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tandfonline.com/action/journalInformation?journalCode=tmma20 Mathematical Modelling and Analysis ISSN: 1392-6292 (Print) 1648-3510 (Online) Journal homepage: https://www.tandfonline.com/loi/tmma20 Asymptotical solutions for a vibrationally relaxing gas R. Arora To cite this article: R. Arora (2009) Asymptotical solutions for a vibrationally relaxing gas, Mathematical Modelling and Analysis, 14:4, 423-434 To link to this article: https://doi.org/10.3846/1392-6292.2009.14.423-434 Published online:

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14 Number 4, 2009, pages 423C434 ISSN 1648-3510 online Doi:10.3846/1392-6292.2009.14.423-434 www.vgtu.lt/mma/ c Vilnius Gediminas Technical University,

2009 Asymptotical Solutions for a Vibrationally Relaxing Gas? R. Arora DPT, IIT Roorkee, Saharanpur Campus Saharanpur-247001, U.P., India E-mail: [email protected], [email protected]. Received

2009 03 06;

revised

2009 08 27;

published online November 10,

2009 Abstract. Using the weakly non-linear geometrical acoustics theory, we obtain the small amplitude high frequency asymptotic solution to the basic equations governing one dimensional unsteady planar, spherically and cylindrically symmetric ?ow in a vibrationally relaxing gas with Van der Waals equation of state. The transport equa- tions for the amplitudes of resonantly interacting waves are derived. The evolutionary behavior of non-resonant wave modes culminating into shock waves is also studied. Key words: Weakly Non-linear hyperbolic waves, Asymptotic Solution, Resonance, Planar and non-planar Shock Waves, Vibrationally Relaxing Gas.

1 Introduction When non-equilibrium e?ects are considered in ?uid-dynamic equations, the analysis becomes considerably more complex than the corresponding classical gas dynamic ?ow because of nonlinear coupling between the relaxing mode and the ?uid ?ow. The occurrence of shock waves in a relaxing gas has been well studied in the past. The propagation of shock waves in a relaxing gas constitutes a problem of great interest for researchers and scientists. It is well known that at high temperatures, the internal energy of the gas molecules consists of translational, rotational and vibrational components. When the gas is in equilibrium, each internal mode is characterized by the same temperature. A rate of transfer of energy from one mode to another mode can be observed by inducing small changes in any of these temperatures and observing the rate of return to equilibrium. For instance, when gas is compressed by the mechanical action of a piston or by the passage of a shock front, the whole energy goes initially to increase the translational energy, and it is followed by a relaxation from translational mode to rotational mode and ? Research funding from DST, India vide Project grant number SR/FTP/MS-12/2008 is gratefully acknowledged

424 R. Arora also from translational mode to vibrational mode until the equilibrium between these modes is re-established. This is called a relaxation process. A number of problems relating to the e?ects of nonlinearity in gases with in- ternal relaxation have been studied previously, in particular, by Clarke and Mc- Chesney [7], Scott and Johannesen [22], Blythe [4], Ockenden and Spence [19], Parker [20], Radha and Sharma [21] among others. There has been widespread interest in the nonlinear wave phenomena. The work of Whitham [28], Moodie et al. [18], He and Moodie ([9, 10]), Shtaras [27], Kalyakin [13], Krylovas and ?iegis ([14, 15]), Sharma and Radha [24], Sharma and Srinivasan [25], Sharma and Arora [23], Arora and Sharma [3], and Arora ([1, 2]) is worth mentioning in the context of nonlinear wave propagation in gas dynamic media. We use asymptotic method for the analysis of weakly nonlinear hyperbolic waves. Choquet-Bruhat [6] proposed a method to discuss shockless solutions of hyperbolic systems which depend upon a single phase function. Germain [8] has given the general discussion of single phase progressive waves. Hunter and Keller [11] established a general non resonant multi-wave mode theory which has led to several interesting generalizations by Majda and Rosales [16] and Hunter et al [12] to include resonantly interacting multi-wave mode features. Krylovas and ?iegis [14] developed a method of averaging for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. We use the method of weakly non-linear geometrical acoustics to obtain the small amplitude high frequency asymptotic solution to the basic equations governing one dimensional unsteady planar, cylindrically and spherically sym- metric ?ow in vibrationally relaxing gas with Van der Waals equation of state. We derive the transport equations for the amplitudes of resonantly interacting waves.