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Journal articles on the topic 'Rankine-Hugoniot'

Author: Grafiati

Published: 6 September 2023

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1

Avramenko, Andriy A., Andrii I. Tyrinov, and Igor V. Shevchuk. "Analytical simulation of normal shock waves in turbulent flow." Physics of Fluids 34, no. 5 (May 2022): 056101. http://dx.doi.org/10.1063/5.0093205.

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The focus of the work is on analytical modeling of normal shock wave propagation in a turbulent adiabatic gas flow. For this, a modified Rankine–Hugoniot model was developed. A solution is obtained for the Rankine–Hugoniot conditions in a turbulent gas flow with different turbulence intensity. Variation of the velocity of an adiabatic turbulent gas flow during its passage through a normal shock wave is elucidated depending on the turbulence intensity. The equation of the modified Hugoniot adiabat is also obtained.

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2

Mellmann, Marcel, and Markus Scholle. "Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism." Symmetry 13, no. 9 (September 9, 2021): 1662. http://dx.doi.org/10.3390/sym13091662.

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By rigorous analysis, it is proven that from discontinuous Lagrangians, which are invariant with respect to the Galilean group, Rankine–Hugoniot conditions for propagating discontinuities can be derived via a straight forward procedure that can be considered an extension of Noether’s theorem. The use of this general procedure is demonstrated in particular for a Lagrangian for viscous flow, reproducing the well known Rankine–Hugoniot conditions for shock waves.

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3

Gedalin, Michael, Nikolai V. Pogorelov, and Vadim Roytershteyn. "Rankine–Hugoniot Relations Including Pickup Ions." Astrophysical Journal 889, no. 2 (January 30, 2020): 116. http://dx.doi.org/10.3847/1538-4357/ab6660.

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4

Kentzer, Czeslaw P. "Quasilinear form of Rankine-Hugoniot jump conditions." AIAA Journal 24, no. 4 (April 1986): 691–93. http://dx.doi.org/10.2514/3.9332.

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5

Gao, Yang, and Chung K. Law. "RANKINE-HUGONIOT RELATIONS IN RELATIVISTIC COMBUSTION WAVES." Astrophysical Journal 760, no. 2 (November 15, 2012): 122. http://dx.doi.org/10.1088/0004-637x/760/2/122.

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6

SATOH, Akira. "Rankine-Hugoniot Relations for Lennard-Jones Liquids." Transactions of the Japan Society of Mechanical Engineers Series B 58, no. 549 (1992): 1419–25. http://dx.doi.org/10.1299/kikaib.58.1419.

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7

Satoh, Akira. "Rankine-Hugoniot Relations for Lennard-Jones Liquids." Journal of Fluids Engineering 116, no. 3 (September 1, 1994): 625–30. http://dx.doi.org/10.1115/1.2910323.

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The purpose of the present study is to clarify the Rankine-Hugoniot relations for Lennard-Jones liquids. First, Monte Carlo simulations are conducted to evaluate the state quantities such as the pressures, the internal energies, and the sound velocities. These computed values are used to obtain the approximate expressions for the state quantities by the method of least squares. The Rankine-Hugoniot relations are then clarified numerically as a function of the shock Mach number by solving the basic equations together with those approximate expressions. For liquid shock waves, not only the pressure but also the temperature increases much larger than those for an ideal gas. The results obtained here enable us to conduct more efficient molecular dynamics simulations such as simulating shock fronts alone for the investigation of the internal structures of liquid shock waves.

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8

Renardy, Michael. "On Rankine—Hugoniot conditions for Maxwell liquids." Journal of Non-Newtonian Fluid Mechanics 32, no. 1 (January 1989): 69–77. http://dx.doi.org/10.1016/0377-0257(89)85041-4.

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9

Szabo, A. "An improved solution to the “Rankine-Hugoniot” problem." Journal of Geophysical Research 99, A8 (1994): 14737. http://dx.doi.org/10.1029/94ja00782.

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10

GAVRILYUK, S. L., and R. SAUREL. "Rankine–Hugoniot relations for shocks in heterogeneous mixtures." Journal of Fluid Mechanics 575 (March 2007): 495–507. http://dx.doi.org/10.1017/s0022112006004496.

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The conservation of mass, momentum and energy are not sufficient to close a system of jump relations for shocks propagating in a heterogeneous mixture of compressible fluids. We propose here a closed set of relations corresponding to a two-stage structure of shock fronts. At the first stage, microkinetic energy due to the relative motion of mixture components is produced at the shock front. At the second stage, this microkinetic energy disappears inducing strong variations in the thermodynamical states that reach mechanical equilibrium. The microkinetic energy produced at the shock front is estimated by using an idea developed earlier for turbulent shocks in compressible fluids. The relaxation zone between the shocked state and the equilibrium state is integrated over a thermodynamic path a justification of which is provided. Comparisons with experiments on shock propagation in a mixture of condensed materials confirm the proposed theory.

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11

GEDALIN, M., and M. BALIKHIN. "Rankine–Hugoniot relations for shocks with demagnetized ions." Journal of Plasma Physics 74, no. 2 (April 2008): 207–14. http://dx.doi.org/10.1017/s0022377807006708.

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AbstractThe width of a quasi-perpendicular collisionless shock front is smaller than the convective ion gyroradius so that ions become demagnetized in the ramp. An approach is proposed for derivation of approximate expressions for the magnetic compression ratio and cross-shock potential from the analysis of the ion motion across the ramp and pressure balance condition, without making assumptions about the ion equation of state. The cross-shock potential and magnetic compression ratio are found as functions of the Mach number for low-Mach-number perpendicular shocks.

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12

EHRT, JULIA, and JÖRG HÄRTERICH. "ASYMPTOTIC BEHAVIOR OF SPATIALLY INHOMOGENEOUS BALANCE LAWS." Journal of Hyperbolic Differential Equations 02, no. 03 (September 2005): 645–72. http://dx.doi.org/10.1142/s0219891605000579.

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We study the longtime behavior of spatially inhomogeneous scalar balance laws with periodic initial data and a convex flux. Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution. This extends results of Lyberopoulos, Sinestrari and Fan and Hale obtained in the spatially homogeneous case. The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory. A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.

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13

Carioli, S. M. "Solutions of the Rankine–Hugoniot relations in relativistic magnetohydrodynamics." Physics of Fluids 29, no. 3 (1986): 672. http://dx.doi.org/10.1063/1.865916.

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14

Viñas, Adolfo F., and Jack D. Scudder. "Fast and optimal solution to the “Rankine-Hugoniot problem”." Journal of Geophysical Research 91, A1 (1986): 39. http://dx.doi.org/10.1029/ja091ia01p00039.

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15

Kennel, C. F., R. D. Blandford, and P. Coppi. "MHD intermediate shock discontinuities. Part 1. Rankine—Hugoniot conditions." Journal of Plasma Physics 42, no. 2 (October 1989): 299–319. http://dx.doi.org/10.1017/s0022377800014379.

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Recent numerical investigations have focused attention once more on the role of intermediate shocks in MHD. Four types of intermediate shock are identified using a graphical representation of the MHD Rankine-Hugoniot conditions. This same representation can be used to exhibit the close relationship of intermediate shocks to switch-on shocks and rotational discontinuities. The conditions under which intermediate discontinuities can be found are elucidated. The variations in velocity, pressure, entropy and magnetic-field jumps with upstream parameters in intermediate shocks are exhibited graphically. The evolutionary arguments traditionally advanced against intermediate shocks may fail because the equations of classical MHD are not strictly hyperbolic.

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16

Wei-Min, Gu, and Lu Ju-Fu. "Standing Rankine–Hugoniot Shocks in Black Hole Accretion Discs." Chinese Physics Letters 21, no. 12 (December 2004): 2551–54. http://dx.doi.org/10.1088/0256-307x/21/12/064.

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17

Jaisankar, S., and S. V. Raghurama Rao. "A central Rankine–Hugoniot solver for hyperbolic conservation laws." Journal of Computational Physics 228, no. 3 (February 2009): 770–98. http://dx.doi.org/10.1016/j.jcp.2008.10.002.

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18

Yuryk, I. I. "Invariant solutions of a system of Euler equations that satisfy the Rankine–Hugoniot conditions." Reports of the National Academy of Sciences of Ukraine, no. 7 (July 24, 2018): 10–19. http://dx.doi.org/10.15407/dopovidi2018.07.010.

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19

Scherer, K., L. R. Baalmann, H. Fichtner, J. Kleimann, D. J. Bomans, K. Weis, S. E. S. Ferreira, and K. Herbst. "MHD-shock structures of astrospheres: λ Cephei -like astrospheres." Monthly Notices of the Royal Astronomical Society 493, no. 3 (February 20, 2020): 4172–85. http://dx.doi.org/10.1093/mnras/staa497.

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ABSTRACT The interpretation of recent observations of bow shocks around O-stars and the creation of corresponding models require a detailed understanding of the associated (magneto-)hydrodynamic structures. We base our study on 3D numerical (magneto-)hydrodynamical models, which are analysed using the dynamically relevant parameters, in particular, the (magneto)sonic Mach numbers. The analytic Rankine–Hugoniot relation for HD and MHD are compared with those obtained by the numerical model. In that context, we also show that the only distance which can be approximately determined is that of the termination shock, if it is an HD shock. For MHD shocks, the stagnation point does not, in general, lie on the inflow line, which is the line parallel to the inflow vector and passing through the star. Thus an estimate via the Bernoulli equation as in the HD case is, in general, not possible. We also show that in O-star astrospheres, distinct regions exist in which the fast, slow, Alfvénic, and sonic Mach numbers become lower than one, implying subslow magnetosonic as well as subfast and subsonic flows. Nevertheless, the analytic MHD Rankine–Hugoniot relations can be used for further studies of turbulence and cosmic ray modulation.

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20

Shelkovich, V. M. "The Rankine-Hugoniot conditions and balance laws for δ-shocks." Journal of Mathematical Sciences 151, no. 1 (May 2008): 2781–92. http://dx.doi.org/10.1007/s10948-008-0173-y.

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21

LeFloch, Philippe G., and Mai Duc Thanh. "Properties of rankine-hugoniot curves for van der Waals fluids." Japan Journal of Industrial and Applied Mathematics 20, no. 2 (June 2003): 211–38. http://dx.doi.org/10.1007/bf03170427.

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22

Scherer, Klaus, Horst Fichtner, Hans Jörg Fahr, Christian Röken, and Jens Kleimann. "GENERALIZED MULTI-POLYTROPIC RANKINE–HUGONIOT RELATIONS AND THE ENTROPY CONDITION." Astrophysical Journal 833, no. 1 (December 7, 2016): 38. http://dx.doi.org/10.3847/1538-4357/833/1/38.

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23

Ketcheson, David I., and Manuel Quezada de Luna. "Effective Rankine–Hugoniot conditions for shock waves in periodic media." Communications in Mathematical Sciences 18, no. 4 (2020): 1023–40. http://dx.doi.org/10.4310/cms.2020.v18.n4.a6.

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24

Pain, J. C. "Shell-structure effects on high-pressure Rankine–Hugoniot shock adiabats." High Energy Density Physics 3, no. 1-2 (May 2007): 204–10. http://dx.doi.org/10.1016/j.hedp.2007.02.013.

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25

Jenny, P., and B. Müller. "Rankine–Hugoniot–Riemann Solver Considering Source Terms and Multidimensional Effects." Journal of Computational Physics 145, no. 2 (September 1998): 575–610. http://dx.doi.org/10.1006/jcph.1998.6037.

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26

Krunić, Tanja, and Marko Nedeljkov. "Shadow wave solutions for a scalar two-flux conservation law with Rankine–Hugoniot deficit." Journal of Hyperbolic Differential Equations 18, no. 03 (September 2021): 539–56. http://dx.doi.org/10.1142/s021989162150017x.

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This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

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27

Wang, Aiju, Wangxun Yu, and Yanyan Zhang. "The Interaction of Waves in the Zero-Pressure Euler Equations with a Coulomb-Like Friction Term." Mathematical Problems in Engineering 2022 (February 28, 2022): 1–6. http://dx.doi.org/10.1155/2022/4837968.

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In this study, the interaction of waves in the zero-pressure Euler equations with a Coulomb-like friction term is considered, which is equivalent to the Riemann problem with three constant initial states for the zero-pressure Euler equations. By solving generalized Rankine–Hugoniot relations under suitable entropy conditions, four different structures of explicit solutions are obtained uniquely, in which the interactions among contact discontinuity, vacuum, and delta shock are presented.

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28

DELMONT, P., and R. KEPPENS. "Parameter regimes for slow, intermediate and fast MHD shocks." Journal of Plasma Physics 77, no. 2 (March 8, 2010): 207–29. http://dx.doi.org/10.1017/s0022377810000115.

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AbstractWe investigate under which parameter regimes the magnetohydrodynamic (MHD) Rankine–Hugoniot conditions, which describe discontinuous solutions to the MHD equations, allow for slow, intermediate and fast shocks. We derive limiting values for the upstream and downstream shock parameters for which shocks of a given shock-type can occur. We revisit this classical topic in nonlinear MHD dynamics, augmenting the recent time reversal duality finding by in the usual shock frame parametrization.

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29

MYONG, R. S., and P. L. ROE. "Shock waves and rarefaction waves in magnetohydrodynamics. Part 2. The MHD system." Journal of Plasma Physics 58, no. 3 (October 1997): 521–52. http://dx.doi.org/10.1017/s0022377897005941.

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In Part 1 of this study, a model set exactly preserving the MHD hyperbolic singularities was considered. By developing the viscosity admissibility condition, it was shown that the intermediate shocks are necessary to ensure that the planar Riemann problem is well-posed. Here in Part 2, the MHD Rankine–Hugoniot condition and rarefaction-wave relations are presented in phase space, which allows construction of analytical solutions of the planar MHD Riemann problem. In this process, a viscosity admissibility condition is proposed to determine physically admissible shocks. A complete account of MHD Hugoniot loci is given, leading to a classification of several subproblems in which the solution patterns are qualitatively same. Finally, it is shown that the planar MHD Riemann problem is well-posed using intermediate shocks that have been considered non-evolutionary.

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30

Zhang, Yanyan, and Yu Zhang. "Delta-Shock Solution to the Eulerian Droplet Model by Variable Substitution Method." Zeitschrift für Naturforschung A 75, no. 3 (March 26, 2020): 201–10. http://dx.doi.org/10.1515/zna-2019-0256.

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AbstractBy introducing a special kind of variable substitution, we skillfully solve the delta-shock and vacuum solutions to the one-dimensional Eulerian droplet model. The position, propagation speed, and strength of the delta shock wave are derived under the generalised Rankine–Hugoniot relation and entropy condition. Moreover, we show that the Riemann solution of the Eulerian droplet model converges to the corresponding the pressureless Euler system solution as the drag coefficient goes to zero.

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31

Jackson, C. R., W. E. Lear, and S. A. Sherif. "Rankine–Hugoniot analysis of two-phase flow with inter-phase slip." Acta Astronautica 45, no. 11 (December 1999): 679–86. http://dx.doi.org/10.1016/s0094-5765(99)00183-6.

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32

Gedalin, Michael, Michal Golan, Nikolai V. Pogorelov, and Vadim Roytershteyn. "Change of Rankine–Hugoniot Relations during Postshock Relaxation of Anisotropic Distributions." Astrophysical Journal 940, no. 1 (November 1, 2022): 21. http://dx.doi.org/10.3847/1538-4357/ac958d.

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Abstract Collisionless shocks channel the energy of the directed plasma flow into the heating of the plasma species and magnetic field enhancement. The kinetic processes at the shock transition cause the ion distributions just behind the shock to be nongyrotropic. Gyrotropization and subsequent isotropization occur at different spatial scales. Accordingly, for a given upstream plasma and magnetic field state, there would be different downstream states corresponding to the anisotropic and isotropic regions. Thus, at least two sets of Rankine–Hugoniot relations are needed, in general, to describe the connection of the downstream measurable parameters to the upstream ones. We establish the relation between the two sets.

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33

Lin, Xiao-Biao. "Generalized Rankine–Hugoniot Condition and Shock Solutions for Quasilinear Hyperbolic Systems." Journal of Differential Equations 168, no. 2 (December 2000): 321–54. http://dx.doi.org/10.1006/jdeq.2000.3889.

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34

PEAKE, N. "A NOTE ON "COMPUTATIONAL AEROACOUSTICS EXAMPLES SHOWING THE FAILURE OF THE ACOUSTIC ANALOGY THEORY TO IDENTIFY THE CORRECT NOISE SOURCES" BY CKW TAM." Journal of Computational Acoustics 12, no. 04 (December 2004): 631–34. http://dx.doi.org/10.1142/s0218396x04002420.

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In a recent paper (J. Computational Acoustics10 (2002) 387–405) Tam has claimed that the famous Lighthill Acoustic Analogy predicts the wrong flow field for the simple problem of the propagation of a normal shock. However, we show that Tam has misinterpreted the results of his analysis, and that when this error is corrected the results of the Acoustic Analogy are brought into exact agreement with the well-known Rankine–Hugoniot solution of the Euler equations.

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35

Shanmugaraju, A., and S. Umapathy. "On the Possibility of Radio Emission from Quasi-parallel and Quasi-perpendicular Propagation of Shocks." International Astronomical Union Colloquium 179 (2000): 259–62. http://dx.doi.org/10.1017/s0252921100064629.

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AbstractA set of 21 solar type II radio bursts observed using Hiraiso radio spectrograph have been analysed to study the direction of propagation of coronal shocks. A simple analysis is carried out to find the approximate angle between the shock normal and magnetic field by solving the Rankine-Hugoniot MHD relation with assumption of Alfven speed and plasma beta. From this analysis, it is suggested that both quasi-parallel shocks (favourable) and quasi-perpendicular shocks can generate type II bursts depending upon the circumstances of the corona.

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36

Chen, Hao, Bin Zhang, and Hong Liu. "Non-Rankine–Hugoniot Shock Zone of Mach Reflection in Hypersonic Rarefied Flows." Journal of Spacecraft and Rockets 53, no. 4 (July 2016): 619–28. http://dx.doi.org/10.2514/1.a33411.

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37

Feng, H. Q., C. C. Lin, J. K. Chao, D. J. Wu, L. H. Lyu, and L. C. Lee. "From Rankine-Hugoniot relation fitting procedure: Tangential discontinuity or intermediate/slow shock?" Journal of Geophysical Research: Space Physics 112, A10 (October 2007): n/a. http://dx.doi.org/10.1029/2007ja012311.

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38

Majorana, A. "Analytical solutions of the Rankine-Hugoniot relations for a relativistic simple gas." Il Nuovo Cimento B 98, no. 2 (April 1987): 111–18. http://dx.doi.org/10.1007/bf02721473.

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39

He, Yong, Xiwei Hu, Yemin Hu, Zhonghe Jiang, and Jianhong Lü. "Rankine-Hugoniot relations of an axial shock in cylindrical non-neutral plasma." Physics of Plasmas 13, no. 9 (September 2006): 092116. http://dx.doi.org/10.1063/1.2355661.

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40

Sugiyama, Masaru, and Toshiyuki Isogai. "Microscopic Approach to Shock Waves in Crystal Solids. II Rankine-Hugoniot Relations." Japanese Journal of Applied Physics 35, Part 1, No. 6A (June 15, 1996): 3505–17. http://dx.doi.org/10.1143/jjap.35.3505.

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41

Guy, Capdeville. "A HLL-Rankine–Hugoniot Riemann solver for complex non-linear hyperbolic problems." Journal of Computational Physics 251 (October 2013): 156–93. http://dx.doi.org/10.1016/j.jcp.2013.05.024.

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42

Davey, K., and R. Darvizeh. "Neglected transport equations: extended Rankine–Hugoniot conditions and J -integrals for fracture." Continuum Mechanics and Thermodynamics 28, no. 5 (March 2, 2016): 1525–52. http://dx.doi.org/10.1007/s00161-016-0493-2.

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43

Lund, Halvor, Florian Müller, Bernhard Müller, and Patrick Jenny. "Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms." Computers & Fluids 101 (September 2014): 1–14. http://dx.doi.org/10.1016/j.compfluid.2014.05.022.

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44

Gedalin, M. "Rankine-Hugoniot Relations in Multispecies Plasma With Gyrotropic Anisotropic Downstream Ion Distributions." Journal of Geophysical Research: Space Physics 122, no. 12 (December 2017): 11,857–11,863. http://dx.doi.org/10.1002/2017ja024757.

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45

Cheng, Hongjun. "Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type." Advances in Mathematical Physics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/958120.

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This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

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46

Liu, Y. Y., H. S. Fu, J. B. Cao, Z. Wang, R. J. He, Z. Z. Guo, and C. X. Du. "Magnetic Discontinuities in the Inner Heliosphere: Do Intermediate Shocks Exist?" Astrophysical Journal 953, no. 1 (August 1, 2023): 34. http://dx.doi.org/10.3847/1538-4357/ace04c.

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Abstract Magnetic discontinuities are fundamental structures in space and laboratory plasmas where the changes in magnetic and velocity fields are constrained by Rankine–Hugoniot relations. Due to the absence of precise measurements for particles, some issues therein are hardly investigated. The nature of discontinuities driven by the magnetohydrodynamics (MHD) turbulence, and the intermediate shock are two puzzles to be solved. The MHD turbulence generates numerous discontinuities with both small normal magnetic fields and nearly constant magnetic field magnitudes in statistics. By utilizing the data from the Parker Solar Probe, we identify among the turbulence-driven discontinuities two components that exhibit diverse statistical characteristics of the plasma density, and reveal that these discontinuities comprise 80.2% rotational and 19.8% tangential discontinuities. Then, we note a special class of discontinuities within 0.35 au that have jump conditions similar to that of the rotational discontinuity and the shock simultaneously, including (1) positively correlated jumps in the plasma density and temperature, (2) a small change in the magnetic field magnitude, and (3) opposite tangential magnetic fields on two sides. These features conform to the theoretical intermediate shock, which previous studies have found to not practically exist due to the breakdown of the evolutionary condition. By the conservation law of the mass flux across a boundary, we calculate their propagation speeds and find three intermediate shock candidates with super-Alfvénic upstream and sub-Alfvénic downstream flows. This work can improve our understanding of plasma intermittencies and suggests reassessing conclusions based on ideal MHD Rankine–Hugoniot relations.

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47

De la cruz, Richard, Juan Galvis, Juan Carlos Juajibioy, and Leonardo Rendón. "Delta Shock Wave for the Suliciu Relaxation System." Advances in Mathematical Physics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/354349.

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We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered3×3system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data inL∞. We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.

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48

Zank, G. P., and J. F. Mckenzie. "The interaction of long-wavelength compressive waves with a cosmic ray shock." Journal of Plasma Physics 37, no. 3 (June 1987): 363–72. http://dx.doi.org/10.1017/s0022377800012241.

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This paper investigates the stability of a cosmic ray shock to long-wavelength perturbations. The problem is formulated in terms of finding the transmission coefficient for compressive waves across a cosmic ray shock by solving the generalized, two-fluid Rankine-Hugoniot relations. For strong shocks, the transmission coefficient confirms that compressive waves can undergo considerable amplification on passage through such shocks. The resonances of the transmission coefficient provides us with the dispersion equation governing the stability of the shock to long-wavelength ripple-like distortions. By using the principle of the argument method, it is established that cosmic ray shocks are stable.

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49

Eliezer, Shalom, Shirly Vinikman Pinhasi, José Maria Martinez Val, Erez Raicher, and Zohar Henis. "Heating in ultraintense laser-induced shock waves." Laser and Particle Beams 35, no. 2 (April 3, 2017): 304–12. http://dx.doi.org/10.1017/s0263034617000192.

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AbstractThis paper considers the heating of a target in a shock wave created in a planar geometry by the ponderomotive force induced by a short laser pulse with intensity higher than 1018 W/cm2. The shock parameters were calculated using the relativistic Rankine–Hugoniot equations coupled to a laser piston model. The temperatures of the electrons and the ions were calculated as a function of time by using the energy conservation separately for ions and electrons. These equations are supplemented by the ideal gas equations of state (with one or three degrees of freedom) separately for ions and electrons. The efficiency of the transition of the work done by the laser piston into internal thermal energy is calculated in the context of the Hugoniot equations by taking into account the binary collisions during the shock wave formation from the target initial condition to the compressed domain. It is shown that for each laser intensity there is threshold pulse duration for the formation of a shock wave. The explicit calculations are done for an aluminum target.

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Liu, Haochen, Hao Chen, Bin Zhang, and Hong Liu. "Effects of Mach Number on Non-Rankine–Hugoniot Shock Zone of Mach Reflection." Journal of Spacecraft and Rockets 56, no. 3 (May 2019): 761–70. http://dx.doi.org/10.2514/1.a34251.

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