An Exact Solution for the Propagation of Cylindrical Shock Waves in a Rotational Axisymmetric Non-Ideal Gas with Axial Magnetic Field and Radiative Heat Flux

An Exact Solution for the Propagation of Cylindrical Shock Waves in a Rotational Axisymmetric Non-Ideal Gas with Axial Magnetic Field and Radiative Heat Flux

G. Nath* P.K. Sahu S. Chaurasia

Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad-211004, India

Department of Mathematics, SPM College, Sitapur-497111, Sarguja University, India

Mechanical Engineering Department, MMMUT Gorakhpur-273010, India

Corresponding Author Email: 
gnath@mnnit.ac.in
Page: 
236-243
|
DOI: 
https://doi.org/10.18280/mmc_b.870404
Received: 
20 May 2018
|
Accepted: 
15 September 2018
|
Published: 
31 December 2018
| Citation

OPEN ACCESS

Abstract: 

Propagation of cylindrical shock wave in a rotational axisymmetric non-ideal gas with axial magnetic field, radiation heat flux and the components of vorticity vectors are investigated. The axial magnetic field and the fluid velocity in the ambient medium are assumed to vary and obey the power laws. An exact similarity solution is obtained. The total energy of the shock wave is not constant but increases with time. The effects of variation of parameter of non-idealness of the gas, the Alfven-Mach number and the adiabatic exponent of the gas are investigated. It is shown that an increase in the non-idealness of the gas or the ratio of specific heats of the gas or strength of initial magnetic field decreases the shock strength but increases the shock velocity. Further it is observed an increase in the value of parameter of non-idealness of the gas and adiabatic exponent of the gas have same behavior on the flow variables and the shock strength.

Keywords: 

shock waves, similarity solution, rotating medium, non-ideal gas, radiation heat flux, magnetogasdynamics

1. Introduction
2. Fundamental Equations of Motions and Boundary Conditions
3. Self-Similarity Transformations
4. Results and Discussion
5. Conclusion
Nomenclature
  References

[1] Chaturani P. (1970). Strong cylindrical shocks in a rotating gas. Appl. Sci. Res 23(1): 197-211. http://doi.org/10.1007/BF00413198

[2] Sakurai A. (1956). Propagation of spherical shock waves in stars. Fluid Mech 1(4): 436–453. http://doi.org/10.1017/S0022112056000275

[3] Nath O, Ojha S, Takhar HS. (1999). Propagation of a shock wave in a rotating interplanetary atmosphere with increasing energy. Mhd. Plasma Res 8(4): 269-282. 

[4] Ganguly A, Jana M. (1998). Propagation of shock waves in a self-gravitating radiative magnetohydrodynamic non-uniform rotating atmosphere. Bull. Cal. Math. Soc 90: 77-82. 

[5] Marshak RE. (1958). Effects of radiation on shock wave behavior. Phys. Fluids 1: 24-29. http://doi.org/10.1063/1.1724332

[6] Elliot LA. (1960). Similarity methods in radiation hydrodynamics. Proc. Roy. Soc 258(1294): 287-301. http://doi.org/10.2307/2413960

[7] Wang KC. (1964). The piston problem with thermal radiation. Fluid Mech 20(3): 447–455. http://doi.org/10.1017/S0022112064001343

[8] Taylor GI. (1950). The formation of a blast wave by a very intense explosion. I, II, Proc. Roy. Soc 201(1065): 159-174. http://doi.org/10.1098/rspa.1950.0049

[9] Ashraf S, Sachdev PL. (1970). An exact similarity solution in radiation-gas-dynamics. Proc. Indian Acd. Sci 71(6): 275-281. http://doi.org/10.1007/BF03049574

[10] Verma BG, Vishwakarma JP. (1978). An exact similarity solution for spherical shock wave in magnetoradiative gas. Astrophys. Space Sci 58(1): 139-147. http://doi.org/10.1007/BF00645381

[11] Anisimov SI, Spiner OM. (1972). Motion of an almost ideal gas in the presence of a strong point explosion. Appl. Math. Mech 36(5): 883–887. http://doi.org/10.1016/0021-8928(72)90144-X

[12] Rao MPR, Purohit NK. (1976). Self-similar piston problem in non-ideal gas. Int. J. Eng. Sci 14(1): 91–97. http://doi.org/10.1016/0020-7225(76)90059-8

[13] Vishwakarma JP, Nath G. (2007). Similarity solutions for the flow behind an exponential shock in a non-ideal gas, Meccanica 42(4): 331-339. http://doi.org/10.1007/s11012-007-9058-6

[14] Vishwakarma JP, Nath G. (2009). A self-similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles. Meccanica 44(3): 239-254. http://doi.org/10.1007/s11012-008-9166-y

[15] Nath G. (2012). Propagation of a cylindrical shock wave in a rotational axisymmetric isothermal flow of a non-ideal gas in magnetogasdynamics. Ain Shams Eng 3(4): 393-401. http://doi.org/10.1016/j.asej.2012.03.009

[16] Wu CC, Roberts PH. (1993). Shock-wave propagation in a sonoluminescing gas bubble, Phys. Rev. Lett 70(22): 3424-3427. http://doi.org/10.1103/PhysRevLett.70.3424

[17] Roberts PH, Wu CC. (1996). Structure and stability of a spherical implosion. Phys. Lett 213(1): 59-64. http://doi.org/10.1016/0375-9601(96)00082-5

[18] Levin VA, Skopina GA. (2004). Detonation wave propagation in rotational gas flows. Appl. Mech. Tech. Phy 45(4): 457-460. http://doi.org/10.1023/b:jamt.0000030320.77965.c1

[19]  Nath G. (2011). Magnetogasdynamic shock wave generated by a moving piston in a rotational axisymmetric isothermal flow of perfect gas with variable density. Adv. Space Res, 47(9): 1463-1471. http://doi.org/10.1016/j.asr.2010.11.032

[20] Nath G. (2015). Similarity solutions for unsteady flow behind an exponential shock in an axisymmetric rotating non-ideal gas. Meccanica 50: 1701-1715. http://doi.org/10.1007/s11012-015-0115-2

[21] Vishwakarm JP, Patel N. (2015). Magnetogasdynamic cylindrical shock waves in a rotating nonideal gas with radiation heat flux. Eng. Phy. Thermophys 88(2): 521-530. http://doi.org/10.1007/s10891-015-1217-3

[22] McVittie GC. Spherically solutions of the equations of gas dynamics. Proc. Roy. Soc 220(1142): 339-355. http://doi.org/ 10.1098/rspa.1953.0191

[23] Nicastro JR. (1970). Similarity analysis of radiative gasdynamics with spherical symmetry. Phys. Fluids, 13: 2000-2006. http://doi.org/10.1063/1.1693197

[24] Greifinger C, Cole D. (1961). On cylindrical magnetohydrodynamic shock waves. Phys. Fluids 4: 527-534. http://doi.org/10.1063/1.1706358

[25] Vishwakarma JP, Maurya AK, Singh KK. (2007). Self-similar adiabatic flow headed by a magnetogasdynamics cylindrical shock wave in a rotating non-ideal gas. Geophys. Astrophys. Fluid Dynamics 101(2): 155-168. http://doi.org/10.1080/03091920701298112

[26] Nath G, Vishwakarma JP. (2014). Similarity solution for the flow behind a shock wave in a non-ideal gas with heat conduction and radiation heat-flux in magnetogasdynamics. Commun. Nonlinear Sci. Numer. Simul 19(5): 1347-1365. https://doi.org/10.1016/j.cnsns.2013.09.009

[27] Vishwakarma JP, Shrivastava RC, Kumar A. (1987). An exact similarity solution in radiation magnetogasdynamics for the flows behind a spherical shock. Astrophys. Space Sci 129(1): 45-52. http://doi.org/10.1007/BF00717856

[28] Vishwakarma JP, Pandey SN. (2004). Magnetogasdynamic cylindrical shock waves in a non-ideal gas with radiation heat flux. Modelling Meas. Control B 88(2): 23-37. http://doi.org/10.1007/s10891-015-1217-3

[29] Rosenau P, Frankenthal S. (1976). Equatorial propagation of axisymmetric magnetohydrodynamic shocks. I. Phys. Fluids 19: 1889-1899. http://doi.org/10.1063/1.861424

[30] Pai SI. (1969). Inviscid flow of radiation gasdynamics (High temperature inviscid flow of ideal radiating gas, analyzing effects of radiation pressure and energy on flow field). Math. Phys. Sci 3: 361-370. 

[31] Singh AK. (2012). Self-similar flow behind a cylindrical shock wave in a self-gravitating rotating gas with heat conduction and radiation heat flux. Modelling, Measurment and Control B 81: 61-81.

[32] Vishwakarma JP, Maurya AK, Singh AK. (2011). Cylindrical shock waves in a non-ideal gas with radiation heat-flux and magnetic field. Modelling, Measurment and Control B 80(1/2): 35-52.

[33] Nath G, Dutta M, Pathak RP. (2017). An exact solution for the propagation of shock waves in self-gravitating medium in the presence of magnetic field and radiative heat flux Modelling. Measurement and Control B 86(4): 907-927. http://iieta.org/sites/default/files/Journals/MMC/MMC_B/86.04_06.pdf