- W. Varnhorn, The Stokes Equations, Akademie Verlag, Berlin, 1994. MR 1282728 (95e:35162)
- W. Varnhorn, The boundary value problems of the Stokes resolvent equations in $n$ dimensions, Math. Nachr., 269-270 (2004), 210-230. MR 2074782 (2005e:35192)
- J. Wloka, Funktionalanalysis und Anwendungen, de Gruyter, 1971. MR 0467224 (57:7088)
- G. K. Youngren, A. Acrivos, Stokes flow past a particle of arbitrary shape: a numerical method of solution, J. Fluid Mech., 69 (1975), 377-403. MR 0398289 (53:2142)
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Additional Information
Mirela Kohr
Affiliation:
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania
Email:
mkohr@math.ubbcluj.ro
G. P. Raja Sekhar
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India
Email:
rajas@maths.iitkgp.ernet.in
Keywords:
Stokes equation,
Brinkman equation,
boundary value problems,
fundamental solution,
potential theory,
boundary integral representations,
existence and uniqueness result
Received by editor(s):
April 17, 2006
Published electronically:
August 28, 2007
Article copyright:
© Copyright 2007
Brown University