Evolution of Weak Shock Waves in Perfectly Conducting Gases

ABSTRACT

This article aims at studying one dimensional unsteady planar and cylindrically symmetric flow involving shocks under the influence of magnetic field. The method of generalized wavefront expansion (GWE) is employed to derive a coupled system of nonlinear transport equations for the jump of field variables and of its spatial derivatives across the shock, which, in turn determine the evolution of wave amplitude and admit a solution that agrees with the classical decay laws of weak shocks. A closed form solution exhibiting the features of nonlinear steepening of the wave front. A general criterion for a compression wave to steepen into a shock is derived. An analytic expression elucidating how the shock formation distance is influenced by the magnetic field strength is obtained. Also, the effects of geometrical spreading and nonlinear convection on the distortion of the waveform are investigated in the presence of magnetic field.

This article aims at studying one dimensional unsteady planar and cylindrically symmetric flow involving shocks under the influence of magnetic field. The method of generalized wavefront expansion (GWE) is employed to derive a coupled system of nonlinear transport equations for the jump of field variables and of its spatial derivatives across the shock, which, in turn determine the evolution of wave amplitude and admit a solution that agrees with the classical decay laws of weak shocks. A closed form solution exhibiting the features of nonlinear steepening of the wave front. A general criterion for a compression wave to steepen into a shock is derived. An analytic expression elucidating how the shock formation distance is influenced by the magnetic field strength is obtained. Also, the effects of geometrical spreading and nonlinear convection on the distortion of the waveform are investigated in the presence of magnetic field.

KEYWORDS

Magnetogasdynamic Flow, Weak Shock, Induced Discontinuity, Generalized Wavefrontexpansion (GWE)

Magnetogasdynamic Flow, Weak Shock, Induced Discontinuity, Generalized Wavefrontexpansion (GWE)

Cite this paper

nullL. Singh, D. Singh and S. Ram, "Evolution of Weak Shock Waves in Perfectly Conducting Gases,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 653-660. doi: 10.4236/am.2011.25086.

nullL. Singh, D. Singh and S. Ram, "Evolution of Weak Shock Waves in Perfectly Conducting Gases,"

References

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[2] J. D. Achenbach, “Wave Propagation in Elastic Solids,” North-Holland American Elsevier, Amsterdam, 1973.

[3] P. J. Chen, “Selected Topics in Wave Propagation,” Noordhoff, Leyden, 1976.

[4] A. Jeffrey, “Quasilinear Hyperbolic System and Waves,” Pitman, London, 1976.

[5] M. F. McCarthy, “Singular Surfaces and Waves,” In: A. C. Eringen, Ed., Continuum Physics, Vol. 2, Academic Press, London, 1975, pp. 449-521.

[6] C. Truesdell and K. R. Rajagopal, “An Introduction to the Mechanics of Fluids,” Birkh?use, Boston, 2000. doi:10.1007/978-0-8176-4846-6

[7] T. Y. Thomas, “The Growth and Decay of Sonic Discontinuities in Ideal Gases,” Journal of Mathematics and Mechanics, Vol. 6, No. 3, 1957, pp. 455-469.

[8] B. D. Coleman and M. E. Gurtin, “Growth and Decay of Discontinuities in Fluids with Internal State Variables,” Physics of Fluids, Vol. 10, No. 7, 1967, pp. 1454-1458. doi:10.1063/1.1762305

[9] V. V. Menon, V. D. Sharma and A. Jeffrey, “On the General Behavior of Acceleration Waves,” Applicable Analysis, Vol. 16, No. 2, 1983, pp. 101-120. doi:10.1080/00036818308839462

[10] H. Lin and A. J. Szeri, “Shock Formation in the Presence of Entropy Gradient,” Journal of Fluid Mechanics, Vol. 431, No. 1, 2001, pp. 161-188. doi:10.1017/S0022112000003104

[11] P. M. Jordan, “Growth and Decay of Shock and Acceleration Waves in a Traffic Flow Model with Relaxation,” Physica D: Nonlinear Phenomena, Vol. 207, No. 3-4, 2005, pp. 220-229. doi:10.1016/j.physd.2005.06.002

[12] D. Bhardwaj, “Formation of Shock Waves in Reactive Magnetogas Dynamic Flow,” International Journal of Engineering Science, Vol. 38, No. 11, 2000, pp. 1197-1206. doi:10.1016/S0020-7225(99)00071-3

[13] M. Tyagi and R. I. Sujith, “The Propagation of Finite Amplitude Gasdynamic Disturbances in a Stratified Atmosphere around a Celestial Body: An Analytical Study,” Physica D: Nonlinear Phenomena, Vol. 211, No. 1-2, 2005, pp. 139-150. doi:10.1016/j.physd.2005.08.006

[14] I. Christov, P. M. Jordan and C. I. Christov, “Nonlinear Acoustic Propagation in Homentropic Perfect Gases: A Numerical Study,” Physics Letters A, Vol. 353, No. 4, 2006, pp. 273-280. doi:10.1016/j.physleta.2005.12.101

[15] T. R. Sekhar and V. D. Sharma, “Evolution of Weak Dis- continuities in Shallow Water Equations,” Applied Mathematics Letters, Vol. 23, No. 3, 2010, pp. 327-330. doi:10.1016/j.aml.2009.10.003

[16] V. P. Maslov, “Propagation of Shock Waves in an Isentropic Non-Viscous Gas,” Journal of Mathematical Sci- ences, Vol. 13, No. 1, 1980, pp. 119-163. doi:10.1007/BF01084111

[17] M. A. Grinfel’d, “Ray Method for Calculating the Wavefront Intensity in Non-Linear Elastic Material,” Journal of Applied Mathematics and Mechanics, Vol. 42, No. 5, 1978, pp. 958-977.

[18] A. M. Anile, “Propagation of Weak Shock Waves,” Wave Motion, Vol. 6, No. 6, 1984, pp. 571-578. doi:10.1016/0165-2125(84)90047-7

[19] G. Russo, “Generalized Wavefront Expansion: Properties and Limitations,” Meccanica, Vol. 21, No. 4, 1986, pp. 191-199. doi:10.1007/BF01556485

[20] A. M. Anile and G. Russo, “Generalized Wavefront Expansion I: Higher Order Corrections for the Propagation of Weak Shock Waves,” Wave Motion, Vol. 8, No. 3, 1986, pp. 243-258. doi:10.1016/S0165-2125(86)80047-6

[21] G. Madhumita and V. D. Sharma, “Imploding Cylindrical and Spherical Shock Waves in a Non-Ideal Medium,” Journal of Hyperbolic Differential Equations, Vol. 1, No. 3, 2004, pp. 521-530. doi:10.1142/S0219891604000184

[22] V. D. Sharma, L. P. Singh and R. Ram, “The Progressive Wave Approach Analyzing the Decay of a Sawtooth Profile in Magnetogasdynamics,” Physics of Fluids, Vol. 30, No. 5, 1987, pp. 1572-1574. doi:10.1063/1.866222

[23] S. Muralidharan and R. I. Sujith, “Shock Formation in the Presence of Entropy Gradients in Fluids Exhibiting Mixed Nonlinearity,” Physics of Fluids, Vol. 6, No. 11, 2004, pp. 4121-4128. doi:10.1063/1.1795272

[24] R. Courant and K. O. Friedrichs, “Supersonic Flow and Shock Waves,” Interscience Inc, New York, 1948.

[25] L. D. Landau, “On Shock Waves at Large Distances from the Place of Their Origin,” Soviet Physics Journal, Vol. 9, 1945, pp. 496-500.

[1] G. B. Whitham, “Linear and Nonlinear Waves,” Wiley- Interscience, New York, 1974.

[2] J. D. Achenbach, “Wave Propagation in Elastic Solids,” North-Holland American Elsevier, Amsterdam, 1973.

[3] P. J. Chen, “Selected Topics in Wave Propagation,” Noordhoff, Leyden, 1976.

[4] A. Jeffrey, “Quasilinear Hyperbolic System and Waves,” Pitman, London, 1976.

[5] M. F. McCarthy, “Singular Surfaces and Waves,” In: A. C. Eringen, Ed., Continuum Physics, Vol. 2, Academic Press, London, 1975, pp. 449-521.

[6] C. Truesdell and K. R. Rajagopal, “An Introduction to the Mechanics of Fluids,” Birkh?use, Boston, 2000. doi:10.1007/978-0-8176-4846-6

[7] T. Y. Thomas, “The Growth and Decay of Sonic Discontinuities in Ideal Gases,” Journal of Mathematics and Mechanics, Vol. 6, No. 3, 1957, pp. 455-469.

[8] B. D. Coleman and M. E. Gurtin, “Growth and Decay of Discontinuities in Fluids with Internal State Variables,” Physics of Fluids, Vol. 10, No. 7, 1967, pp. 1454-1458. doi:10.1063/1.1762305

[9] V. V. Menon, V. D. Sharma and A. Jeffrey, “On the General Behavior of Acceleration Waves,” Applicable Analysis, Vol. 16, No. 2, 1983, pp. 101-120. doi:10.1080/00036818308839462

[10] H. Lin and A. J. Szeri, “Shock Formation in the Presence of Entropy Gradient,” Journal of Fluid Mechanics, Vol. 431, No. 1, 2001, pp. 161-188. doi:10.1017/S0022112000003104

[11] P. M. Jordan, “Growth and Decay of Shock and Acceleration Waves in a Traffic Flow Model with Relaxation,” Physica D: Nonlinear Phenomena, Vol. 207, No. 3-4, 2005, pp. 220-229. doi:10.1016/j.physd.2005.06.002

[12] D. Bhardwaj, “Formation of Shock Waves in Reactive Magnetogas Dynamic Flow,” International Journal of Engineering Science, Vol. 38, No. 11, 2000, pp. 1197-1206. doi:10.1016/S0020-7225(99)00071-3

[13] M. Tyagi and R. I. Sujith, “The Propagation of Finite Amplitude Gasdynamic Disturbances in a Stratified Atmosphere around a Celestial Body: An Analytical Study,” Physica D: Nonlinear Phenomena, Vol. 211, No. 1-2, 2005, pp. 139-150. doi:10.1016/j.physd.2005.08.006

[14] I. Christov, P. M. Jordan and C. I. Christov, “Nonlinear Acoustic Propagation in Homentropic Perfect Gases: A Numerical Study,” Physics Letters A, Vol. 353, No. 4, 2006, pp. 273-280. doi:10.1016/j.physleta.2005.12.101

[15] T. R. Sekhar and V. D. Sharma, “Evolution of Weak Dis- continuities in Shallow Water Equations,” Applied Mathematics Letters, Vol. 23, No. 3, 2010, pp. 327-330. doi:10.1016/j.aml.2009.10.003

[16] V. P. Maslov, “Propagation of Shock Waves in an Isentropic Non-Viscous Gas,” Journal of Mathematical Sci- ences, Vol. 13, No. 1, 1980, pp. 119-163. doi:10.1007/BF01084111

[17] M. A. Grinfel’d, “Ray Method for Calculating the Wavefront Intensity in Non-Linear Elastic Material,” Journal of Applied Mathematics and Mechanics, Vol. 42, No. 5, 1978, pp. 958-977.

[18] A. M. Anile, “Propagation of Weak Shock Waves,” Wave Motion, Vol. 6, No. 6, 1984, pp. 571-578. doi:10.1016/0165-2125(84)90047-7

[19] G. Russo, “Generalized Wavefront Expansion: Properties and Limitations,” Meccanica, Vol. 21, No. 4, 1986, pp. 191-199. doi:10.1007/BF01556485

[20] A. M. Anile and G. Russo, “Generalized Wavefront Expansion I: Higher Order Corrections for the Propagation of Weak Shock Waves,” Wave Motion, Vol. 8, No. 3, 1986, pp. 243-258. doi:10.1016/S0165-2125(86)80047-6

[21] G. Madhumita and V. D. Sharma, “Imploding Cylindrical and Spherical Shock Waves in a Non-Ideal Medium,” Journal of Hyperbolic Differential Equations, Vol. 1, No. 3, 2004, pp. 521-530. doi:10.1142/S0219891604000184

[22] V. D. Sharma, L. P. Singh and R. Ram, “The Progressive Wave Approach Analyzing the Decay of a Sawtooth Profile in Magnetogasdynamics,” Physics of Fluids, Vol. 30, No. 5, 1987, pp. 1572-1574. doi:10.1063/1.866222

[23] S. Muralidharan and R. I. Sujith, “Shock Formation in the Presence of Entropy Gradients in Fluids Exhibiting Mixed Nonlinearity,” Physics of Fluids, Vol. 6, No. 11, 2004, pp. 4121-4128. doi:10.1063/1.1795272

[24] R. Courant and K. O. Friedrichs, “Supersonic Flow and Shock Waves,” Interscience Inc, New York, 1948.

[25] L. D. Landau, “On Shock Waves at Large Distances from the Place of Their Origin,” Soviet Physics Journal, Vol. 9, 1945, pp. 496-500.