Quantum Effects on the Rayleigh-Taylor Instability of Viscoelastic Plasma Model through a Porous Medium

Author(s)
Gamal A. Hoshoudy

ABSTRACT

The stability of stratified of incompressible, viscoelastic plasma through a porous medium in the presence of the quantum mechanism is considered. The dispersion relation is obtained using the normal mode technique. The behavior of growth rate with respect to the quantum effect, strain retardation time and stress relaxation time are examined in the presence of porosity of the porous medium, the medium permeability, kinematic viscosity. It is shown that, the presence of quantum term stabilizes a certain wave number band, whereas the system is unstable for all wave numbers in the absence of quantum term. The considered parameters beside the quantum term will bring about more stability on the considered system.

The stability of stratified of incompressible, viscoelastic plasma through a porous medium in the presence of the quantum mechanism is considered. The dispersion relation is obtained using the normal mode technique. The behavior of growth rate with respect to the quantum effect, strain retardation time and stress relaxation time are examined in the presence of porosity of the porous medium, the medium permeability, kinematic viscosity. It is shown that, the presence of quantum term stabilizes a certain wave number band, whereas the system is unstable for all wave numbers in the absence of quantum term. The considered parameters beside the quantum term will bring about more stability on the considered system.

Cite this paper

nullG. Hoshoudy, "Quantum Effects on the Rayleigh-Taylor Instability of Viscoelastic Plasma Model through a Porous Medium,"*Journal of Modern Physics*, Vol. 2 No. 10, 2011, pp. 1146-1155. doi: 10.4236/jmp.2011.210142.

nullG. Hoshoudy, "Quantum Effects on the Rayleigh-Taylor Instability of Viscoelastic Plasma Model through a Porous Medium,"

References

[1] L. Rayleigh, “Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density,” Proceedings of the London Mathematical Society, Vol. 14, 1882, pp. 170-177. doi:10.1112/plms/s1-14.1.170

[2] G. I. Taylor, “The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to Their Planes,” Proceedings of the Royal Society of London. Series A, Vol. 201, 1950, pp. 192-196. doi:10.1098/rspa.1950.0052

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

[4] J. G. Oldroyd, “On the Formulation of Rheological Equations of State,” Proceedings of the Royal Society of London. Series A, Vol. 200, 1950, pp. 523-541. doi:10.1098/rspa.1950.0035

[5] J. G. Oldroyd, “The Elastic and Viscous Properties of Emulsions and Suspensions,” Proceedings of the Royal Society of London. Series A, Vol. 218, No. 1132, 1953, pp. 122-132. doi:10.1098/rspa.1953.0092

[6] J. G. Oldroyd, “Non-Newtonian Effects in Steady Motion of Some Idealized Elastico-Viscous Liquids,” Proceedings of the Royal Society of London. Series A, Vol. 245, No. 1241, 1958, pp. 278-279. doi:10.1098/rspa.1958.0083

[7] R. C. Sharma and P. Kumar, “Rayleigh-Taylor Instability of Viscous-Viscoelastic Fluids through Porous Medium,” Indian Journal of Pure and Applied Mathematics, Vol. 24, No. 9, 1993, pp. 563-569.

[8] S. C. Agrawal and A. K. Goel, “Shear Flow Instability of Viscoelastic Fluid in a Porous Medium,” Indian Journal of Pure and Applied Mathematics, Vol. 29, 1998, pp. 969-981.

[9] P. K. Bhatia and R. P Mathur, “Instability of Viscoelastic Superposed Fluids in a Vertical Magnetic Field through Porous Medium,” Ganita Sandesh (India), Vol. 17, 2003, pp. 21-32.

[10] P. Kumar, H. Mohan and G. J. Singh, “Rayleigh—Taylor Instability of Rotating Oldroydian Viscoelastic Fluids in Porous Medium in Presence of a Variable Magnetic Field,” Transport in Porous Media, Vol. 56, No. 2, 2004, pp. 19-208. doi:10.1023/B:TIPM.0000021828.01346.57

[11] G. L. Kalra and S. P. Talwar, “Slipping Stream Instability of a Self-Gravitating Hydromagnetic Gas Cloud,” Monthly Notices of the Royal Astronomical Society, Vol. 135, 1967, pp. 391-398.

[12] S. Singh and J. N. Tnadon, “On the Rayleigh-Taylor Instability in Hydromagnetics with Finite Electrical Resistivity and Hall Current,” Journal of Plasma Physics, Vol. 3, No. 4, 1969, pp. 633-642. doi:10.1017/S0022377800004670

[13] P. K. Bhatia and IL Nuovo Cimento, “Rayleigh-Taylor Instability of Viscous Superposed Conducting Fluids,” Il Nuovo Cimento B (1971-1996), Vol. 19 B, No. 2, 1974, pp. 161-168.

[14] P. K. Bhatia and R. P. S. Chhonkar, “Rayleigh-Taylor Instability of Two Viscous Superposed Rotating and Conducting Fluids,” Astrophysics and Space Science, Vol. 114, No. 2, 1985, pp. 271-276. doi:10.1007/BF00653970

[15] P. K. Sharma and R. K. Chhajlani, “Effect of Rotation on the Rayleigh-Taylor Instability of Superposed Magnetic Conducting Plasma,” Physics of plasmas, Vol. 5, No. 6, 1998, pp. 2203-2209. doi:10.1063/1.872893

[16] A. Ali and P. K. Bhatia, “Rayleigh-Taylor Instability of a Stratified Hall Plasma in Two-Dimensional Horizontal Magnetic Field,” Physica Scripta, Vol. 47, 1993, pp. 567 -570. doi:10.1088/0031-8949/47/4/016

[17] A. Khan and P. K. Bhatia, “Gravitational Instability of a Rotating Fluid in an Oblique Magnetic Field,” Physica Scripta, Vol. 47, 1993, pp. 230-234. doi:10.1088/0031-8949/47/2/018

[18] R. J. Goldston and P. H. Rutherford, “Introduction to Plasma Physics,” Institute of Physics, London, 1997.

[19] P. K. Bhatia and A. Sharma, “Rayleigh-Taylor Instability of a Stratified Fluid/Plasma in an Inhomogeneous Magnetic Field,” Proceedings of the National Academy of Sciences, India. Section A. Physical, Vol. 68 (A) III, 1998, pp. 239-247.

[20] Z. Wu, W. Zhang, D. Li and W. Yang, “Effect of Magnetic Field and Equilibrium Flow on Rayleigh-Taylor Instability,” Chinese Physics Letters, Vol. 21, No. 10, 2004, pp. 2001-2004. doi:10.1088/0256-307X/21/10/038

[21] A. M. Al-Khateeb and N. M. Laham, “Magnetic Field Effects on the Rayleigh-Taylor Instability in Inhomogeneous Plasma,” The Arabian Journal for Science and Engineering, Vol. 27, No. 1A, 2002, pp. 75-83.

[22] A. M. Al-Khateeb and N. M. Laham, “Stability Criteria of Rayleigh-Taylor Modes in Magnetized Plasmas,” Contributions to Plasma Physics, Vol. 43, No. 1, 2003, pp. 25-32. doi:10.1002/ctpp.200310003

[23] D. Banerjee M. S. Janaki, N. Chakrabarti and M. Chaudhuri, “Viscosity Gradient-Driven Instability of ‘Shear Mode’ in a Strongly Coupled Plasma,” New Journal of Physics, Vol. 12, 2010, p. 123031. doi:10.1088/1367-2630/12/12/123031

[24] M. F. El-Sayed and R. A. Mohamed, “Gravitational Instability of Rotating Viscoelastic Partially Ionized Plasma in the Presence of an Oblique Magnetic Field and Hall Current,” International Scholarly Research Network ISRN Mechanical Engineering, 2011, Article ID 597172.

[25] G. Manfredi, “How to Model Quantum Plasmas,” Fields Institute Communications Series, Vol. 46, 2005, pp. 263- 287.

[26] C. Gardner, “The Quantum Hydrodynamic Model for Semiconductor Devices,” SIAM Journal on Applied Mathematics, Vol. 54, 1994, pp. 409-427. doi:10.1137/S0036139992240425

[27] F. Haas, “Quantum Magnetohydrodynamics,” Physics of Plasmas, Vol. 12, 2005, p. 062117. doi:10.1063/1.1939947

[28] B. Vitaly, M. Marklund and M. Modestov, “The Rayleigh-Taylor Instability and Internal Waves in Quantum Plasmas,” Physics Letters A, Vol. 372, No. 17, 2008, pp. 3042-3045. doi:10.1016/j.physleta.2007.12.065

[29] J. T. Cao, H. J. Ren, Z. W. Wu and P. K. Chu, “Quantum Effects on Rayleigh-Taylor Instability in Magnetized Plasma,” Physics of Plasmas, Vol. 15, 2008, p. 012110. doi:10.1063/1.2833588

[30] G. A. Hoshoudy, “Quantum Effects on Rayleigh-Taylor Instability in a Vertical Inhomogeneous Rotating Plasma,” Physics of Plasmas, Vol. 16, 2009, p. 024501. doi:10.1063/1.3080202

[31] G. A. Hoshoudy, “Quantum Effects on Rayleigh-Taylor Instability in a Horizontal Inhomogeneous Rotating Plas- ma,” Physics of Plasmas, Vol.16, 2009, p. 064501. doi:10.063/1.3140038

[32] M. Modestov, V. Bychkov and M. Marklund, “The Rayleigh-Taylor Instability in Quantum Magnetized Plasma with Para- and Ferromagnetic Properties,” Physics of Plasmas, Vol. 16, 2009, p. 032106. doi:10.1063/1.3085796

[33] G. A. Hoshoudy, “Quantum Effects on the Rayleigh- Taylor Instability of Stratified Fluid/Plasma through Porous Media,” Physics Letters A, Vol. 373, No. 30, 2009, pp. 2560-2567. doi:10.1016/j.physleta.2009.05.036

[34] S. Ali, Z. Ahmed, M. Arshad Mirza and I. Ahmad, “Rayleigh-Taylor/Gravitational Instability in Dense Mag- netoplasmas,” Physics Letters A, Vol. 373, 2009, pp. 2940-2946. doi:10.1016/j.physleta.2009.06.021

[35] L. C. Gardner, “The Quantum Hydrodynamic Model for Semiconductor Devices,” SIAM Journal on Applied Mathematics, Vol. 54, No. 2, 1994, pp. 409-427. doi:10.1137/S0036139992240425

[1] L. Rayleigh, “Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density,” Proceedings of the London Mathematical Society, Vol. 14, 1882, pp. 170-177. doi:10.1112/plms/s1-14.1.170

[2] G. I. Taylor, “The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to Their Planes,” Proceedings of the Royal Society of London. Series A, Vol. 201, 1950, pp. 192-196. doi:10.1098/rspa.1950.0052

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

[4] J. G. Oldroyd, “On the Formulation of Rheological Equations of State,” Proceedings of the Royal Society of London. Series A, Vol. 200, 1950, pp. 523-541. doi:10.1098/rspa.1950.0035

[5] J. G. Oldroyd, “The Elastic and Viscous Properties of Emulsions and Suspensions,” Proceedings of the Royal Society of London. Series A, Vol. 218, No. 1132, 1953, pp. 122-132. doi:10.1098/rspa.1953.0092

[6] J. G. Oldroyd, “Non-Newtonian Effects in Steady Motion of Some Idealized Elastico-Viscous Liquids,” Proceedings of the Royal Society of London. Series A, Vol. 245, No. 1241, 1958, pp. 278-279. doi:10.1098/rspa.1958.0083

[7] R. C. Sharma and P. Kumar, “Rayleigh-Taylor Instability of Viscous-Viscoelastic Fluids through Porous Medium,” Indian Journal of Pure and Applied Mathematics, Vol. 24, No. 9, 1993, pp. 563-569.

[8] S. C. Agrawal and A. K. Goel, “Shear Flow Instability of Viscoelastic Fluid in a Porous Medium,” Indian Journal of Pure and Applied Mathematics, Vol. 29, 1998, pp. 969-981.

[9] P. K. Bhatia and R. P Mathur, “Instability of Viscoelastic Superposed Fluids in a Vertical Magnetic Field through Porous Medium,” Ganita Sandesh (India), Vol. 17, 2003, pp. 21-32.

[10] P. Kumar, H. Mohan and G. J. Singh, “Rayleigh—Taylor Instability of Rotating Oldroydian Viscoelastic Fluids in Porous Medium in Presence of a Variable Magnetic Field,” Transport in Porous Media, Vol. 56, No. 2, 2004, pp. 19-208. doi:10.1023/B:TIPM.0000021828.01346.57

[11] G. L. Kalra and S. P. Talwar, “Slipping Stream Instability of a Self-Gravitating Hydromagnetic Gas Cloud,” Monthly Notices of the Royal Astronomical Society, Vol. 135, 1967, pp. 391-398.

[12] S. Singh and J. N. Tnadon, “On the Rayleigh-Taylor Instability in Hydromagnetics with Finite Electrical Resistivity and Hall Current,” Journal of Plasma Physics, Vol. 3, No. 4, 1969, pp. 633-642. doi:10.1017/S0022377800004670

[13] P. K. Bhatia and IL Nuovo Cimento, “Rayleigh-Taylor Instability of Viscous Superposed Conducting Fluids,” Il Nuovo Cimento B (1971-1996), Vol. 19 B, No. 2, 1974, pp. 161-168.

[14] P. K. Bhatia and R. P. S. Chhonkar, “Rayleigh-Taylor Instability of Two Viscous Superposed Rotating and Conducting Fluids,” Astrophysics and Space Science, Vol. 114, No. 2, 1985, pp. 271-276. doi:10.1007/BF00653970

[15] P. K. Sharma and R. K. Chhajlani, “Effect of Rotation on the Rayleigh-Taylor Instability of Superposed Magnetic Conducting Plasma,” Physics of plasmas, Vol. 5, No. 6, 1998, pp. 2203-2209. doi:10.1063/1.872893

[16] A. Ali and P. K. Bhatia, “Rayleigh-Taylor Instability of a Stratified Hall Plasma in Two-Dimensional Horizontal Magnetic Field,” Physica Scripta, Vol. 47, 1993, pp. 567 -570. doi:10.1088/0031-8949/47/4/016

[17] A. Khan and P. K. Bhatia, “Gravitational Instability of a Rotating Fluid in an Oblique Magnetic Field,” Physica Scripta, Vol. 47, 1993, pp. 230-234. doi:10.1088/0031-8949/47/2/018

[18] R. J. Goldston and P. H. Rutherford, “Introduction to Plasma Physics,” Institute of Physics, London, 1997.

[19] P. K. Bhatia and A. Sharma, “Rayleigh-Taylor Instability of a Stratified Fluid/Plasma in an Inhomogeneous Magnetic Field,” Proceedings of the National Academy of Sciences, India. Section A. Physical, Vol. 68 (A) III, 1998, pp. 239-247.

[20] Z. Wu, W. Zhang, D. Li and W. Yang, “Effect of Magnetic Field and Equilibrium Flow on Rayleigh-Taylor Instability,” Chinese Physics Letters, Vol. 21, No. 10, 2004, pp. 2001-2004. doi:10.1088/0256-307X/21/10/038

[21] A. M. Al-Khateeb and N. M. Laham, “Magnetic Field Effects on the Rayleigh-Taylor Instability in Inhomogeneous Plasma,” The Arabian Journal for Science and Engineering, Vol. 27, No. 1A, 2002, pp. 75-83.

[22] A. M. Al-Khateeb and N. M. Laham, “Stability Criteria of Rayleigh-Taylor Modes in Magnetized Plasmas,” Contributions to Plasma Physics, Vol. 43, No. 1, 2003, pp. 25-32. doi:10.1002/ctpp.200310003

[23] D. Banerjee M. S. Janaki, N. Chakrabarti and M. Chaudhuri, “Viscosity Gradient-Driven Instability of ‘Shear Mode’ in a Strongly Coupled Plasma,” New Journal of Physics, Vol. 12, 2010, p. 123031. doi:10.1088/1367-2630/12/12/123031

[24] M. F. El-Sayed and R. A. Mohamed, “Gravitational Instability of Rotating Viscoelastic Partially Ionized Plasma in the Presence of an Oblique Magnetic Field and Hall Current,” International Scholarly Research Network ISRN Mechanical Engineering, 2011, Article ID 597172.

[25] G. Manfredi, “How to Model Quantum Plasmas,” Fields Institute Communications Series, Vol. 46, 2005, pp. 263- 287.

[26] C. Gardner, “The Quantum Hydrodynamic Model for Semiconductor Devices,” SIAM Journal on Applied Mathematics, Vol. 54, 1994, pp. 409-427. doi:10.1137/S0036139992240425

[27] F. Haas, “Quantum Magnetohydrodynamics,” Physics of Plasmas, Vol. 12, 2005, p. 062117. doi:10.1063/1.1939947

[28] B. Vitaly, M. Marklund and M. Modestov, “The Rayleigh-Taylor Instability and Internal Waves in Quantum Plasmas,” Physics Letters A, Vol. 372, No. 17, 2008, pp. 3042-3045. doi:10.1016/j.physleta.2007.12.065

[29] J. T. Cao, H. J. Ren, Z. W. Wu and P. K. Chu, “Quantum Effects on Rayleigh-Taylor Instability in Magnetized Plasma,” Physics of Plasmas, Vol. 15, 2008, p. 012110. doi:10.1063/1.2833588

[30] G. A. Hoshoudy, “Quantum Effects on Rayleigh-Taylor Instability in a Vertical Inhomogeneous Rotating Plasma,” Physics of Plasmas, Vol. 16, 2009, p. 024501. doi:10.1063/1.3080202

[31] G. A. Hoshoudy, “Quantum Effects on Rayleigh-Taylor Instability in a Horizontal Inhomogeneous Rotating Plas- ma,” Physics of Plasmas, Vol.16, 2009, p. 064501. doi:10.063/1.3140038

[32] M. Modestov, V. Bychkov and M. Marklund, “The Rayleigh-Taylor Instability in Quantum Magnetized Plasma with Para- and Ferromagnetic Properties,” Physics of Plasmas, Vol. 16, 2009, p. 032106. doi:10.1063/1.3085796

[33] G. A. Hoshoudy, “Quantum Effects on the Rayleigh- Taylor Instability of Stratified Fluid/Plasma through Porous Media,” Physics Letters A, Vol. 373, No. 30, 2009, pp. 2560-2567. doi:10.1016/j.physleta.2009.05.036

[34] S. Ali, Z. Ahmed, M. Arshad Mirza and I. Ahmad, “Rayleigh-Taylor/Gravitational Instability in Dense Mag- netoplasmas,” Physics Letters A, Vol. 373, 2009, pp. 2940-2946. doi:10.1016/j.physleta.2009.06.021

[35] L. C. Gardner, “The Quantum Hydrodynamic Model for Semiconductor Devices,” SIAM Journal on Applied Mathematics, Vol. 54, No. 2, 1994, pp. 409-427. doi:10.1137/S0036139992240425