Blast Waves in Multi-Component Medium with Thermal Relaxation
Author(s)
Vyacheslav O. Vakhnenko*
ABSTRACT
The mathematical models of relaxing media with a structure for describing nonlinear long-wave processes are explored. The wave processes in non-equilibrium heterogeneous media are studied in terms of the suggested asymptotic averaged model. On the microstructure level of the medium, the dynamical behavior is governed only by the laws of thermodynamics, while, on the macrolevel, the motion of the medium can be described by the wave-dynamical laws. It is proved rigorously that on the acoustic level, the propagation of long waves can be properly described only in terms of dispersive dissipative properties of the medium, and in this case, the dynamical behavior of the medium can be modeled by a homogeneous relaxing medium. At the same time, the dynamical behavior of the medium cannot be modeled by a homogeneous medium even for long waves, if they are nonlinear. For a finite-amplitude wave, the structure of medium produces nonlinear effects even if the individual components of the medium are described by a linear law. The heterogeneity of the structure of medium always introduces additional nonlinearity. It is shown that the solution of many problems for multi-component media with incompressible phases can be obtained through the known solution of a similar problem for a homogeneous compressible medium by means of the suggested transformation. It is not necessary to solve directly the problem for the medium with incompressible component, and it is sufficient just to transform the known solution of the similar problem for a homogeneous medium. The scope for the suggested transformation is demonstrated by the reference to the strong explosion state in a two-phase medium. The special attention is focused on the research of blast waves in multi-component media with thermal relaxation. The dependence of the shock damping parameters on the thermal relaxation time is analyzed in order to provide a deeper understanding of the damping of shock waves in such media and to determine their effectiveness as localizing media. This problem attracts the interest also in view of the practical possibility to estimate the efficiency of medium for damping the shock wave action. To find the nature of the relaxation interaction between the components of medium and to estimate the attenuation of shock waves generated by solid explosives, we have studied experimentally both the velocity field of shock waves and the pressure at front in an air foam. The comparison of experimental and theoretical investigations of the relaxation phenomena which accompany the propagation of shock waves in foam indicates that within the scope of relaxation hydrodynamics it is possible to explain the observed phenomena and estimate the efficiency of medium as localizer of the shock wave action.
Cite this paper
Vakhnenko, V. (2014) Blast Waves in Multi-Component Medium with Thermal Relaxation. Natural Science, 6, 1055-1092. doi: 10.4236/ns.2014.612096.
Vakhnenko, V. (2014) Blast Waves in Multi-Component Medium with Thermal Relaxation. Natural Science, 6, 1055-1092. doi: 10.4236/ns.2014.612096.
References
[1] Bonner, B.P. and Wanamaker, B.J. (1991) Acoustic Nonlinearities Produced by a Single Macroscopic Fracture in Granite. In: Thompson, D.O. and Chimenti, D.E., Eds., Review of Progress in Quantitative Nondestructive Evaluation, Springer, New York, 1861-1867.
http://dx.doi.org/10.1007/978-1-4615-3742-7_94 [2] Guyer, R.A. and Johnson, P.A. (2009) Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Rocks, Soil, Concrete. WILEY-VCH Verlag GmbH and Co. KgaA, Weinheim.
http://dx.doi.org/10.1002/9783527628261 [3] Johnson, P.A. and McCall, K.R. (1994) Observation and Implications of Nonlinear Elastic Wave Response in Rock. Geophysical Research Letters, 21, 165-168.
http://dx.doi.org/10.1029/93GL03162 [4] Johnson, P.A., Shankland, T.J., O’Connell, R.J. and Albright, J.N. (1987) Nonlinear Generation of Elastic Waves in Crystalline Rock. Journal of Geophysical Research, 92, 3597-3602.
http://dx.doi.org/10.1029/JB092iB05p03597 [5] Rodionov, V.N., Sizov, I.A. and Tsvetkov, V.M. (1986) Fundamentals of Geomechanics. Nedra Press, Moscow. (in Russian) [6] Sadovsky, M.A. and Pisarenko, G.F. (1991) Seismic Process in Block Medium. Nauka, Moscow. (in Russian) [7] Sadovsky, M.A., Ed. (1989) Discrete Properties of Geophysics Medium. Nauka, Moscow. (in Russian) [8] Achenbach, J.D. (1973) Wave Propagation in Plastic Solids. North-Holland, Amsterdam. [9] Aki, K. and Richards, P.G. (1980) Quantitative Seismology. Theory and Methods, Vol. I and II. W.H. Freeman, San Francisco. [10] Truesdell, C. (1984) Rational Thermodynamics. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5206-1 [11] Biot, M.A. (1956) Theory of Propagation of Elastic Waves in a Fluid-Saturated Solid. I. Low-Frequency Range. Journal of the Acoustical Society of America, 28, 168-178.
http://dx.doi.org/10.1121/1.1908239 [12] Kutateladze, S.S. and Nakoryakov, V.E. (1984) Heat Exchange and Waves in Gas-Liquid Systems. Nauka, Novosibisk. (in Russian) [13] Lyakhov, G.M. (1982) Waves in Soils and Porous Multicomponent Media. Nauka, Moscow. (in Russian) [14] Raats, P.A.C. (1984) Applications of the Theory of Mixtures in Soil Physics. In: Truesdell, C., Ed., Rational Thermodynamics, Springer, New York, 326-343.
http://dx.doi.org/10.1007/978-1-4612-5206-1_16 [15] Rajagopal, K.R. and Tao, L. (1995) Machanics of Mixtures. World Scientific Publishing, Singapore. [16] Nikolaevskii, V.N. (1985) Viscoelasticity with Internal Oscillators as a Possible Model of Seismoactive Medium. Doklady Akademii Nauk SSSR, 283, 1321-1324. (in Russian) [17] Nigmatulin, R.I. (1987) Dynamics of Multiphase Media, Vol. I and II. Nauka, Moscow. (in Russian) [18] Struminskii, V.V. (1980) Mechanics and Technical Progress. Nauka, Moscow. (in Russian) [19] Vakhnenko, V.O., Danylenko, V.A. and Michtchenko, A.V. (1999) An Asymptotic Averaged Model of Nonlinear Long Waves Propagation in Media with a Regular Structure. International Journal of Non-Linear Mechanics, 34, 643-654.
http://dx.doi.org/10.1016/S0020-7462(98)00014-6 [20] Vakhnenko, V.O., Danylenko, V.A. and Michtchenko, A.V. (2000) Diagnostics of the Medium Structure by Long Wave of Finite Amplitude. International Journal of Non-Linear Mechanics, 35, 1105-1113.
http://dx.doi.org/10.1016/S0020-7462(99)00082-7 [21] Vakhnenko, V.A., Danylenko, V.A. and Kulich, V.V. (1993) Averaged Description of Wave Processes in Geophysical Medium. Geophysics Journal (Ukraine), 15, 66-71. [22] Vakhnenko, V.A., Danylenko, V.A. and Kulich, V.V. (1994) Averaged Description of Shock-Wave Processes in Periodic Media. Soviet Journal of Chemical Physics, 12, 534-546. [23] Vakhnenko, V.A. and Kulich, V.V. (1992) Long-Wave Processes in Periodic Media. Journal of Applied Mechanics and Technical Physics, 32, 814-820. [24] Landau, L.D. and Lifshitz, E.M. (1988) Fluids Mechanics. Pergamon, New York. [25] Lavrentiev, M. and Chabat, B. (1980) Effets Hyrdodynamiques et Modéles Mathématiques. Traduction franc. éditions, Mir, Moscow. [26] Clarke, J.E. (1984) Lectures on Plane Waves in Reacting Gases. Annals of Physics, 9, 211-306.
http://dx.doi.org/10.1051/anphys:0198400902021100 [27] Yasnikov, G.P. and Belousov, V.S. (1978) Effective Thermodynamic Functions of a Gas with Solid Particles. Journal of Engineering Physics, 34, 734-737.
http://dx.doi.org/10.1007/BF00860421 [28] Danylenko, V.A., Sorokina, V.V. and Vladimirov, V.A. (1993) On the Governing Equations in Relaxing Models and Self-Similar Quasiperiodic Solutions. Journal of Physics A, 26, 7125-7135.
http://dx.doi.org/10.1088/0305-4470/26/23/047 [29] Mandel’shtam, L.I. and Leonovich, M.A. (1937) To the Theory of a Sound Attenuation in Liquids. Journal of Experimental and Theoretical Physics, 3, 438-449. (in Russian) [30] Bakhvalov, N.S. and Panasenko, G.P. (1984) Averaging of Processes in Periodic Media. Nauka, Moscow. (in Russian) [31] Bakhvalov, N.S. and Eglit, M.E. (1983) Processes in Periodic Media Not Described in Terms of Averaged Characteristics. Doklady Akademii Nauk SSSR, 268, 836-840. [32] Berdichevsky, V.L. (2010) Variational Principles of Continuum Mechanics. Springer-Verlag, Berlin-Heidelberg. [33] Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory. Springer, New York. [34] Korn, G. and Korn, T. (1961) Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York-Toronto- London. [35] Kudinov, V.M. and Palamarchuk, B.I. (1976) Parameters of Shock Waves under Explosion of Explosive Charge in Foam. Soviet Physics Doklady, 228, 555-557. [36] Kudinov, V.M., Palamarchuk, B.I. and Vakhnenko, V.A. (1983) Attenuation of a Strong Shock Wave in a Two-Phase Medium. Soviet Physics Doklady, 272, 1080-1083. [37] Kudinov, V.M., Palamarchuk, B.I., Vakhnenko, V.A., et al. (1983) Relaxation Phenomena in a Foamy Structure. In: Bowen, J.R., Manson, N., Oppenheim, A.K. and Soloukhin, R.I., Eds., Shook Waves, Explosions, and Detonations, American Institute of Aeronautics and Astronautics, New York, 96-118. [38] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1982) Effect of Thermal Relaxation of Attenuation of Shock Waves in Two-Phase Medium. Soviet Applied Mechanics, 18, 1126-1133.
http://dx.doi.org/10.1007/BF00882225 [39] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1983) Analogy of Motion of the Two-Phase Media Containing Incompressible and Gaseous Phases with Gas Motion. Doklady Akademii Nauk Ukrainskoj SSR Serija A, 6, 22-24. (in Russian) [40] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1984) Damping of Strong Shocks in Relaxing Media. Combustion, Explosion and Shock Waves, 20, 97-103.
http://dx.doi.org/10.1007/BF00749928 [41] Vakhnenko, V.A. and Palmarchuk, B.I. (1984) Description of Shock-Wave Processes in a Two-Phase Medium Containing an Incompressible Phase. Journal of Applied Mechanics and Technical Physics, 25, 101-107.
http://dx.doi.org/10.1007/BF00916876 [42] Vakhnenko, V.A. and Palmarchuk, B.I. (1986) Evolution of Strong Shock Waves in a Medium with Thermal Relaxation. Soviet Applied Mechanics, 22, 267-272.
http://dx.doi.org/10.1007/BF00887250 [43] Rudinger, G. (1965) Some Effects of Finite Particle Volume on the Dynamics of Gas-Particle Mixtures. AIAA Journal, 3, 1217-1222.
http://dx.doi.org/10.2514/3.3112 [44] Rudinger, G. (1964) Some Properties of Shook Relaxation in Gas Plows Garring Small Particles. Physics of Fluids, 7, 658-663. http://dx.doi.org/10.1063/1.1711265 [45] Rudinger, G. and Chang, A. (1964) Analysis of Nonsteady Two-Phase Flow. Physics of Fluids, 7, 1747-1754.
http://dx.doi.org/10.1063/1.2746772 [46] Pai, I., Menon, S. and Fan, Z.Q. (1980) Similarity Solution of a Strong Shock Wave Propagating in a Mixture of Gas and Dusty Particles. International Journal of Engineering Science, 18, 1365-1378.
http://dx.doi.org/10.1016/0020-7225(80)90093-2 [47] Suzuki, T., Ohyagi, S., Higashino, F., et al. (1976) The Propagation of Reacting Blast Waves through Inert Particle Clouds. Acta Astronautica, 3, 517-529.
http://dx.doi.org/10.1016/0094-5765(76)90158-2 [48] Vakhnenko, V.O. (2010) An Analogy of the Self-Similar Flows of a Gas and a Two-Phase Medium with Noncompressive Component. Reports of the National Academy of Sciences of Ukraine, 12, 97-104. (in Ukrainian) [49] Vakhnenko, V.O. (2011) Similarity in Stationary Motions of Gas and Two-Phase Medium with Incompressible Component. International Journal of Non-Linear Mechanics, 46, 1356-1360.
http://dx.doi.org/10.1016/j.ijnonlinmec.2011.07.009 [50] Korobeinikov, V.P. (1991) Problems of Point-Blast Theory. American Institute of Physics Press, New York. [51] Sedov, L.I. (1993) Similarity and Dimensional Methods in Mechanics. CRC Press, Boca Raton. [52] Kestenboim, Kh.S., Roslyakov, G.S. and Chudov, L.A. (1974) Point Explosion. Methods of Calculation. Tables. Nauka Press, Moscow. [53] Gel’fand, B.E., Gubanov, A.V. and Timofeev, E.I. (1981) Features of Shock-Wave Propagation in Foams. Fizika Goreniya i Vzryva, 17, 129-136. [54] Gel’fand, B.E., Gubanov, A.V. and Gubin, S.A. (1977) Attenuation of Shock Waves in a Two-Phase Liquid-Gas-Bubble Medium. Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 1, 173-176. [55] Tseitlin, Ya.I., Gil’manov, R.A. and Nilov, V.G. (1980) On Localization of Action of the Blast Hydro-Shock Wave by Bubble Screen. In: Korenistov, A.V., Ed., Vzryvnoe delo (Explosives), 82/39, Nedra Press, Moscow, 264-272. [56] Kudinov, V.M., Palamarchuk, B.I., Gel’fand, B.E. and Gubin, S.A. (1976) The Use of Foam for Damping Shock- Waves at Welding and Cutting by Explosion. Avtomaticheskaya Svarka (Automatic Welding), 69, 12-16. [57] Kherrmann, V. (1976) Governing Equations for Compressible Porous Materials. In: Mechanics (New Results in Foreign Science). Problems in the Theory of Plasticity, 7, Mir, Moscow, 178-216. (in Russian) [58] Nakoryakov, V.E., Pokusaev, B.G. and Shreieber, I.R. (1993) Wave Propogation in Gas-Liquid Media. Begell House, New York. [59] Noordrij, L. and Van Wijngaarden, L. (1979) Relaxation Effects Caused by Relative Motion on Shook Waves in Gas-Bubble-Liguid Mixtures. Journal of Fluid Mechanics, 66, 1-9. [60] Parkin, B.R., Gilmore, F.R. and Broud, G.L. (1974) Shock Waves in Water with Bubbles of Air. In: Underwater and Underground Explosions (Russian Translation), Mir, Moscow, 152-258. [61] Held, M. (1983) Blast Waves in Free Air. Propellants, Explosives, Pyrotechnics, 8, 1-7.
http://dx.doi.org/10.1002/prep.19830080102 [62] Adushkin, V.V. (1963) On Shock Wave Forming and Explosion Products Flying Away. Journal of Applied Mechanics and Technical Physics, 5, 107-120. (in Russian) [63] Adushkin, V.V. and Korotkov, A.I. (1961) Shock Wave Parameters near Chemical Explosion in Air. Journal of Applied Mechanics and Technical Physics, 5, 119-123. (in Russian) [64] Baker, W.E., Cox, P.A., Westine, P.S., et al. (1983) Explosion Hazards and Evaluation. Elsevier Scientific Publishing Company, Amsterdam-Oxford-New York. [65] Brode, H.I. (1955) Numerical Solutions of Spherical Blast Waves. Journal of Applied Physics, 26, 766-775.
http://dx.doi.org/10.1063/1.1722085 [66] Brode, H.I. (1976) Point Explosion in Air. In: Ishlinsky, A.Yu. and Chornyy, G.G., Eds., Mechanics, Numerical Solutions of Explosions, Mir, Moscow, 4, 7-70. [67] Gekber, N. and Bartos, J.M. (1974) Strong Spherical Blast Waves in a Dust-Laden Gas. AIAA Journal, 12, 120-122.
http://dx.doi.org/10.2514/3.49176
[1] Bonner, B.P. and Wanamaker, B.J. (1991) Acoustic Nonlinearities Produced by a Single Macroscopic Fracture in Granite. In: Thompson, D.O. and Chimenti, D.E., Eds., Review of Progress in Quantitative Nondestructive Evaluation, Springer, New York, 1861-1867.
http://dx.doi.org/10.1007/978-1-4615-3742-7_94 [2] Guyer, R.A. and Johnson, P.A. (2009) Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Rocks, Soil, Concrete. WILEY-VCH Verlag GmbH and Co. KgaA, Weinheim.
http://dx.doi.org/10.1002/9783527628261 [3] Johnson, P.A. and McCall, K.R. (1994) Observation and Implications of Nonlinear Elastic Wave Response in Rock. Geophysical Research Letters, 21, 165-168.
http://dx.doi.org/10.1029/93GL03162 [4] Johnson, P.A., Shankland, T.J., O’Connell, R.J. and Albright, J.N. (1987) Nonlinear Generation of Elastic Waves in Crystalline Rock. Journal of Geophysical Research, 92, 3597-3602.
http://dx.doi.org/10.1029/JB092iB05p03597 [5] Rodionov, V.N., Sizov, I.A. and Tsvetkov, V.M. (1986) Fundamentals of Geomechanics. Nedra Press, Moscow. (in Russian) [6] Sadovsky, M.A. and Pisarenko, G.F. (1991) Seismic Process in Block Medium. Nauka, Moscow. (in Russian) [7] Sadovsky, M.A., Ed. (1989) Discrete Properties of Geophysics Medium. Nauka, Moscow. (in Russian) [8] Achenbach, J.D. (1973) Wave Propagation in Plastic Solids. North-Holland, Amsterdam. [9] Aki, K. and Richards, P.G. (1980) Quantitative Seismology. Theory and Methods, Vol. I and II. W.H. Freeman, San Francisco. [10] Truesdell, C. (1984) Rational Thermodynamics. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5206-1 [11] Biot, M.A. (1956) Theory of Propagation of Elastic Waves in a Fluid-Saturated Solid. I. Low-Frequency Range. Journal of the Acoustical Society of America, 28, 168-178.
http://dx.doi.org/10.1121/1.1908239 [12] Kutateladze, S.S. and Nakoryakov, V.E. (1984) Heat Exchange and Waves in Gas-Liquid Systems. Nauka, Novosibisk. (in Russian) [13] Lyakhov, G.M. (1982) Waves in Soils and Porous Multicomponent Media. Nauka, Moscow. (in Russian) [14] Raats, P.A.C. (1984) Applications of the Theory of Mixtures in Soil Physics. In: Truesdell, C., Ed., Rational Thermodynamics, Springer, New York, 326-343.
http://dx.doi.org/10.1007/978-1-4612-5206-1_16 [15] Rajagopal, K.R. and Tao, L. (1995) Machanics of Mixtures. World Scientific Publishing, Singapore. [16] Nikolaevskii, V.N. (1985) Viscoelasticity with Internal Oscillators as a Possible Model of Seismoactive Medium. Doklady Akademii Nauk SSSR, 283, 1321-1324. (in Russian) [17] Nigmatulin, R.I. (1987) Dynamics of Multiphase Media, Vol. I and II. Nauka, Moscow. (in Russian) [18] Struminskii, V.V. (1980) Mechanics and Technical Progress. Nauka, Moscow. (in Russian) [19] Vakhnenko, V.O., Danylenko, V.A. and Michtchenko, A.V. (1999) An Asymptotic Averaged Model of Nonlinear Long Waves Propagation in Media with a Regular Structure. International Journal of Non-Linear Mechanics, 34, 643-654.
http://dx.doi.org/10.1016/S0020-7462(98)00014-6 [20] Vakhnenko, V.O., Danylenko, V.A. and Michtchenko, A.V. (2000) Diagnostics of the Medium Structure by Long Wave of Finite Amplitude. International Journal of Non-Linear Mechanics, 35, 1105-1113.
http://dx.doi.org/10.1016/S0020-7462(99)00082-7 [21] Vakhnenko, V.A., Danylenko, V.A. and Kulich, V.V. (1993) Averaged Description of Wave Processes in Geophysical Medium. Geophysics Journal (Ukraine), 15, 66-71. [22] Vakhnenko, V.A., Danylenko, V.A. and Kulich, V.V. (1994) Averaged Description of Shock-Wave Processes in Periodic Media. Soviet Journal of Chemical Physics, 12, 534-546. [23] Vakhnenko, V.A. and Kulich, V.V. (1992) Long-Wave Processes in Periodic Media. Journal of Applied Mechanics and Technical Physics, 32, 814-820. [24] Landau, L.D. and Lifshitz, E.M. (1988) Fluids Mechanics. Pergamon, New York. [25] Lavrentiev, M. and Chabat, B. (1980) Effets Hyrdodynamiques et Modéles Mathématiques. Traduction franc. éditions, Mir, Moscow. [26] Clarke, J.E. (1984) Lectures on Plane Waves in Reacting Gases. Annals of Physics, 9, 211-306.
http://dx.doi.org/10.1051/anphys:0198400902021100 [27] Yasnikov, G.P. and Belousov, V.S. (1978) Effective Thermodynamic Functions of a Gas with Solid Particles. Journal of Engineering Physics, 34, 734-737.
http://dx.doi.org/10.1007/BF00860421 [28] Danylenko, V.A., Sorokina, V.V. and Vladimirov, V.A. (1993) On the Governing Equations in Relaxing Models and Self-Similar Quasiperiodic Solutions. Journal of Physics A, 26, 7125-7135.
http://dx.doi.org/10.1088/0305-4470/26/23/047 [29] Mandel’shtam, L.I. and Leonovich, M.A. (1937) To the Theory of a Sound Attenuation in Liquids. Journal of Experimental and Theoretical Physics, 3, 438-449. (in Russian) [30] Bakhvalov, N.S. and Panasenko, G.P. (1984) Averaging of Processes in Periodic Media. Nauka, Moscow. (in Russian) [31] Bakhvalov, N.S. and Eglit, M.E. (1983) Processes in Periodic Media Not Described in Terms of Averaged Characteristics. Doklady Akademii Nauk SSSR, 268, 836-840. [32] Berdichevsky, V.L. (2010) Variational Principles of Continuum Mechanics. Springer-Verlag, Berlin-Heidelberg. [33] Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory. Springer, New York. [34] Korn, G. and Korn, T. (1961) Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York-Toronto- London. [35] Kudinov, V.M. and Palamarchuk, B.I. (1976) Parameters of Shock Waves under Explosion of Explosive Charge in Foam. Soviet Physics Doklady, 228, 555-557. [36] Kudinov, V.M., Palamarchuk, B.I. and Vakhnenko, V.A. (1983) Attenuation of a Strong Shock Wave in a Two-Phase Medium. Soviet Physics Doklady, 272, 1080-1083. [37] Kudinov, V.M., Palamarchuk, B.I., Vakhnenko, V.A., et al. (1983) Relaxation Phenomena in a Foamy Structure. In: Bowen, J.R., Manson, N., Oppenheim, A.K. and Soloukhin, R.I., Eds., Shook Waves, Explosions, and Detonations, American Institute of Aeronautics and Astronautics, New York, 96-118. [38] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1982) Effect of Thermal Relaxation of Attenuation of Shock Waves in Two-Phase Medium. Soviet Applied Mechanics, 18, 1126-1133.
http://dx.doi.org/10.1007/BF00882225 [39] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1983) Analogy of Motion of the Two-Phase Media Containing Incompressible and Gaseous Phases with Gas Motion. Doklady Akademii Nauk Ukrainskoj SSR Serija A, 6, 22-24. (in Russian) [40] Vakhnenko, V.A., Kudinov, V.M. and Palmarchuk, B.I. (1984) Damping of Strong Shocks in Relaxing Media. Combustion, Explosion and Shock Waves, 20, 97-103.
http://dx.doi.org/10.1007/BF00749928 [41] Vakhnenko, V.A. and Palmarchuk, B.I. (1984) Description of Shock-Wave Processes in a Two-Phase Medium Containing an Incompressible Phase. Journal of Applied Mechanics and Technical Physics, 25, 101-107.
http://dx.doi.org/10.1007/BF00916876 [42] Vakhnenko, V.A. and Palmarchuk, B.I. (1986) Evolution of Strong Shock Waves in a Medium with Thermal Relaxation. Soviet Applied Mechanics, 22, 267-272.
http://dx.doi.org/10.1007/BF00887250 [43] Rudinger, G. (1965) Some Effects of Finite Particle Volume on the Dynamics of Gas-Particle Mixtures. AIAA Journal, 3, 1217-1222.
http://dx.doi.org/10.2514/3.3112 [44] Rudinger, G. (1964) Some Properties of Shook Relaxation in Gas Plows Garring Small Particles. Physics of Fluids, 7, 658-663. http://dx.doi.org/10.1063/1.1711265 [45] Rudinger, G. and Chang, A. (1964) Analysis of Nonsteady Two-Phase Flow. Physics of Fluids, 7, 1747-1754.
http://dx.doi.org/10.1063/1.2746772 [46] Pai, I., Menon, S. and Fan, Z.Q. (1980) Similarity Solution of a Strong Shock Wave Propagating in a Mixture of Gas and Dusty Particles. International Journal of Engineering Science, 18, 1365-1378.
http://dx.doi.org/10.1016/0020-7225(80)90093-2 [47] Suzuki, T., Ohyagi, S., Higashino, F., et al. (1976) The Propagation of Reacting Blast Waves through Inert Particle Clouds. Acta Astronautica, 3, 517-529.
http://dx.doi.org/10.1016/0094-5765(76)90158-2 [48] Vakhnenko, V.O. (2010) An Analogy of the Self-Similar Flows of a Gas and a Two-Phase Medium with Noncompressive Component. Reports of the National Academy of Sciences of Ukraine, 12, 97-104. (in Ukrainian) [49] Vakhnenko, V.O. (2011) Similarity in Stationary Motions of Gas and Two-Phase Medium with Incompressible Component. International Journal of Non-Linear Mechanics, 46, 1356-1360.
http://dx.doi.org/10.1016/j.ijnonlinmec.2011.07.009 [50] Korobeinikov, V.P. (1991) Problems of Point-Blast Theory. American Institute of Physics Press, New York. [51] Sedov, L.I. (1993) Similarity and Dimensional Methods in Mechanics. CRC Press, Boca Raton. [52] Kestenboim, Kh.S., Roslyakov, G.S. and Chudov, L.A. (1974) Point Explosion. Methods of Calculation. Tables. Nauka Press, Moscow. [53] Gel’fand, B.E., Gubanov, A.V. and Timofeev, E.I. (1981) Features of Shock-Wave Propagation in Foams. Fizika Goreniya i Vzryva, 17, 129-136. [54] Gel’fand, B.E., Gubanov, A.V. and Gubin, S.A. (1977) Attenuation of Shock Waves in a Two-Phase Liquid-Gas-Bubble Medium. Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 1, 173-176. [55] Tseitlin, Ya.I., Gil’manov, R.A. and Nilov, V.G. (1980) On Localization of Action of the Blast Hydro-Shock Wave by Bubble Screen. In: Korenistov, A.V., Ed., Vzryvnoe delo (Explosives), 82/39, Nedra Press, Moscow, 264-272. [56] Kudinov, V.M., Palamarchuk, B.I., Gel’fand, B.E. and Gubin, S.A. (1976) The Use of Foam for Damping Shock- Waves at Welding and Cutting by Explosion. Avtomaticheskaya Svarka (Automatic Welding), 69, 12-16. [57] Kherrmann, V. (1976) Governing Equations for Compressible Porous Materials. In: Mechanics (New Results in Foreign Science). Problems in the Theory of Plasticity, 7, Mir, Moscow, 178-216. (in Russian) [58] Nakoryakov, V.E., Pokusaev, B.G. and Shreieber, I.R. (1993) Wave Propogation in Gas-Liquid Media. Begell House, New York. [59] Noordrij, L. and Van Wijngaarden, L. (1979) Relaxation Effects Caused by Relative Motion on Shook Waves in Gas-Bubble-Liguid Mixtures. Journal of Fluid Mechanics, 66, 1-9. [60] Parkin, B.R., Gilmore, F.R. and Broud, G.L. (1974) Shock Waves in Water with Bubbles of Air. In: Underwater and Underground Explosions (Russian Translation), Mir, Moscow, 152-258. [61] Held, M. (1983) Blast Waves in Free Air. Propellants, Explosives, Pyrotechnics, 8, 1-7.
http://dx.doi.org/10.1002/prep.19830080102 [62] Adushkin, V.V. (1963) On Shock Wave Forming and Explosion Products Flying Away. Journal of Applied Mechanics and Technical Physics, 5, 107-120. (in Russian) [63] Adushkin, V.V. and Korotkov, A.I. (1961) Shock Wave Parameters near Chemical Explosion in Air. Journal of Applied Mechanics and Technical Physics, 5, 119-123. (in Russian) [64] Baker, W.E., Cox, P.A., Westine, P.S., et al. (1983) Explosion Hazards and Evaluation. Elsevier Scientific Publishing Company, Amsterdam-Oxford-New York. [65] Brode, H.I. (1955) Numerical Solutions of Spherical Blast Waves. Journal of Applied Physics, 26, 766-775.
http://dx.doi.org/10.1063/1.1722085 [66] Brode, H.I. (1976) Point Explosion in Air. In: Ishlinsky, A.Yu. and Chornyy, G.G., Eds., Mechanics, Numerical Solutions of Explosions, Mir, Moscow, 4, 7-70. [67] Gekber, N. and Bartos, J.M. (1974) Strong Spherical Blast Waves in a Dust-Laden Gas. AIAA Journal, 12, 120-122.
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