Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

  • Authors:

  • Anders Rønne Rassmusen;Mads Peter Sørensen;Yuri Borisovich Gaididei;Peter Leth Christiansen

  • Affiliations:

  • Department of Mathematics, Technical University of Denmark, Kongens Lyngby, Denmark 2800 and IRD Fuel Cells, Svendborg, Denmark 5700;Department of Mathematics, Technical University of Denmark, Kongens Lyngby, Denmark 2800;Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine 03143;Department of Informatics and Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark 2800

  • Venue:
  • Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
  • Year:
  • 2011

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Abstract

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.