scholarly journals Thermal solitons: travelling waves in combustion

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120587 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Travelling Waves ◽  
Reaction System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Governing Equations ◽  
Competitive Reaction ◽  
Chemical Reagent ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

A competitive reaction system is considered, under which some chemical reagent decays by means of two simultaneous chemical reactions to form two separate inert products. One reaction is exothermic, and the other is endothermic. The governing equations for the model are presented, and a weakly nonlinear theory is then generated using the method of strained coordinates. Travelling-wave solutions are possible in the model, and the temperature is found to have a classical sech-squared profile. The stability of these moderate-amplitude temperature solitons is confirmed both analytically and numerically.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

scholarly journals Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 950 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa Durán ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Dispersion Equations ◽  
Different Dimensions ◽  
Double Dispersion ◽  
Long Nonlinear Wave ◽  
Line Soliton

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

scholarly journals Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Mohammed Rizwan Sadiq Iqbal
Keyword(s):  
Magnetic Field ◽  
Inertial Force ◽  
Transverse Magnetic ◽  
Nonlinear Analyses ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Strength ◽  
Long Time ◽  
The Stability ◽  
The One

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.

scholarly journals Morphological instability of a non-equilibrium ice–colloid interface

2009 ◽  
Vol 466 (2113) ◽  
pp. 177-194 ◽  
Author(s):  
S. S. L. Peppin ◽  
A. Majumdar ◽  
J. S. Wettlaufer
Keyword(s):  
Travelling Waves ◽  
Colloidal Suspension ◽  
Bentonite Clay ◽  
Travelling Wave ◽  
Liquidus Curve ◽  
Colloid Interface ◽  
Colloidal Systems ◽  
Weakly Nonlinear ◽  
Highly Nonlinear ◽  
Non Equilibrium

We assess the morphological stability of a non-equilibrium ice–colloidal suspension interface, and apply the theory to bentonite clay. An experimentally convenient scaling is employed that takes advantage of the vanishing segregation coefficient at low freezing velocities, and when anisotropic kinetic effects are included, the interface is shown to be unstable to travelling waves. The potential for travelling-wave modes reveals a possible mechanism for the polygonal and spiral ice lenses observed in frozen clays. A weakly nonlinear analysis yields a long-wave evolution equation for the interface shape containing a new parameter related to the highly nonlinear liquidus curve in colloidal systems. We discuss the implications of these results for the frost susceptibility of soils and the fabrication of microtailored porous materials.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

Viscous film flow coating the interior of a vertical tube. Part 1. Gravity-driven flow

2014 ◽  
Vol 745 ◽  
pp. 682-715 ◽  
Author(s):  
Roberto Camassa ◽  
H. Reed Ogrosky ◽  
Jeffrey Olander
Keyword(s):  
Time Evolution ◽  
Film Flow ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Basic Solution ◽  
Travelling Wave Solutions ◽  
Liquid Plug ◽  
Wave Solutions ◽  
Gravity Driven ◽  
Gravity Driven Flow

AbstractThe gravity-driven flow of a viscous liquid film coating the inside of a tube is studied both theoretically and experimentally. As the film moves downward, small perturbations to the free surface grow due to surface tension effects and can form liquid plugs. A first-principles strongly nonlinear model based on long-wave asymptotics is developed to provide simplified governing equations for the motion of the film flow. Linear stability analysis on the basic solution of the model predicts the speed and wavelength of the most unstable mode, and whether the film is convectively or absolutely unstable. These results are found to be in remarkable agreement with the experiments. The model is also solved numerically to follow the time evolution of instabilities. For relatively thin films, these instabilities saturate as a series of small-amplitude travelling waves, while thicker films lead to solutions whose amplitude becomes large enough for the liquid surface to approach the centre of the tube in finite time, suggesting liquid plug formation. Next, the model’s periodic travelling wave solutions are determined by a continuation algorithm using the results from the time evolution code as initial seed. It is found that bifurcation branches for these solutions exist, and the critical turning points where branches merge determine film mean thicknesses beyond which no travelling wave solutions exist. These critical thickness values are in good agreement with those for liquid plug formations determined experimentally and numerically by the time-evolution code.

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

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