Abstract
The general forms of self-similar solutions for two-dimensional weak shock waves in fluid dynamics are obtained. The functional form describing the area under the initial pulse is characterized under which the general system of PDEs admits similarity solutions. It is shown how one can construct new solutions with shock discontinuity from these self-similar solutions. In particular, a plane-wave solution is joined with a self-similar solution across a non-trivial shock. Furthermore, a new class of non-trivial simple wave solutions is presented.
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References
Olver PJ, Rosenau P (1987) Group-invariant solutions of differential equations. SIAM J Appl Math 47: 263–278
Bluman GW, Kumei S, Reid GJ (1988) New classes of symmetries of partial differential equations. J Math Phys 29: 806–811
Lie S (1891) Vorlesungen uber differential gleichungen mit bekannten infinitesimalen transformationen GB Teuber, Leipzing (reprinted 1967 by Chelsea, New York)
Bluman GW, Cole JD (1969) The general similarity solution of the heat equation. J Math Mech 18: 1025–1042
Bluman GW, Cole JD (1974) Similarity methods for differential equations. Appl Math Sci no. 13. Springer-Verlag, New York
Ovsjannikov LV (1962) Group properties of differential equations (trans: Bluman GW of Gruppovye Svoystva Differentsialny Uravneni, Novosibirsk, USSR) (reprinted 1982 Group Analysis of Differential Equations, Academic Press, New York)
Clarkson PA, Kruskal MD (1989) New similarity reductions of the Boussineq equations. J Math Phys 36: 2201–2213
Levi D, Winternitz P (1989) Non-classical symmetry reduction: example of the Boussinesq equation. J Phys A 22: 2915–2924
Ibragimov, NH (eds) (1994) Handbook of Lie group analysis of differential equations. Volume 1: Symmetries, exact solutions and conservation laws. CRC Press, Boca Raton
Ibragimov, NH (eds) (1995) Handbook of Lie group analysis of differential equations. Volume 2: Applications in engineering and physical sciences. CRC Press, Boca Raton
Ibragimov, NH (eds) (1996) Handbook of Lie group analysis of differential equations. Volume 3: New trends in theoretical developments and computational methods. CRC Press, Boca Raton
Dauenhauer EC, Majdalani J (2003) Exact self-similarity solution of the Navier Stokes equations for a porous channel with orthogonally moving walls. Phys Fluids 15: 1485–1495
Galaktionov VA, Williams JF (2004) On very singular similarity solutions of a higher-order semilinear parabolic equation. Nonlinearity 17: 1075–1099
Ludlow DK, Clarkson PA, Bassom AP (2000) New similarity solutions of the unsteady incompressible boundary-layer equations. Q J Mech Appl Math 53: 175–206
Sharma VD, Arora R (2005) Similarity solutions for strong shocks in an ideal gas. Stud Appl Math 114: 375–394
Nikitin AG, Boyko VM, Popovych RO (2001) In: Proceedings of the fourth international conference: symmetry in nonlinear mathematical physics, vol 43, Kyiv, Ukraine
Fridman VE (1982) Self-refraction and small amplitude shock waves. Wave Motion 4: 151–161
Zakeri GA (1988) Numerical results for wavefront tracking. J Wave Mater Interact 3: 127–134
Zakeri GA (1989) Geometric structure of 2D weak shock waves. Appl Math Comput 33: 161–183
Whitham GB (1957) A new approach to problems of shock dynamics, part 1, two-dimensional problems. J Fluid Mech 2: 146–171
Anile AM (1984) Propagation of weak shock waves. Wave Motion 6: 571–578
Henshaw WD, Smyth NF, Schwendeman DW (1986) Numerical shock propagation using geometrical shock dynamics. J Fluid Mech 171: 519–545
Schwendeman DW (1988) Numerical shock propagation in non-uniform media. J Fluid Mech 188: 383–410
Baskar S, Prasad P (2005) Propagation of curved shock fronts using shock ray theory and comparison with other theories. J Fluid Mech 523: 171–198
Whitham GB (1974) Linear and nonlinear waves. Wiley, New York
Coppens AB, Beyer RT, Seiden MB, Donohue J, Guepin F, Hodson R, Townsend C (1965) Parameter of nonlinearity in fluids II. J Acoust Soc Am 38: 797–804
Obermeier F (1981) On the propagation of weak and moderately strong curved shock waves. Bericht 116, Max-Planck Institut fur Stromungforschung
Cole RH (1948) Underwater explosions. Princeton University Press, Princeton
Stoker JJ (1992) Water waves: the mathematical theory with applications. Wiley, New York
Zakeri GA (2008) Non-simple waves solutions via conservation laws for 2D weak shock waves. Commun Appl Nonlinear Anal 15: 9–25
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Zakeri, GA. Similarity solutions for two-dimensional weak shock waves. J Eng Math 67, 275–288 (2010). https://doi.org/10.1007/s10665-009-9337-4
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DOI: https://doi.org/10.1007/s10665-009-9337-4