Analytical and numerical study of travelling waves using the Maxwell-Cattaneo relaxation model extended to reaction-advection-diffusion systems

V. A. Sabelnikov, N. N. Petrova, and A. N. Lipatnikov
Phys. Rev. E 94, 042218 – Published 21 October 2016

Abstract

Within the framework of the Maxwell-Cattaneo relaxation model extended to reaction-diffusion systems with nonlinear advection, travelling wave (TW) solutions are analytically investigated by studying a normalized reaction-telegraph equation in the case of the reaction and advection terms described by quadratic functions. The problem involves two governing parameters: (i) a ratio φ2 of the relaxation time in the Maxwell-Cattaneo model to the characteristic time scale of the reaction term, and (ii) the normalized magnitude N of the advection term. By linearizing the equation at the leading edge of the TW, (i) necessary conditions for the existence of TW solutions that are smooth in the entire interval of <ζ< are obtained, (ii) the smooth TW speed is shown to be less than the maximal speed φ1 of the propagation of a substance, (iii) the lowest TW speed as a function of φ and N is determined. If the necessary condition of N>φφ1 does not hold, e.g., if the magnitude N of the nonlinear advection is insufficiently high in the case of φ2>1, then, the studied equation admits piecewise smooth TW solutions with sharp leading fronts that propagate at the maximal speed φ1, with the substance concentration or its spatial derivative jumping at the front. An increase in N can make the solution smooth in the entire spatial domain. Moreover, an explicit TW solution to the considered equation is found provided that N>φ. Subsequently, by invoking a principle of the maximal decay rate of TW solution at its leading edge, relevant TW solutions are selected in a domain of (φ,N) that admits the smooth TWs. Application of this principle to the studied problem yields transition from pulled (propagation speed is controlled by the TW leading edge) to pushed (propagation speed is controlled by the entire TW structure) TW solutions at N=Ncr=1+φ2, with the pulled (pushed) TW being relevant at smaller (larger) N. An increase in the normalized relaxation time φ2 results in increasing Ncr, thus promoting the pulled TW solutions. The domains of (φ,N) that admit either the smooth or piecewise smooth TWs are not overlapped and, therefore, the selection problem does not arise for these two types of solutions. All the aforementioned results and, in particular, the maximal-decay-rate principle or appearance of the piecewise smooth TW solutions, are validated by numerically solving the initial boundary value problem for the reaction-telegraph equation with natural initial conditions localized to a bounded spatial region.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
9 More
  • Received 11 February 2016

DOI:https://doi.org/10.1103/PhysRevE.94.042218

©2016 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

V. A. Sabelnikov1,*, N. N. Petrova1, and A. N. Lipatnikov2,†

  • 1ONERA–The French Aerospace Laboratory, F-91761 Palaiseau, France
  • 2Department of Applied Mechanics, Chalmers University of Technology, Gothenburg 412–96, Sweden

  • *vladimir.sabelnikov@onera.fr
  • lipatn@chalmers.se

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 94, Iss. 4 — October 2016

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×