Combined Effect of Magnetic Field and Compressibility on Rayleigh Taylor Instability
Affiliation(s)
1 Dept. of Instrumentation Science & Centre for Plasma Studies, Jadavpur University, Kolkata, India. 2 Dept. of Physics, Budge Budge Institute of Technology, Kolkata, India. 3 Advanced Centre for Nonlinear & Complex Phenomena, Kolkata, India.
1 Dept. of Instrumentation Science & Centre for Plasma Studies, Jadavpur University, Kolkata, India. 2 Dept. of Physics, Budge Budge Institute of Technology, Kolkata, India. 3 Advanced Centre for Nonlinear & Complex Phenomena, Kolkata, India.
ABSTRACT
The nonlinear analysis of the combined effect of magnetic field and compressibility on the growth rate of Rayleigh-Taylor (RT) instability has been investigated for inviscid two fluid interface. We have considered an interface-parallel density dependent magnetic field and used Layzer’s approach to analyze the problem. We have also investigated the relative effect of magnetic pressure and hydrodynamic pressure on RT instability through the variation of the ratio of hydromagnetic pressure to magnetic pressure (β). Dynamics of bubble and spike has been studied analytically and numerically. Finally, we have obtained the stability conditions of our result through linear stability analysis
The nonlinear analysis of the combined effect of magnetic field and compressibility on the growth rate of Rayleigh-Taylor (RT) instability has been investigated for inviscid two fluid interface. We have considered an interface-parallel density dependent magnetic field and used Layzer’s approach to analyze the problem. We have also investigated the relative effect of magnetic pressure and hydrodynamic pressure on RT instability through the variation of the ratio of hydromagnetic pressure to magnetic pressure (β). Dynamics of bubble and spike has been studied analytically and numerically. Finally, we have obtained the stability conditions of our result through linear stability analysis
KEYWORDS
Rayleigh-Taylor Instability, Bubbles, Spikes, Self Generated Magnetic Field, Compressibility, Density Dependent Magnetic Field, ICF
Rayleigh-Taylor Instability, Bubbles, Spikes, Self Generated Magnetic Field, Compressibility, Density Dependent Magnetic Field, ICF
Cite this paper
Mitra, A. , Mandal, L. , Roychoudhury, R. and Khan, M. (2015) Combined Effect of Magnetic Field and Compressibility on Rayleigh Taylor Instability. Open Journal of Fluid Dynamics, 5, 322-338. doi: 10.4236/ojfd.2015.54033.
Mitra, A. , Mandal, L. , Roychoudhury, R. and Khan, M. (2015) Combined Effect of Magnetic Field and Compressibility on Rayleigh Taylor Instability. Open Journal of Fluid Dynamics, 5, 322-338. doi: 10.4236/ojfd.2015.54033.
References
[1] Goncharov, V.N. (2002) Analytical Model of Nonlinear, Single-Mode, Classical Rayleigh-Taylor Instability at Arbitrary Atwood Numbers. Physical Review Letters, 88, Article ID: 134502.
http://dx.doi.org/10.1103/PhysRevLett.88.134502 [2] Gupta, M.R., Banerjee, B., Mandal, L.K., Bhar, R., Pant, H.C., Khan, M. and Srivastava, M.K. (2012) Effect of Viscosity and Surface Tension on the Growth of Rayleigh-Taylor Instability and Richtmyer-Meshkov Instability Induced Two Fluid Interfacial Nonlinear Structure. Indian Journal of Physics, 86, 471-479.
http://dx.doi.org/10.1007/s12648-012-0077-3 [3] Gupta, N.K. and Lawande, S.V. (1986) Rayleigh-Taylor Instability in Multi-Structured Spherical Targets. Plasma Physics and Controlled Fusion, 28, 925-941.
http://dx.doi.org/10.1088/0741-3335/28/6/008 [4] Hazak, G. (1996) Lagrangian Formalism for the Rayleigh-Taylor Instability. Physical Review Letters, 76, 4167-4170.
http://dx.doi.org/10.1103/PhysRevLett.76.4167 [5] Hecht, J., Alon, U. and Shvarts, D. (1994) Potential Flow Models of Rayleigh-Taylor and Richtmyer-Meshkov Bubble Fronts. Physics of Fluids, 6, 4019-4030.
http://dx.doi.org/10.1063/1.868391 [6] Piriz, A.R., Lpez Cela, J.J. and Cortzar, O.D. (2005) Rayleigh-Taylor Instability in Elastic Solids. Physical Review E, 72, Article ID: 056313.
http://dx.doi.org/10.1103/PhysRevE.72.056313 [7] Ramaprabhu, P. and Dimonte, G. (2005) Single-Mode Dynamics of the Rayleigh-Taylor Instability at Any Density Ratio. Physical Review E, 71, Article ID: 036314.
http://dx.doi.org/10.1103/PhysRevE.71.036314 [8] Sohn, S.-I. (2003) Simple Potential-Flow Model of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities for All Density Ratios. Physical Review E, 67, Article ID: 026301. [9] Velikovich, A.L. and Dimonte, G. (1996) Nonlinear Perturbation Theory of the Incompressible Richtmyer-Meshkov Instability. Physical Review Letters, 76, 3112.
http://dx.doi.org/10.1103/PhysRevLett.76.3112 [10] Zhang, Q. (1998) Analytical Solutions of Layzer-Type Approach to Unstable Interfacial Fluid Mixing. Physical Review Letters, 81, 3391-3394.
http://dx.doi.org/10.1103/PhysRevLett.81.3391 [11] Banerjee, R., Mandal, L.K., Khan, M. and Gupta, M.R. (2013) Bubble and Spike Growth Rate of Rayleigh Taylor and Richtmeyer Meshkov Instability in Finite Layers. Indian Journal of Physics, 87, 929-937.
http://dx.doi.org/10.1007/s12648-013-0300-x [12] Smalyuk, V.A., Barrios, M., Caggiano, J.A., et al. (2014) Hydrodynamic Instability Growth and Mix Experiments at the National Ignition Facilitya. Physics of Plasmas, 21, Article ID: 056301.
http://dx.doi.org/10.1063/1.4872026 [13] Edwards, M.J., et al. (2013) Progress towards Ignition on the National Ignition Facility. Physics of Plasmas, 20, Article ID: 070501. [14] Mason, R.J. and Tabak, M. (1998) Magnetic Field Generation in High-Intensity-Laser-Matter Interactions. Physical Review Letters, 80, 524-527.
http://dx.doi.org/10.1103/PhysRevLett.80.524 [15] Sudan, R.N. (1993) Mechanism for the Generation of 10 9 G Magnetic Fields in the Interaction of Ultraintense Short Laser Pulse with an Overdense Plasma Target. Physical Review Letters, 70, 3075-3078.
http://dx.doi.org/10.1103/PhysRevLett.70.3075 [16] Wilks, S.C., Kruer, W.L., Tabak, M. and Langdon, A.B. (1992) Absorption of Ultra-Intense Laser Pulses. Physical Review Letters, 69, 1383-1386.
http://dx.doi.org/10.1103/PhysRevLett.69.1383 [17] Srivastava, M.K., Lawande, S.V., Khan, M., Das, C. and Chakraborty, B. (1992) Axial Magnetic Field Generation by Ponderomotive Force in a Laser-Produced Plasma. Physics of Fluids B, 4, 4086.
http://dx.doi.org/10.1063/1.860315 [18] Khan, M., Das, C., Chakraborty, B., et al. (1998) Self-Generated Magnetic Field and Faraday Rotation in a Laser- Produced Plasma. Physical Review E, 58, 925-930.
http://dx.doi.org/10.1103/PhysRevE.58.925 [19] Evans, R.G. (1986) The Influence of Self-Generated Magnetic Fields on the Rayleigh-Taylor Instability. Plasma Physics and Controlled Fusion, 28, 1021.
http://dx.doi.org/10.1088/0741-3335/28/7/006 [20] Ghezzi, C.R., de Gouveia Dal Pino, E.M. and Horvath, J.E. (2001) Magnetic Field Effects on the Thermonuclear Combustion Front of Chandrasekhar Mass White Dwarfs. The Astrophysical Journal Letters, 548, L193.
http://dx.doi.org/10.1086/319091 [21] de Gouveia Dal Pino, E.M. and Benz, W. (1993) Three-Dimensional Simulations of Protostellar Jets. Astrophysical Journal, 410, 686-695.
http://dx.doi.org/10.1086/172785 [22] Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Dover Publications, Inc., New York. [23] Liberatore, S. and Bouquet, S. (2008) Analytical Modeling of Magnetic Rayleigh-Taylor Instabilities in Compressible Fluids. Physics of Fluids, 20, Article ID: 116101.
http://dx.doi.org/10.1063/1.3025832 [24] Samtaney, R. (2003) Suppression of the Richtmyer-Meshkov Instability in the Presence of a Magnetic Field. Physics of Fluids, 15, L53.
http://dx.doi.org/10.1063/1.1591188 [25] Jun, B.I., Norman, M.L. and Stone, J.M. (1995) A Numerical Study of Rayleigh-Taylor Instability in Magnetic Fluids. The Astrophysical Journal, 453, 322.
http://dx.doi.org/10.1086/176393 [26] Gupta, M.R., Roy, S., Khan, M., et al. (2009) Effect of Compressibility on the Rayleigh-Taylor and Richtmyer-Meshkov Instability Induced Nonlinear Structure at Two Fluid Interface. Physics of Plasmas, 16, Article ID: 032303.
http://dx.doi.org/10.1063/1.3074789 [27] Gupta, M.R., Mandal, L.K., Roy, S. and Khan, M. (2010) Effect of Magnetic Field on Temporal Development of Rayleigh-Taylor Instability Induced Interfacial Nonlinear Structure. Physics of Plasmas, 17, Article ID: 012306.
http://dx.doi.org/10.1063/1.3293120 [28] Layzer, D. (1955) On the Instability of Superposed Fluids in a Gravitational Field. The Astrophysical Journal, 122, 1.
http://dx.doi.org/10.1086/146048 [29] Drake, R.P. (2006) High Energy-Density Physics. Springer, New York. [30] Casali, R., Castro, L.B. and Menezes, D.P. (2014) Hadronic and Hybrid Stars Subject to Density-Dependent Magnetic Fields. Physical Review C, 89, Article ID: 015805.
http://dx.doi.org/10.1103/PhysRevC.89.015805 [31] Lamb, H. (1932) Hydrodynamics. Cambridge University Press, Cambridge. [32] Bernstein, I.B. and Book, D.L. (1983) Effect of Compressibility on the Rayleigh-Taylor Instability. Physics of Fluids, 26, 453.
http://dx.doi.org/10.1063/1.864158
[1] Goncharov, V.N. (2002) Analytical Model of Nonlinear, Single-Mode, Classical Rayleigh-Taylor Instability at Arbitrary Atwood Numbers. Physical Review Letters, 88, Article ID: 134502.
http://dx.doi.org/10.1103/PhysRevLett.88.134502 [2] Gupta, M.R., Banerjee, B., Mandal, L.K., Bhar, R., Pant, H.C., Khan, M. and Srivastava, M.K. (2012) Effect of Viscosity and Surface Tension on the Growth of Rayleigh-Taylor Instability and Richtmyer-Meshkov Instability Induced Two Fluid Interfacial Nonlinear Structure. Indian Journal of Physics, 86, 471-479.
http://dx.doi.org/10.1007/s12648-012-0077-3 [3] Gupta, N.K. and Lawande, S.V. (1986) Rayleigh-Taylor Instability in Multi-Structured Spherical Targets. Plasma Physics and Controlled Fusion, 28, 925-941.
http://dx.doi.org/10.1088/0741-3335/28/6/008 [4] Hazak, G. (1996) Lagrangian Formalism for the Rayleigh-Taylor Instability. Physical Review Letters, 76, 4167-4170.
http://dx.doi.org/10.1103/PhysRevLett.76.4167 [5] Hecht, J., Alon, U. and Shvarts, D. (1994) Potential Flow Models of Rayleigh-Taylor and Richtmyer-Meshkov Bubble Fronts. Physics of Fluids, 6, 4019-4030.
http://dx.doi.org/10.1063/1.868391 [6] Piriz, A.R., Lpez Cela, J.J. and Cortzar, O.D. (2005) Rayleigh-Taylor Instability in Elastic Solids. Physical Review E, 72, Article ID: 056313.
http://dx.doi.org/10.1103/PhysRevE.72.056313 [7] Ramaprabhu, P. and Dimonte, G. (2005) Single-Mode Dynamics of the Rayleigh-Taylor Instability at Any Density Ratio. Physical Review E, 71, Article ID: 036314.
http://dx.doi.org/10.1103/PhysRevE.71.036314 [8] Sohn, S.-I. (2003) Simple Potential-Flow Model of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities for All Density Ratios. Physical Review E, 67, Article ID: 026301. [9] Velikovich, A.L. and Dimonte, G. (1996) Nonlinear Perturbation Theory of the Incompressible Richtmyer-Meshkov Instability. Physical Review Letters, 76, 3112.
http://dx.doi.org/10.1103/PhysRevLett.76.3112 [10] Zhang, Q. (1998) Analytical Solutions of Layzer-Type Approach to Unstable Interfacial Fluid Mixing. Physical Review Letters, 81, 3391-3394.
http://dx.doi.org/10.1103/PhysRevLett.81.3391 [11] Banerjee, R., Mandal, L.K., Khan, M. and Gupta, M.R. (2013) Bubble and Spike Growth Rate of Rayleigh Taylor and Richtmeyer Meshkov Instability in Finite Layers. Indian Journal of Physics, 87, 929-937.
http://dx.doi.org/10.1007/s12648-013-0300-x [12] Smalyuk, V.A., Barrios, M., Caggiano, J.A., et al. (2014) Hydrodynamic Instability Growth and Mix Experiments at the National Ignition Facilitya. Physics of Plasmas, 21, Article ID: 056301.
http://dx.doi.org/10.1063/1.4872026 [13] Edwards, M.J., et al. (2013) Progress towards Ignition on the National Ignition Facility. Physics of Plasmas, 20, Article ID: 070501. [14] Mason, R.J. and Tabak, M. (1998) Magnetic Field Generation in High-Intensity-Laser-Matter Interactions. Physical Review Letters, 80, 524-527.
http://dx.doi.org/10.1103/PhysRevLett.80.524 [15] Sudan, R.N. (1993) Mechanism for the Generation of 10 9 G Magnetic Fields in the Interaction of Ultraintense Short Laser Pulse with an Overdense Plasma Target. Physical Review Letters, 70, 3075-3078.
http://dx.doi.org/10.1103/PhysRevLett.70.3075 [16] Wilks, S.C., Kruer, W.L., Tabak, M. and Langdon, A.B. (1992) Absorption of Ultra-Intense Laser Pulses. Physical Review Letters, 69, 1383-1386.
http://dx.doi.org/10.1103/PhysRevLett.69.1383 [17] Srivastava, M.K., Lawande, S.V., Khan, M., Das, C. and Chakraborty, B. (1992) Axial Magnetic Field Generation by Ponderomotive Force in a Laser-Produced Plasma. Physics of Fluids B, 4, 4086.
http://dx.doi.org/10.1063/1.860315 [18] Khan, M., Das, C., Chakraborty, B., et al. (1998) Self-Generated Magnetic Field and Faraday Rotation in a Laser- Produced Plasma. Physical Review E, 58, 925-930.
http://dx.doi.org/10.1103/PhysRevE.58.925 [19] Evans, R.G. (1986) The Influence of Self-Generated Magnetic Fields on the Rayleigh-Taylor Instability. Plasma Physics and Controlled Fusion, 28, 1021.
http://dx.doi.org/10.1088/0741-3335/28/7/006 [20] Ghezzi, C.R., de Gouveia Dal Pino, E.M. and Horvath, J.E. (2001) Magnetic Field Effects on the Thermonuclear Combustion Front of Chandrasekhar Mass White Dwarfs. The Astrophysical Journal Letters, 548, L193.
http://dx.doi.org/10.1086/319091 [21] de Gouveia Dal Pino, E.M. and Benz, W. (1993) Three-Dimensional Simulations of Protostellar Jets. Astrophysical Journal, 410, 686-695.
http://dx.doi.org/10.1086/172785 [22] Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Dover Publications, Inc., New York. [23] Liberatore, S. and Bouquet, S. (2008) Analytical Modeling of Magnetic Rayleigh-Taylor Instabilities in Compressible Fluids. Physics of Fluids, 20, Article ID: 116101.
http://dx.doi.org/10.1063/1.3025832 [24] Samtaney, R. (2003) Suppression of the Richtmyer-Meshkov Instability in the Presence of a Magnetic Field. Physics of Fluids, 15, L53.
http://dx.doi.org/10.1063/1.1591188 [25] Jun, B.I., Norman, M.L. and Stone, J.M. (1995) A Numerical Study of Rayleigh-Taylor Instability in Magnetic Fluids. The Astrophysical Journal, 453, 322.
http://dx.doi.org/10.1086/176393 [26] Gupta, M.R., Roy, S., Khan, M., et al. (2009) Effect of Compressibility on the Rayleigh-Taylor and Richtmyer-Meshkov Instability Induced Nonlinear Structure at Two Fluid Interface. Physics of Plasmas, 16, Article ID: 032303.
http://dx.doi.org/10.1063/1.3074789 [27] Gupta, M.R., Mandal, L.K., Roy, S. and Khan, M. (2010) Effect of Magnetic Field on Temporal Development of Rayleigh-Taylor Instability Induced Interfacial Nonlinear Structure. Physics of Plasmas, 17, Article ID: 012306.
http://dx.doi.org/10.1063/1.3293120 [28] Layzer, D. (1955) On the Instability of Superposed Fluids in a Gravitational Field. The Astrophysical Journal, 122, 1.
http://dx.doi.org/10.1086/146048 [29] Drake, R.P. (2006) High Energy-Density Physics. Springer, New York. [30] Casali, R., Castro, L.B. and Menezes, D.P. (2014) Hadronic and Hybrid Stars Subject to Density-Dependent Magnetic Fields. Physical Review C, 89, Article ID: 015805.
http://dx.doi.org/10.1103/PhysRevC.89.015805 [31] Lamb, H. (1932) Hydrodynamics. Cambridge University Press, Cambridge. [32] Bernstein, I.B. and Book, D.L. (1983) Effect of Compressibility on the Rayleigh-Taylor Instability. Physics of Fluids, 26, 453.
http://dx.doi.org/10.1063/1.864158