Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface in Presence of Magnetic Field

Affiliation(s)

Department of Mathematics, Government First Grade College, Yellapur, India.

Department of Mathematics, Manipal Institute of Technology, Manipal, India.

Department of Mathematics, Maharani’s Science College for Women, Bangalore, India.

Department of Mathematics, Government First Grade College, Yellapur, India.

Department of Mathematics, Manipal Institute of Technology, Manipal, India.

Department of Mathematics, Maharani’s Science College for Women, Bangalore, India.

ABSTRACT

We study the stability of an interface between two fluids of different densities flowing parallel to each other in the presence of a transverse magnetic field. A simple theory based on fully developed flow approximations is used to de-rive the dispersion relation for the growth rate of KHI. We replace the effect of boundary layer with Beavers and Joseph slip condition. The dispersion relation is derived using suitable boundary and surface conditions and results are discussed graphically. The magnetic field is found to be stabilizing and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed. These are favorable to control the surface instabilities in many practical applications discussed in this paper.

We study the stability of an interface between two fluids of different densities flowing parallel to each other in the presence of a transverse magnetic field. A simple theory based on fully developed flow approximations is used to de-rive the dispersion relation for the growth rate of KHI. We replace the effect of boundary layer with Beavers and Joseph slip condition. The dispersion relation is derived using suitable boundary and surface conditions and results are discussed graphically. The magnetic field is found to be stabilizing and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed. These are favorable to control the surface instabilities in many practical applications discussed in this paper.

Cite this paper

K. Chavaraddi, N. Katagi and V. Awati, "Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface in Presence of Magnetic Field,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 564-570. doi: 10.4236/am.2012.36086.

K. Chavaraddi, N. Katagi and V. Awati, "Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface in Presence of Magnetic Field,"

References

[1] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Dover Publications, New York, 1961.

[2] J. F. Lyon, “The Electrodynamic Kelvin-Helmholtz Instability,” M.Sc Thesis, MIT, Cambridge, 1962.

[3] P. G. Drazin, “Kelvin-Helmholtz Instability of Finite Amplitude,” Journal of Fluid Mechanics, Vol. 42, 1970, pp. 321-335. doi:10.1017/S0022112070001295

[4] M. A. Weissman, “Nonlinear Wave Packets in the Kelvin-Helmholtz Instability,” Philosophical Transactions of the Royal Society A, Vol. 290, No. 1377, 1979, pp. 639-681.

[5] D. Y. Hsieh and F. Chen, “A Nonlinear Study of the Kelvin-Helmholtz Instability,” Physics of Fluids, Vol. 28, 1985, p. 1253. doi:10.1063/1.865008

[6] A. R. F. Elhefnawy, “Nonlinear Electrohydrodynamic Kelvin-Helmholtz Instability under the Influence of an Oblique Electric Field,” Physica A, Vol. 182, 1992, pp. 419-435. doi:10.1016/0378-4371(92)90352-Q

[7] G. M. Moatimid, “Dynamic Instability of an Excited Horizontal Interface Supporting a Surface Charge and Admitting Mass and Heat Transfer,” International Journal of Engineering, Vol. 32, No. 3, 1994, pp. 535-543. doi:10.1016/0020-7225(94)90140-6

[8] G. M. Moatimid, “Dynamic Instability of an Excited Cylindrical Interface Supporting Surface Charges and Admitting Mass and Heat Transfer,” Journal of Physics D: Applied Physics, Vol. 27, No. 7, 1994, pp. 1390-1398. doi:10.1088/0022-3727/27/7/009

[9] R. C. Sharma and T. J. T. Spanos, “The Instability of Streaming Fluids in a Porous Medium,” Canadian Journal of Physics, Vol. 60, No. 10, 1982, pp. 1391-1395. doi:10.1139/p82-187

[10] R. C. Sharma and V. Kumari, “Hydromagnetic Instability of Streaming Fluids in Porous Medium,” Czechoslovak Journal of Physics, Vol. 41, No. 5, 1991, pp. 459-465. doi:10.1007/BF01597949

[11] R. C. Sharma and N. D. Sharma, “The Instability of Streaming Fluids with Fine Dust in Porous Medium,” Czechoslovak Journal of Physics, Vol. 42, 1992, pp. 907-918. doi:10.1007/BF01605167

[12] H. H. Bau, “Kelvin-Helmholtz Instability Parallel Flow in Porous Media; A Linear Theory,” Physics of Fluids, Vol. 25, No. 10, 1982, pp. 1719-1722. doi:10.1063/1.863642

[13] P. Kumar, “Rayeligh-Taylor of Viscous-Viscoelastic Fluids in Presence of Suspended Particles through Porous Medium,” Zeitschrift für Naturforschung, Vol. 51A, 1996, p. 17.

[14] M. F. El-Sayed, “Electrohydrodynamic Instability of Two Superposed Viscous Streaming Fluids through Porous Medium,” Canadian Journal of Physics, Vol. 75, No. 7, 1997, pp. 499-508.

[15] V. V. Gogosov and G. A. Shaposhnikova, “Electrohydrodynamics of Surface Phenomena,” International Journal of Applied Electromagnetics in Materials, Vol. 1, No. 1, 1990, pp. 45-48.

[16] J. R. Melcher, “Field Coupled Surface Waves,” MIT Press, Cambridge, 1963.

[17] J. R. Melcher, “Continuum Electromechanics,” MIT Press, Cambridge, 1981. doi:10.1016/0169-5983(89)90016-6

[18] A. A. Mohamed and E. F. Elshehawey, “Nonlinear Electrohydrodynamic Rayleigh-Taylor Instability,” Fluid Dynamic Research, Vol. 5, 1989, pp. 117-133. doi:10.1016/0377-0427(94)00048-6

[19] A. A. Mohamed, E. F. Elshehawey and M. F. El-Sayed, “Electrohydrodynamic Kelvin-Helmholtz Instability for a Velocity Stratified Fluid,” Journal of Computational and Applied Mathematics, Vol. 60, No. 3, 1995, pp. 331-346. doi:10.1139/p97-008

[20] M. F. El-Sayed, “EHD KHI in Viscous Porous Medium Permeated with Suspended Particles,” Czechoslovak Journal of Physics, Vol. 49, No. 4, 1999, p. 473. doi:10.1023/A:1022864808337

[21] K. Zakaria, “Nonlinear Kelvin-Helmholtz Instability of a Subsonic Gas-Liquid Interface in the Presence of a Normal Magnetic Field,” Physica A, Vol. 273, No. 3, 1999, pp. 248-271. doi:10.1016/S0378-4371(99)00201-0

[22] M. F. El-Sayed, “Effect of Variable Magnetic Field on the Stability of a Stratified Rotating Fluid Layer in Porous Medium,” Czechoslovak Journal of Physics, Vol. 50, 2002, p. 607. doi:10.1023/A:1022854217365

[23] P. K. Bhatia and A. Sharma, “KHI of Two Viscous Superposed Conducting Fluids,” Proceedings of the National Academy of Sciences, Vol. 73(A) , No. 4, 2003, p. 497.

[24] A. J. Babchin, A. L. Frenkel, B. G. Levich and G. I. Shivashinsky, “Nonlinear Saturation of Rayleigh-Taylor Instability in Thin Films,” Physics of Fluids, Vol. 26, 1983, pp. 3159-3161. doi:10.1063/1.864083

[25] N. Rudraiah, R. D. Mathad and H. Betigeri, “The RTI of Viscous Fluid Layer with Viscosity Stratification,” Current Science, Vol. 72, No. 6, 1997, p. 391.

[26] G. S. Beavers and D. D. Joseph, “Boundary Conditions at a Naturally Permeable Wall,” Journal of Fluid Mechanics, Vol. 30, No. 1, 1967, pp. 197-207. doi:10.1017/S0022112067001375

[27] Y. O. El-Dib and R. T. Matoog, “Electrorheological Kelvin-Helmholtz Instability of a Fluid Sheet,” Journal of Colloid and Interface Science, Vol. 289, No. 1, 2005, pp. 223-241. doi:10.1016/j.jcis.2005.03.054

[28] R. Asthana and G. S. Agrawal, “Viscous Potential Flow Analysis of Kelvin-Helmholtz Instability with Mass Transfer and Vaporization,” Physica A, Vol. 382, 2007, pp. 389-404. doi:10.1016/j.physa.2007.04.037

[29] A. E. Khalil Elcoot, “New Analytical Approximation Forms Fornon-Linear Instability of Electric Porous Media,” International Journal of Non-Linear Mechanics, Vol. 45, No. 1, 2010, pp. 1-11. doi:10.1016/j.ijnonlinmec.2009.08.011

[30] K. B. Chavaraddi, N. N. Katagi and N. P. Pai, “Electrohydrodynamic Kelvin-Helmholtz Instability in a Fluid Layer Bounded above by a Porous Layer and below by a Rigid Surface,” International Journal of Engineering and Technoscience, Vol. 2, No. 4, 2011, pp. 281-288.

[1] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Dover Publications, New York, 1961.

[2] J. F. Lyon, “The Electrodynamic Kelvin-Helmholtz Instability,” M.Sc Thesis, MIT, Cambridge, 1962.

[3] P. G. Drazin, “Kelvin-Helmholtz Instability of Finite Amplitude,” Journal of Fluid Mechanics, Vol. 42, 1970, pp. 321-335. doi:10.1017/S0022112070001295

[4] M. A. Weissman, “Nonlinear Wave Packets in the Kelvin-Helmholtz Instability,” Philosophical Transactions of the Royal Society A, Vol. 290, No. 1377, 1979, pp. 639-681.

[5] D. Y. Hsieh and F. Chen, “A Nonlinear Study of the Kelvin-Helmholtz Instability,” Physics of Fluids, Vol. 28, 1985, p. 1253. doi:10.1063/1.865008

[6] A. R. F. Elhefnawy, “Nonlinear Electrohydrodynamic Kelvin-Helmholtz Instability under the Influence of an Oblique Electric Field,” Physica A, Vol. 182, 1992, pp. 419-435. doi:10.1016/0378-4371(92)90352-Q

[7] G. M. Moatimid, “Dynamic Instability of an Excited Horizontal Interface Supporting a Surface Charge and Admitting Mass and Heat Transfer,” International Journal of Engineering, Vol. 32, No. 3, 1994, pp. 535-543. doi:10.1016/0020-7225(94)90140-6

[8] G. M. Moatimid, “Dynamic Instability of an Excited Cylindrical Interface Supporting Surface Charges and Admitting Mass and Heat Transfer,” Journal of Physics D: Applied Physics, Vol. 27, No. 7, 1994, pp. 1390-1398. doi:10.1088/0022-3727/27/7/009

[9] R. C. Sharma and T. J. T. Spanos, “The Instability of Streaming Fluids in a Porous Medium,” Canadian Journal of Physics, Vol. 60, No. 10, 1982, pp. 1391-1395. doi:10.1139/p82-187

[10] R. C. Sharma and V. Kumari, “Hydromagnetic Instability of Streaming Fluids in Porous Medium,” Czechoslovak Journal of Physics, Vol. 41, No. 5, 1991, pp. 459-465. doi:10.1007/BF01597949

[11] R. C. Sharma and N. D. Sharma, “The Instability of Streaming Fluids with Fine Dust in Porous Medium,” Czechoslovak Journal of Physics, Vol. 42, 1992, pp. 907-918. doi:10.1007/BF01605167

[12] H. H. Bau, “Kelvin-Helmholtz Instability Parallel Flow in Porous Media; A Linear Theory,” Physics of Fluids, Vol. 25, No. 10, 1982, pp. 1719-1722. doi:10.1063/1.863642

[13] P. Kumar, “Rayeligh-Taylor of Viscous-Viscoelastic Fluids in Presence of Suspended Particles through Porous Medium,” Zeitschrift für Naturforschung, Vol. 51A, 1996, p. 17.

[14] M. F. El-Sayed, “Electrohydrodynamic Instability of Two Superposed Viscous Streaming Fluids through Porous Medium,” Canadian Journal of Physics, Vol. 75, No. 7, 1997, pp. 499-508.

[15] V. V. Gogosov and G. A. Shaposhnikova, “Electrohydrodynamics of Surface Phenomena,” International Journal of Applied Electromagnetics in Materials, Vol. 1, No. 1, 1990, pp. 45-48.

[16] J. R. Melcher, “Field Coupled Surface Waves,” MIT Press, Cambridge, 1963.

[17] J. R. Melcher, “Continuum Electromechanics,” MIT Press, Cambridge, 1981. doi:10.1016/0169-5983(89)90016-6

[18] A. A. Mohamed and E. F. Elshehawey, “Nonlinear Electrohydrodynamic Rayleigh-Taylor Instability,” Fluid Dynamic Research, Vol. 5, 1989, pp. 117-133. doi:10.1016/0377-0427(94)00048-6

[19] A. A. Mohamed, E. F. Elshehawey and M. F. El-Sayed, “Electrohydrodynamic Kelvin-Helmholtz Instability for a Velocity Stratified Fluid,” Journal of Computational and Applied Mathematics, Vol. 60, No. 3, 1995, pp. 331-346. doi:10.1139/p97-008

[20] M. F. El-Sayed, “EHD KHI in Viscous Porous Medium Permeated with Suspended Particles,” Czechoslovak Journal of Physics, Vol. 49, No. 4, 1999, p. 473. doi:10.1023/A:1022864808337

[21] K. Zakaria, “Nonlinear Kelvin-Helmholtz Instability of a Subsonic Gas-Liquid Interface in the Presence of a Normal Magnetic Field,” Physica A, Vol. 273, No. 3, 1999, pp. 248-271. doi:10.1016/S0378-4371(99)00201-0

[22] M. F. El-Sayed, “Effect of Variable Magnetic Field on the Stability of a Stratified Rotating Fluid Layer in Porous Medium,” Czechoslovak Journal of Physics, Vol. 50, 2002, p. 607. doi:10.1023/A:1022854217365

[23] P. K. Bhatia and A. Sharma, “KHI of Two Viscous Superposed Conducting Fluids,” Proceedings of the National Academy of Sciences, Vol. 73(A) , No. 4, 2003, p. 497.

[24] A. J. Babchin, A. L. Frenkel, B. G. Levich and G. I. Shivashinsky, “Nonlinear Saturation of Rayleigh-Taylor Instability in Thin Films,” Physics of Fluids, Vol. 26, 1983, pp. 3159-3161. doi:10.1063/1.864083

[25] N. Rudraiah, R. D. Mathad and H. Betigeri, “The RTI of Viscous Fluid Layer with Viscosity Stratification,” Current Science, Vol. 72, No. 6, 1997, p. 391.

[26] G. S. Beavers and D. D. Joseph, “Boundary Conditions at a Naturally Permeable Wall,” Journal of Fluid Mechanics, Vol. 30, No. 1, 1967, pp. 197-207. doi:10.1017/S0022112067001375

[27] Y. O. El-Dib and R. T. Matoog, “Electrorheological Kelvin-Helmholtz Instability of a Fluid Sheet,” Journal of Colloid and Interface Science, Vol. 289, No. 1, 2005, pp. 223-241. doi:10.1016/j.jcis.2005.03.054

[28] R. Asthana and G. S. Agrawal, “Viscous Potential Flow Analysis of Kelvin-Helmholtz Instability with Mass Transfer and Vaporization,” Physica A, Vol. 382, 2007, pp. 389-404. doi:10.1016/j.physa.2007.04.037

[29] A. E. Khalil Elcoot, “New Analytical Approximation Forms Fornon-Linear Instability of Electric Porous Media,” International Journal of Non-Linear Mechanics, Vol. 45, No. 1, 2010, pp. 1-11. doi:10.1016/j.ijnonlinmec.2009.08.011

[30] K. B. Chavaraddi, N. N. Katagi and N. P. Pai, “Electrohydrodynamic Kelvin-Helmholtz Instability in a Fluid Layer Bounded above by a Porous Layer and below by a Rigid Surface,” International Journal of Engineering and Technoscience, Vol. 2, No. 4, 2011, pp. 281-288.