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1Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo, Egypt
2Department of Physics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo, Egypt
?Corresponding author.
Address: Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Buraidah 51452, Saudi Arabia Email: mfahmye@yahoo.com
Received September 10, 2011; accepted January 11, 2012
Abstract
The instability of two superposed homogeneous streamingfluids is discussed under gravitational force and uniform magneticfield in porous medium. The two streams are moving in opposite directions with equal velocities parallel to the horizontal plane. The solution has been obtained through the normal mode technique, and the most general dispersion relation has been obtained as 20th-orderequation for the growth rate with quite complicated coefficients. Solving numerically the dispersion relation for appropriate boundary conditions with high Alfv′en and sound velocities, it is found thatfluid velocities and porosity of porous medium have stabilizing effects, and Alfv′en and sound velocities have destabilizing effects, while medium permeability has a slightly stabilizing effect, and the dynamic viscosities have slightly destabilizing effect. The limiting cases of non-porous medium have also been studied for both streaming and stationaryfluids.
Key words
Hydrodynamic stability; Conductingfluids; Flows through porous media; Gravitational force; Magnetohydrodynamics M. F. El-Sayed; D. F. Hussein (2012). Magnetogravitational Vortex-Sheet Instability of Two Superposed Conducting Fluids in Porous Medium Under Strong Magnetic Field. Progress in Applied Mathematics, 3(1), 1-15. Available from: URL: http://www.cscanada.net/ index.php/pam/article/view/j.pam.1925252820120301.1665DOI: http://dx.doi.org/10.3968/j.pam.1925252820120301.1665
Note that, in the limiting case of non-porous medium, i.e. when k1→∞or Lr =σr (r = 1, 2), Eqs. (24)-(26), and the boundary conditions (30)-(33), reduce to the same equations obtained earlier by Singh and Khare[20], and their results are, therefore, recovered. They obtained their dispersion relation, and they not discussed the stability analysis for this general dispersion relation, but discussed the stability conditions only for some of its limiting cases. Here, in the present work, we shall obtain the general dispersion relation for the considered system including the effect of porous medium, and discuss the effects of various parameters on the stability of the system due to the obtained dispersion relation in its general form in presence (or absence) of porous medium andfluid velocities.
Numerical solutions, in the presence of porous medium, may lead to the values which are related to the instability criterion for physical problem. In order to study the effects of various physical parameters on the growth rate instability, we have performed numerical calculations of the dispersion relation (39), using Mathematica 9, to locate the roots of the growth rate n againt the wave number k for various values of the parameters included in the analysis. These calculations are presented in Figs. (1)-(6) to show the variation of the growthrate with wave numberof the consideredsystem for differentvalues of Alfv′en velocities,fluid viscosities, sound velocities, medium permeability, streaming velocity, and the porosity of porous medium, respectively.
Figure 5
Variation of growth rate n with wave number k for various values of the stream velocity V in the systemρ1 = 0.01,ρ2 = 0.02, k1 = 0.5, M1 = 100, M2 = 150, G = 6.6×10?11,ε= 0.3,μ1= 0.1,μ2= 0.2, C1= 100, C2= 150, for the cases V = 100, 120 and 150
Fig. (5) shows the variationof the negativereal part of growthrate n with the wave numberk for various values of streaming velocity V. It is clear from thisfigure that, for wave number values k≥0.75, the negative Re(n) increases by increasing the streaming velocity, which indicates that the streaming velocity for the medium V has a stabilizing effect. Note that, for very small wave number values 0 < k < 0.75, the obtained curves are coincide, and this means that the streaming velocity has no effect on the stability of the considered system in this wave numbers range. It is seen also from Fig. (5) that, forfixed value of V, the negative Re(n) decreases by increasing the wave nymber k till afixed critical wave number value after which the negative Re(n) increases for higher wave number values, which indicates that the system is unstable for small wave number values and then it is stable afterwards. Note that this critical wave number values increases by increasing the streaming velocity values.
6.CONCLUSIONS
In this paper, we have studied the hydromagnetic instability of two superposed homogeneous conducting fluids streaming in opposite directions with equal horizontal velocities under gravitational force and high Alfv′enand sound velocities in porousmedium. The perturbationpropagationis taken simultaneouslyalong and perpendicular to streaming motion in the horizontal interface. The obtained results can be summarized as follows:
(1) The Alfv′en and sound velocities have destabilizing effects.
(2) Thefluid velocities and porosity of porous medium have stabilizing effects.
(3)Themediumpermeabilityhas aslightlystabilizingeffect, whilethedynamicviscositieshaveslightly destabilizing effect.
(4) The growth rate varies linearly with wave number for differents values of medium permeability or dynamic viscosities.
(5) Finally, the limiting cases of non-porous medium have also been studied for both streaming and stationaryfluids, and show that the critical wave numbers lying on elliptic orbit and a circular path in the first quadrant, respectively.
ACKNOWLEDGMENTS
We would like to thank Prof. M. A. Kamel (Egypt) for his critical reading of the manuscript and his useful comments that improved its original version.
[1]Boller, B. R., & Stolov, H. L. (1970). Explorer 18 Study of the Stability of the Magnetopause Using a Kelvin-Helmholtz Instability Criterion. J. Geophys. Res., 78, 8078-8086.
[2]Ray, T. P. (1982). Kelvin-Helmholtz Instabilities in Cometary Ion Tails. Planet. Space Sci., 30, 245-250.
[3]Ferrari, A., Trussoni, E., & Zaninetti, L. (1978). Relativistic Kelvin-Helmholtz Instabilities in Extragalactic Radio Sources. Astron. Astrophys., 64, 43-52.
[4]Ray, T. P. (1981). Kelvin-Helmholtz Instabilities in Radio Jets. Mon. Notes R. Astron. Soc., 196, 195-207.
[5]Blondin, J. M., Fryxell, B. A., & K¨onrgl, A. (1990). The Structure and Evolution of Radiatively Cooling Jets. Astrophys. J., 360, 370-386.
[6]Bodo, G., Massaglia, S., Rossi, P., Trussoni, E., & Ferrari, A. (1993). Kelvin-Helmholtz Instabilities in Radiating Flows. Phys. Fluids, 5, 405-411.
[7]S. Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon.
[8]Sen, A. K. (1964). Effect of Compressibility on Kelvin-Helmholtz Instability in a Plasma. Phys. Fluids, 7, 1293-1298.
neous Streaming Dusty Plasma. Pramana J. Phys., 61, 109-120.
[31] Shukla, P. K., & Stenflo, L. (2006). Jeans Instability in a Self-Gravitating Dusty Plasma. Proc. Roy. Soc. London A, 462, 403-407.
[32] Yang, X.-S., Liu, S.-Q., & Li, X.-Q. (2010). Gravito-Magneto-ModulationalInstability of Self- Gravitating System. Gravitation and Cosmology, 16, 316-322.
[33] Pop, I., & Ingham, D. B. (2001). Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluid and Porous Medium, Oxford: Pergamon.
[34] Dullien, F. A. I. (1992). Porous Media, New York: Academic Press.
[35] Ingham, D. B., & I. Pop, I. (1998). Transport Phenomena in Porous Media, New York: SpringerVerlag.
[36] Nield, D. A., & A. Bejan, A. (2005). Convection in Porous Media, New York: Springer-Verlag.
[37] Vafai, K. (2005). Handbook of Porous Media, 2nd Ed., New York: Taylor and Francis.
[38] Raghavan, R., & Marsden, S. S. (1973). A Theoretical Study of the Instability in the Parallel Flow of Immiscible Liquids in a Porous Medium, Q. J. Mech. Appl. Math., 26, 205-216.
[39] Sharma, R. C., & Spanos, J. T. (1982). The Instability of Streaming Fluids in a Porous Medium. Can. J. Phys., 60, 1391-1395.
[40] Bau, H. H. (1982). Kelvin-Helmholtz Instability for Parallel Flow in Porous Media: A Linear Theory. Phys. Fluids, 25, 1719-1722.
[41] Melcher, J. R. (1981). Cintinuum Electromechanics, Mass.: MIT Press.
[42] Blokhin, A., & Trakhinin, Yu. (2003). Stability of Strong Discontinuities in Magnetohydrodynamics and Electrohydrodynamics. New York: Nova Sience Publishers.
[43] Geindreau,C., & Auriault,J. L. (2001).MagnetohydrodynamicFlowThroughPorousMedia.Comltes Rendus Acad. Sci. Ser. II B, 329, 445-450.
[44] Geindreau, C., & Auriault, J. L. (2002).MagnetohydrodynamicFlow ThroughPorous Media. J. Fluid Mech., 466, 343-363.
[45] Zakaria, Z., Sirwah, M. A., & Alkharashi, S. (2009). Instability Through Porous Media of Three Layers Superposed Conducting Fluids. Eur. J. Mech. B/Fluids, 28, 259-270.
2Department of Physics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo, Egypt
?Corresponding author.
Address: Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Buraidah 51452, Saudi Arabia Email: mfahmye@yahoo.com
Received September 10, 2011; accepted January 11, 2012
Abstract
The instability of two superposed homogeneous streamingfluids is discussed under gravitational force and uniform magneticfield in porous medium. The two streams are moving in opposite directions with equal velocities parallel to the horizontal plane. The solution has been obtained through the normal mode technique, and the most general dispersion relation has been obtained as 20th-orderequation for the growth rate with quite complicated coefficients. Solving numerically the dispersion relation for appropriate boundary conditions with high Alfv′en and sound velocities, it is found thatfluid velocities and porosity of porous medium have stabilizing effects, and Alfv′en and sound velocities have destabilizing effects, while medium permeability has a slightly stabilizing effect, and the dynamic viscosities have slightly destabilizing effect. The limiting cases of non-porous medium have also been studied for both streaming and stationaryfluids.
Key words
Hydrodynamic stability; Conductingfluids; Flows through porous media; Gravitational force; Magnetohydrodynamics M. F. El-Sayed; D. F. Hussein (2012). Magnetogravitational Vortex-Sheet Instability of Two Superposed Conducting Fluids in Porous Medium Under Strong Magnetic Field. Progress in Applied Mathematics, 3(1), 1-15. Available from: URL: http://www.cscanada.net/ index.php/pam/article/view/j.pam.1925252820120301.1665DOI: http://dx.doi.org/10.3968/j.pam.1925252820120301.1665
Note that, in the limiting case of non-porous medium, i.e. when k1→∞or Lr =σr (r = 1, 2), Eqs. (24)-(26), and the boundary conditions (30)-(33), reduce to the same equations obtained earlier by Singh and Khare[20], and their results are, therefore, recovered. They obtained their dispersion relation, and they not discussed the stability analysis for this general dispersion relation, but discussed the stability conditions only for some of its limiting cases. Here, in the present work, we shall obtain the general dispersion relation for the considered system including the effect of porous medium, and discuss the effects of various parameters on the stability of the system due to the obtained dispersion relation in its general form in presence (or absence) of porous medium andfluid velocities.
Numerical solutions, in the presence of porous medium, may lead to the values which are related to the instability criterion for physical problem. In order to study the effects of various physical parameters on the growth rate instability, we have performed numerical calculations of the dispersion relation (39), using Mathematica 9, to locate the roots of the growth rate n againt the wave number k for various values of the parameters included in the analysis. These calculations are presented in Figs. (1)-(6) to show the variation of the growthrate with wave numberof the consideredsystem for differentvalues of Alfv′en velocities,fluid viscosities, sound velocities, medium permeability, streaming velocity, and the porosity of porous medium, respectively.
Figure 5
Variation of growth rate n with wave number k for various values of the stream velocity V in the systemρ1 = 0.01,ρ2 = 0.02, k1 = 0.5, M1 = 100, M2 = 150, G = 6.6×10?11,ε= 0.3,μ1= 0.1,μ2= 0.2, C1= 100, C2= 150, for the cases V = 100, 120 and 150
Fig. (5) shows the variationof the negativereal part of growthrate n with the wave numberk for various values of streaming velocity V. It is clear from thisfigure that, for wave number values k≥0.75, the negative Re(n) increases by increasing the streaming velocity, which indicates that the streaming velocity for the medium V has a stabilizing effect. Note that, for very small wave number values 0 < k < 0.75, the obtained curves are coincide, and this means that the streaming velocity has no effect on the stability of the considered system in this wave numbers range. It is seen also from Fig. (5) that, forfixed value of V, the negative Re(n) decreases by increasing the wave nymber k till afixed critical wave number value after which the negative Re(n) increases for higher wave number values, which indicates that the system is unstable for small wave number values and then it is stable afterwards. Note that this critical wave number values increases by increasing the streaming velocity values.
6.CONCLUSIONS
In this paper, we have studied the hydromagnetic instability of two superposed homogeneous conducting fluids streaming in opposite directions with equal horizontal velocities under gravitational force and high Alfv′enand sound velocities in porousmedium. The perturbationpropagationis taken simultaneouslyalong and perpendicular to streaming motion in the horizontal interface. The obtained results can be summarized as follows:
(1) The Alfv′en and sound velocities have destabilizing effects.
(2) Thefluid velocities and porosity of porous medium have stabilizing effects.
(3)Themediumpermeabilityhas aslightlystabilizingeffect, whilethedynamicviscositieshaveslightly destabilizing effect.
(4) The growth rate varies linearly with wave number for differents values of medium permeability or dynamic viscosities.
(5) Finally, the limiting cases of non-porous medium have also been studied for both streaming and stationaryfluids, and show that the critical wave numbers lying on elliptic orbit and a circular path in the first quadrant, respectively.
ACKNOWLEDGMENTS
We would like to thank Prof. M. A. Kamel (Egypt) for his critical reading of the manuscript and his useful comments that improved its original version.
[1]Boller, B. R., & Stolov, H. L. (1970). Explorer 18 Study of the Stability of the Magnetopause Using a Kelvin-Helmholtz Instability Criterion. J. Geophys. Res., 78, 8078-8086.
[2]Ray, T. P. (1982). Kelvin-Helmholtz Instabilities in Cometary Ion Tails. Planet. Space Sci., 30, 245-250.
[3]Ferrari, A., Trussoni, E., & Zaninetti, L. (1978). Relativistic Kelvin-Helmholtz Instabilities in Extragalactic Radio Sources. Astron. Astrophys., 64, 43-52.
[4]Ray, T. P. (1981). Kelvin-Helmholtz Instabilities in Radio Jets. Mon. Notes R. Astron. Soc., 196, 195-207.
[5]Blondin, J. M., Fryxell, B. A., & K¨onrgl, A. (1990). The Structure and Evolution of Radiatively Cooling Jets. Astrophys. J., 360, 370-386.
[6]Bodo, G., Massaglia, S., Rossi, P., Trussoni, E., & Ferrari, A. (1993). Kelvin-Helmholtz Instabilities in Radiating Flows. Phys. Fluids, 5, 405-411.
[7]S. Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon.
[8]Sen, A. K. (1964). Effect of Compressibility on Kelvin-Helmholtz Instability in a Plasma. Phys. Fluids, 7, 1293-1298.
neous Streaming Dusty Plasma. Pramana J. Phys., 61, 109-120.
[31] Shukla, P. K., & Stenflo, L. (2006). Jeans Instability in a Self-Gravitating Dusty Plasma. Proc. Roy. Soc. London A, 462, 403-407.
[32] Yang, X.-S., Liu, S.-Q., & Li, X.-Q. (2010). Gravito-Magneto-ModulationalInstability of Self- Gravitating System. Gravitation and Cosmology, 16, 316-322.
[33] Pop, I., & Ingham, D. B. (2001). Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluid and Porous Medium, Oxford: Pergamon.
[34] Dullien, F. A. I. (1992). Porous Media, New York: Academic Press.
[35] Ingham, D. B., & I. Pop, I. (1998). Transport Phenomena in Porous Media, New York: SpringerVerlag.
[36] Nield, D. A., & A. Bejan, A. (2005). Convection in Porous Media, New York: Springer-Verlag.
[37] Vafai, K. (2005). Handbook of Porous Media, 2nd Ed., New York: Taylor and Francis.
[38] Raghavan, R., & Marsden, S. S. (1973). A Theoretical Study of the Instability in the Parallel Flow of Immiscible Liquids in a Porous Medium, Q. J. Mech. Appl. Math., 26, 205-216.
[39] Sharma, R. C., & Spanos, J. T. (1982). The Instability of Streaming Fluids in a Porous Medium. Can. J. Phys., 60, 1391-1395.
[40] Bau, H. H. (1982). Kelvin-Helmholtz Instability for Parallel Flow in Porous Media: A Linear Theory. Phys. Fluids, 25, 1719-1722.
[41] Melcher, J. R. (1981). Cintinuum Electromechanics, Mass.: MIT Press.
[42] Blokhin, A., & Trakhinin, Yu. (2003). Stability of Strong Discontinuities in Magnetohydrodynamics and Electrohydrodynamics. New York: Nova Sience Publishers.
[43] Geindreau,C., & Auriault,J. L. (2001).MagnetohydrodynamicFlowThroughPorousMedia.Comltes Rendus Acad. Sci. Ser. II B, 329, 445-450.
[44] Geindreau, C., & Auriault, J. L. (2002).MagnetohydrodynamicFlow ThroughPorous Media. J. Fluid Mech., 466, 343-363.
[45] Zakaria, Z., Sirwah, M. A., & Alkharashi, S. (2009). Instability Through Porous Media of Three Layers Superposed Conducting Fluids. Eur. J. Mech. B/Fluids, 28, 259-270.