Traveling Wave Speed Research Articles

Early development and quorum sensing in bacterial biofilms.

We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions.

Condensed phase combustion travelling waves with sequential exothermic or endothermic reactions

The one-dimensional propagation of a combustion wave through a premixed solid fuel for two-stage kinetics is studied. We re-examine the analysis of a single reaction travelling-wave and extend it to the case of two-stage reactions. We derive an expression for the travelling wave speed in the limit of large activation energy for both reactions. The analysis shows that when both reactions are exothermic, the wave structure is similar to the single reaction case. However, when the second reaction is endothermic, the wave structure can be significantly different from single reaction case. In particular, as might be expected, a travelling wave does not necessarily exist in this case. We establish conditions in the limiting large activation energy limit for the non-existence, and for monotonicity of the temperature profile in the travelling wave.

Dispersion Relation and Stability Analysis of Flow-Induced Wave of a Flexible Circular Ring Subjected to Swirling Fluid Flow

This paper deals with a theoretical stability analysis of unstable wave generated in a flexible circular ring subjected to swirling fluid flow. The flexible circular ring is set in a rotating circular outer casing with a narrow gap filled with fluid and is subjected to the swirling fluid flow caused by the rotation of the outer casing. In the stability analysis, a wave equation is derived from the equation of motion of the flexible circular ring coupled with the swirling fluid flow. The equation of motion of the flexible circular ring is based on the Kirchhoff-Love’s thin-shell model, and the equations of motion of the swirling fluid flow are based on the Navier-Stokes equations. Moreover, the dispersion relation of the wave generated in the flexible circular ring is derived from the wave equation as a function of the rotational speed of the outer casing. The analytical results show that an unstable wave occurs as a type of traveling forward wave due to the swirling fluid flow when the rotational speed of the outer casing becomes high, and that the most unstable wave mode, phase speed (traveling-wave speed), and growth rate of the wave vary greatly with increasing rotational speed of the outer casing.

Traveling wave speed and solution in reaction-diffusion equation in one dimension

By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.

Traveling Wave Speed and Solution in Reaction-Diffusion Equation in One Dimension

Traveling Wave Speed and Solution in Reaction-Diffusion Equation in One Dimension

Stability Analysis of Flow-Induced Waves of a Flexible Sheet Moving in a Fluid-Filled Narrow Space with Shear Fluid Flow. 1st Report. In a Case of Moving Infinitely Long Sheet.

This paper deals with a theoretical stability analysis of flow-induced waves induced in a flexible sheet moving in a fluid-filled narrow space with shear fluid flow. In the stability analysis, wave equation of the system is derived from the equation of motion of the moving flexible sheet coupled with the shear fluid flow. The equation of motion of the moving flexible sheet is based on the Kirchhoff-Love's thin-plate model and the equations of motion of the shear fluid flow are based on Navier-Stokes equations. The dispersion relation of the waves are derived as a function of the moving speed of the sheet. As a result, the analytical results show that flow-induced waves occur in the moving flexible sheet due to the shear fluid flow when the moving speed becomes high, and clarify the phase speed (traveling-wave speed) and the growth rate of the flow-induced waves.

Dispersion Relation and Stability Analysis of Flow-Induced Wave of a Flexible Circular Ring Subjected to Swirling Fluid Flow.

This paper deals with a theoretical stability analysis of flow-induced waves induced in a flexible circular ring subjected to swirling fluid flow. In the stability analysis, wave equation of the system is derived from the equation of motion of the flexible circular ring coupled with the swirling fluid flow. The equation of motion of the flexible circular ring is based on the Kirchhoff-Love's thin-plate model and the equations of motion of the fluid flow are based on Navier-Stokes equations. The dispersion relation of the waves are calculated as a function of the ring rotational speed. As a result, the analytical results show that unstable wave occurs in the circular ring due to the swirling fluid flow and clarifies the traveling-wave speed and the growth rate of the flow-induced waves.

A PHASE-PLANE DESCRIPTION OF NONLINEAR TRAVELING WAVES IN BUBBLY LIQUIDS

One-dimensional traveling wave solutions to the fully nonlinear continuity and Euler equations in a bubbly liquid are considered. The elimination of velocity from the two equations leaves a single nonlinear algebraic relation between the pressure and density profiles in the mixture. On assuming the bubbles to have identical size and taking the volume fraction of bubbles in the medium to be small, an equation of state which relates the mixture pressure to the density and its first two material time-derivatives is derived. When this equation of state is linearized and combined with the laws of conservation of mass and momentum, a nonlinear, second-order, ordinary differential equation is obtained for the density as a function of the single traveling wave coordinate. A phase-plane analysis of this equation reveals the existence of two fixed points, one of which is a saddle and the other a node. A single trajectory connects the two fixed points and corresponds to a traveling shock wave solution when the Mach number of the wave, defined as the ratio of traveling wave speed to the low-frequency speed of sound in the bubbly liquid, exceeds unity. The analysis provides a qualitative explanation of the oscillations behind shocks seen in experiments on bubbly liquids.

Surge impedance of independent transmission towers characterized by the direct method

Measurement of the surge impedance of a transmission tower by the direct method is studied with the help of NBC-2. The arrangement of the measuring wires in the experiment influences the result, and it is possible to correct the measured tower surge impedance to that for a standard arrangement with a vertical current injection wire and a horizontal voltage measuring wire. By reviewing various experimental results in this way, it is concluded that an independent double-circuit transmission tower has an impedance of about 150 Ω with the traveling wave speed in the tower being equal to that of light. This value can be applied to the case of a direct lightning hit on the tower top, regardless of the type of lightning, upward or downward. © 1998 Scripta Technica. Electr Eng Jpn, 122(4): 17–24, 1998

Existence of Traveling Waves in a Biodegradation Model for Organic Contaminants

We study a biodegradation model for the time evolution of concentrations of contaminant, nutrient, and bacteria. The bacteria has a natural concentration which will increase when the nutrient reaches the substrate (contaminant). The growth utilizes nutrients and degrades the substrate. Eventually, such a process removes all the substrate and can be described by traveling wave solutions. The model consists of advection-reaction-diffusion equations for the substrate and nutrient concentrations and a rate equation (ODE) for the bacteria concentration. We first show the existence of approximate traveling wave solutions to an elliptically regularized system posed on a finite domain using degree theory and the elliptic maximum principle. To prove that the approximate solutions do not converge to trivial solutions, we construct comparison functions for each component and employ integral identities of the governing equations. This way, we derive a priori estimates of solutions independent of the length of the finite domain and the regularization parameter. The integral identities take advantage of the forms of coupling in the system and help us obtain optimal bounds on the traveling wave speed. We then extend the domain to the infinite line limit, remove the regularization, and construct a traveling wave solution for the original set of equations satisfying the prescribed boundary conditions at spatial infinities. The contaminant and nutrient profiles of the traveling waves are strictly monotone functions, while the biomass profile has a pulse shape.

Time-Asymptotic Limit of Solutions of a Combustion Problem

A combustion model which captures the interacting among nonlinear convection, chemical reaction and radiative heat transfer is studied. New phenomena are found with radiative heat transfer present. In particular, there is a detonation wave solution in which there is a sonic point inside the reaction zone. As a consequence, the traveling wave speed cannot be determined before the problem is solved. The shooting method is used to prove the existence of the traveling wave. The condition we shoot at is the compatibility condition at the sonic point. Furthermore, the speed of the detonation wave decreases as the heat loss coefficient increases, as expected physically. We study the time-asymptotic limit of solutions of initial value problem for the same problem. We prove that the solution exists globally and the solution converges uniformly, away from the shock, to a shifted traveling wave solution as t → + ∞ for certain “compact support” initial data.

Traveling Waves as Limits of Solutions on Bounded Domains

This paper is concerned with the asymptotic behavior as $\epsilon \to 0$ of solutions of the reaction-diffusion equation $u_t = \epsilon ^2 u_{xx} - (u + a)(u^2 - 1)$ defined in $( - 1,1)$ with Neumann boundary conditions. For $a = 0$, this equation has a monotone equilibrium solution $u^\epsilon $ with the property that $u^\epsilon (x) \to - 1$ (resp. $ + 1$) on $[ { - 1,0} )$ (resp. $( {0,1} ]$) as $\epsilon \to 0$; that is, the solution has a sharp transition layer if $a = 0$. Also, it is known that $u^\epsilon $ has a one-dimensional unstable manifold $\mathcal{M}(u^\epsilon )$. Solutions near $\mathcal{M}(u^\epsilon )$ decrease exponentially to $\mathcal{M}(u^\epsilon )$ and move with a speed $O(e^{{{ - c} / \epsilon }} )$ along $\mathcal{M}(u^\epsilon )$. This paper considers the case where a is small and fixed. For each fixed $\epsilon $, $a \ne 0$, small, there is an equilibrium solution $u^{\epsilon a} $ with unstable manifold of dimension one, but $u^{\epsilon a} $ approaches either the function 1 or $ - 1$ as $\epsilon \to 0$; that is, there is no monotone equilibrium solution with a sharp transition layer. If we rescale x to $\epsilon x$ and consider the rescaled equation on $( - \infty ,\infty )$, then there is a unique (except for translation) monotone traveling-wave solution on $( - \infty ,\infty )$ with wave speed $ - \sqrt 2 a$. Using a geometric approach, we prove that there are positive constants $\epsilon _0 $ and $a_0 $ such that, for $0 < \epsilon < \epsilon _0 $ and $|a| < a_0 $, solutions of the rescaled equations on $( - \frac{1}{\epsilon },\frac{1}{\epsilon })$ in a neighborhood of size $C\sqrt {a_0 } $ of a monotone traveling-wave solution decrease exponentially fast before they enter a neighborhood of size $O(\epsilon ^k )$ of such a solution, where k can be any positive integer. Along the traveling-wave direction, solutions move with the traveling-wave speed plus an error term $O(\epsilon ^k )$. It also is proved that the $L^\infty $-norm between the solution and a translation of the traveling wave is of order $O(\epsilon ^k )$ for $C_1 k\log \frac{1}{\epsilon } < t < \frac{{C_2 }}{\epsilon }$.

Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion

When degenerate nonlinear diffusion is introduced into the Fisher equation, giving u t = ( uu x ) x + u(1 − u), the travelling wave structure changes so that there is a sharp-front wave for one particular wave speed, with smooth-front waves for all faster speeds. The sharp-front solution has been studied by a number of previous authors; the present paper is concerned with the smooth-front waves. The authors use heuristic arguments to derive a relationship between initial data and the travelling wave speed to which this initial data evolves. The relationship compares very well with the results of numerical simulations. The authors go on to consider the form of smooth-front waves with speeds close to that of the sharp-front solution. Using singular perturbation theory, they derive an asymptotic approximation to the wave which gives valuable information about the structure of the smooth-front solutions.

Hyperbolic reaction-diffusion equations and irreversible thermodynamics: Cubic reversible reaction model

In this paper we study chemical oscillations and wave formation in a cubic chemical reaction model (a modified Selkov model) by using hyperbolic partial differential equations, i.e., hyperbolic reaction-diffusion equations, instead of the usual parabolic reaction-diffusion equations. We have carried out a comparative investigation for the hyperbolic and parabolic reaction-diffusion equations in order to help determine which system of partial differential equations is more suitable for describing chemical oscillations and wave phenomena. It is shown that the hyperbolic system approaches the parabolic system as a parameter called the reaction-diffusion number increases to a large value. In contrast to the case of parabolic differential equations, the travelling wave speed has no singular behavior if the hyperbolic system is used. The hyperbolic system predicts that there is no necessity to take different diffusion coefficients to have the Turing instability unlike the parabolic system. Numerical solutions are presented for a one-dimensional case to show that the patterns change as the reaction-diffusion number is changed from the hyperbolic regime to the parabolic regime. The entropy productions are also calculated which show that, given the same boundary and initial conditions, different patterns have the same global entropy production, but the system gets organized to local structures of some particular frequencies and wave numbers of high entropy productions at the expense of energy and matter on the part of the rest of the frequency and wave number modes. Some modes appropriate to themselves most of energy and matter available to the system as a whole and get organized to local structures. It is also found that as the trajectories become chaotic the entropy production becomes relatively smaller than the ordered structures. The second law of thermodynamics, however, does not have direct control over such local organizations except for providing thermodynamically consistent evolution equations. In the case of a bistable system, the entropy production is found to make a rather abrupt change as the system makes transition from one stable state to another.

One-way blocks in cardiac tissue: A mechanism for propagation failure in purkinje fibres

The concept of a one-way block, arising from a region of depressed tissue, has remained central to theories for cardiac arrhythmias. We show that both the geometry of a depressed region and spatial heterogeneities in depression are key factors for inducing such a block. By using an asymptotic approximation, known as the eikonal equation, to model qualitatively the movement of a depolarization wave-front down a Purkinje fibre bundle, we show how a one-way block in conduction may result from asymmetric constriction in the width of a depressed bundle. We demonstrate that this theory is valid for biologically relevant parameters and simulate a one-way block by numerically solving the eikonal approximation. We consider the case of non-uniform depression, where the planar travelling wave speed is spatially dependent. Here, numerical simulations indicate that such a spatial dependency may, in itself, be sufficient to produce a one-way block.

One-way blocks in cardiac tissue: A mechanism for propagation failure in Purkinje fibres

The concept of a one-way block, arising from a region of depressed tissue, has remained central to theories for cardiac arrhythmias. We show that both the geometry of a depressed region and spatial heterogeneities in depression are key factors for inducing such a block. By using an asymptotic approximation, known as the eikonal equation, to model qualitatively the movement of a depolarization wave-front down a Purkinje fibre bundle, we show how a one-way block in conduction may result from asymmetric constriction in the width of a depressed bundle. We demonstrate that this theory is valid for biologically relevant parameters and simulate a one-way block by numerically solving the eikonal approximation. We consider the case of non-uniform depression, where the planar travelling wave speed is spatially dependent. Here, numerical simulations indicate that such a spatial dependency may, in itself, be sufficient to produce a one-way block.

Non-uniqueness of supercritical flow in the spherical couette system.

Non-uniqueness of supercritical flow has been experimentally considered in the spherical Couette system with only the inner sphere rotating. The types, wavenumbers and fundamental frequencies of the disturbances, which appear in the quasi-static transitions started from three different initial flow states with zero, two or four toroidal Taylor-Gortler (TG) vortices and in different flow histories, respectively, have been investigated by the simultaneous spectral and flow visualization measurements. The type and wavenumber of the disturbance are found to depend on the number N of the toroidal TG vortices. The wave speed of travelling azimuthal waves is independent of N in the range of reduced Reynolds number R* of 1.77≤R*<lsime>2.1, but not in the other R* ranges. The wave speed of shear waves is almost independent of R*, but depends on N. As for the history of the travelling wave speed, the initial state with four vortices makes a closed hysteresis loop, but the one with two vortices does not.