Incompressible reacting flows

We establish steady-state convergence results for a system of
reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.

     

Travelling waves for a biological reaction diffusion model with spatio-temporal delay

In this paper, we consider the reaction diffusion equations with spatio-temporal delay, which models the microbial growth in a flow reactor. Nonlocal spatial term, a weighted average in space, arises when the individuals have not necessarily been at the same point in space at previous time. By employing linear chain technique, geometric singular perturbation, and the center manifold theorem, we prove that the steady travelling wave does not only persist, but also it looks qualitatively the same as it do with no delay at all, under the introduction of delays, at least for small delay.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Properties of simple model problems for reacting gas flows

The compressible Navier–Stokes equations for reacting gases are extremely complex. Simpler models have been considered, and for these completely non-physical propagation speeds have been observed. These model problems are stiff, meaning that several different scales are present in the solution. Numerical solution of non-reacting flows almost always involves addition of extra dissipation. It will be shown that this action will render a totally wrong propagation speed for a simple model equation of reacting flows. This problem will be accentuated by increasing stiffness of the problem. Existence and uniqueness of a solution to this model equation is proved. The dependence of the propagation speed on the viscosity and a term governing the stiffness (comparable to the reaction rate for a more complete model) is investigated. A remedy for the wrong propagation speed for this simple model equation is proposed such that the speed is correct although the front is smeared out.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Travelling waves in lattice dynamical systems

For a class of one‐dimensional lattice dynamical systems we prove the existence of periodic travelling waves with prescribed speed and arbitrary period. Then we study asymptotic behaviour of such waves for big values of period and show that they converge, in an appropriate topology, to a solitary travelling wave. Copyright © 2000 John Wiley & Sons. Ltd.     

Discontinuous solutions in supersonic reacting gas flows

Zusammenfassung Es werden für Überschallströmungen reagierender Gase verallgemeinerte Lösungen mit schwachen und starken Unstetigkeiten (Sprünge in den Ableitungen bzw. in den Variablen selbst) betrachtet. Es wird gezeigt, dass die Kurven, an denen solche Unstetigkeiten auftreten, die Charakteristiken des Gleichungssystems für schwache Unstetigkeiten sind. Das Zusammenwirken von Unstetigkeiten verschiedener Veränderlichen wird diskutiert.

---

---

Supported by NASA under grant NsG633 and Canadian National Research Council and Defence Research Board.

---

Visiting Research Associate. (On leave from the National Aeronautical Laboratory, Bangalore, India.) Work done while visiting at the Institute for Aerospace Studies, University of Toronto, Toronto, Canada during 1963–65.     

Travelling waves in dilatant non-Newtonian thin films

We prove the existence of a travelling wave solution for a gravity-driven thin film of a viscous and incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second order partial differential equations for the film's height h and the surfactant's concentration γ are derived by means of lubrication theory applied to the non-Newtonian Navier–Stokes equations.     

Travelling waves for reaction-diffusion equations with time depending nonlinearities

The existence of travelling wave with given end points for parabolic system of nonlinear equations is proven. The nonlinear term depends also on a·x−ct where x is the multidimensional space variable, t—time, c—the speed of the wave and a—the direction of travel.     

Travelling waves of the KP equations with transverse modulations

Kadomtsev-Petviashvili (KP) equations arise genetically in modelling nonlinear wave propagation for primarily unidirectional long waves of small amplitude with weak transverse dependence. In the case when transverse dispersion is positive (such as for water waves with large surface tension) we investigate the existence of transversely modulated travelling waves near one-dimensional solitary waves. Using bifurcation theory we show the existence of a unique branch of periodically modulated solitary waves. Then, we briefly discuss the case when the transverse dispersion is negative (such as for water waves with zero surface tension).     

Travelling waves of phase transitions in porous media

Liquid–vapour phase changes for a fluid flow through a porous medium are considered; in particular, the model allows for phase mixtures and includes an equilibrium pressure. Existence and uniqueness of travelling waves is established in a wide range of situations; the end states may be formed either by pure phases or mixtures; in the latter case the pressure equals the equilibrium pressure. A formal asymptotic analysis for vanishing relaxation time is carried out to show that the friction and reaction source terms have smoothing effect when the pressure is close to the equilibrium pressure and pure phases are avoided.     

Rogue waves in baroclinic flows

We investigate an AB system, which can be used to describe marginally unstable baroclinic wave packets in a geophysical fluid. Using the generalized Darboux transformation, we obtain higher-order rogue wave solutions and analyze rogue wave propagation and interaction. We obtain bright rogue waves with one and two peaks. For the wave packet amplitude and the mean-flow correction resulting from the self-rectification of the nonlinear wave, the positions and values of the wave crests and troughs are expressed in terms of a parameter describing the state of the basic flow, in terms of a parameter responsible for the interaction of the wave packet and the mean flow, and in terms of the group velocity. We show that the interaction of the wave packet and mean flow and also the group velocity affect the propagation and interaction of the amplitude of the wave packet and the self-rectification of the nonlinear wave.     

Travelling waves in an ionic quadratic autocatalytic chemical system

The influence of a constant electric field on the travelling waves that are initiated in an ionic autocatalytic reaction-diffusion system with quadratic rate law is considered. The necessary conditions for the existence of minimum speed travelling waves are established and examples of such waves have been generated using numerical simulation. The behaviour of the system when the necessary conditions are not satisfied is also examined. This behaviour involves electrophoretic separation and reaction termination, each of which is discussed using examples obtained from numerical simulation.     

Travelling waves in plasma sustained by a laser beam

By means of the Conley connection index theory we prove existence of travelling wave solutions to a system of equations in a two temperature model of laser maintained plasma. These solutions describe locally the motion of plasma boundaries. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Travelling waves in a reaction-diffusion system modelling fungal mycelia

We discuss travelling-wave solutions of a system of coupledreaction—diffusion equations used by the authors to describethe macroscopic behaviour of fungal mycelia. Such systems havebeen used in a multitude of applications; and, in particular,Merkin and Needham (1990, Proc. R. Soc. Lond. A 430, 315–45)have studied a certain formulation as a model for generic isothermalchemical reactions. We show that an alternative analysis providesmore complete results in ascertaining the conditions under whichtravelling-wave solutions exist and that it allows a wider rangeof parameter values to be considered; this is essential to theapplication considered in the present case. Numerical investigationsof travelling-wave solutions and the related initial-value problemare included to motivate and extend the analysis.     

Energy flows of gravitational waves

The problem of the energy transfer produced by weak gravitational waves on a static background is solved by using the Killing vectors without introducing the momentum-energy pseudotensor. The Lagrangian, which is of the second order in metric excitations and which determines the flow and density of the energy of gravitational waves on a static spherically symmetrical space-time background, is derived. Integration over angular variables in the action integral thus reduces the problem to a two-dimensional effective problem. Energy flows on the Schwarzschild metric background are calculated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 168–176, July, 1999.