This paper studies many fundamental aspects of a newly proposed multi-class traffic flow model. For the first time it presents a complete discussion on the hyperbolicity of the system; and based upon this, the admissible waves of the Riemann problem are deeply investigated. Many important conclusions are made in discussions, and the related physical meanings are interpreted. For the confirmation of main conclusions, numerical examples are given at the end.

The propagation of a solitary wave in a horizontal fluid layer is studied. There is an interfacial free surface above and below this intrusion layer, which is moving at constant speed through a stationary density-stratified fluid system. A weakly nonlinear asymptotic theory is presented, leading to a Korteweg–de Vries equation in which the two fluid interfaces move oppositely. The intrusion layer solitary wave system thus forms a widening bulge that propagates without change of form. These results are confirmed and extended by a fully nonlinear solution, in which a boundary-integral formulation is used to solve the problem numerically. Limiting profiles are approached, for which a corner forms at the crest of the solitary wave, on one or both of the interfaces.

In this paper we investigate the model for the motion of a contact line over a smooth solid surface developed by Shikhmurzaev, [24]. We show that the formulation is incomplete as it stands, since the mathematical structure of the model indicates that an additional condition is required at the contact line. Recent work by Bedeaux, [4], provides this missing condition, and we examine the consequences of this for the relationship between the contact angle and contact line speed for Stokes flow, using asymptotic methods to investigate the case of small capillary number, and a boundary integral method to find the solution for general capillary number, which allows us to include the effect of viscous bending. We compare the theory with experimental data from a plunging tape experiment with water/glycerol mixtures of varying viscosities [11]. We find that we are able to obtain a reasonable fit using Shikhmurzaev's model, but that it remains unclear whether the linearized surface thermodynamics that underlies the theory provide an adequate description for the motion of a contact line.

We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in $L^2$ are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.

This paper is concerned with the positive solutions of the boundary-value problem \[ \left\{{\begin{array}{@{}l@{}}{\varepsilon u^{''} - \sigma (u) = -\gamma,}\\{u(0) = u(1) = 0,}\\ \end{array}}\right.\] where $\varepsilon$ is a small positive parameter and $\gamma$ is a positive constant. The nonlinear term $\sigma(u)$ behaves like a cubic; it vanishes only at $u=0$, where $\sigma'(0) > 0$ and $\sigma''(0) < 0$. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given $\gamma$, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of $\gamma$. Our approach is again based on the shooting method used previously by Ou & Wong (Stud. Appl. Math.112 (2004), 161-200).

Droplet profiles under the influence of van der Waals forces

Euro. Jnl. of Applied Maths.12, 367–393

ELI MINKOV and AMY NOVICK-COHEN

The notion of polarization tensor is employed for the derivation of the leading-order boundary perturbations in the steady-state voltage potentials that are due to the presence of conductivity inclusions of small diameter. Recently, Capdeboscq and Vogelius obtained optimal bounds of Hashin-Shtrikman type for the trace of the polarization tensor, showing that every pair satisfying these optimal bounds arises as the eigenvalues of a polarization tensor associated with a coated ellipse. In this paper, we give numerical evidence of the fact that the set of possible polarization tensor eigenvalue pairs can also be obtained using simply connected domains. Our numerical computations are based on a boundary integral method.

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics (9(1998) 527–542) are presented.

The scalar initial value problem \[ u_t = \rho Du + f(u), \] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $\Omega \in \mathbb{R}^n$ and time $t$, and $u(\cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=\Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=\Delta$). We develop a technique for proving that indeed convergence does hold for small $\rho$ and show by constructing a counter-example that this result does not hold in general for all $\rho$.

Sufficient conditions are obtained for the existence of positive periodic solutions of a class of neutral delay differential equations of the form \begin{equation*} \left\{\begin{array}{@{}l@{}} {\normalsize N}^{\prime}{\normalsize (t)=N(t)F[t,N(t),N(t-\tau (t,N(t))),N}^{\prime}{\normalsize (t-\gamma (t)),P(t),P(t-\mu (t))]}\\ {\normalsize P}^{\prime}{\normalsize (t)=-}e(t)P{\normalsize (t)+k(t)N(t)+h(t)N(t-\sigma (t))}\end{array}\right. \end{equation*} by using the theory of topological degree. These results extend substantially the existing relevant existence results in the literature. As a demonstration, applying the obtained analysis results to a real complex neutral Lotka-Volterra population model, the existence criterion for positive periodic solutions is easily obtained and an example is used to give an impression of how restrictive these conditions are. Especially, this method is more suitable to state-dependent delay, and which have further applications in many fields.

We study surface tension effects for two-dimensional Darcy flow with a free boundary in a corner between two non-parallel walls. The analytic solution is based on two governing expressions constructed in an auxiliary parameter domain, namely a complex velocity and a derivative of the complex potential. These expressions admit a general solution for the problem in a corner geometry for the flow generated by a source/sink at the corner vertex or at infinity. We derive an integral equation in terms of the velocity modulus and angle at the free surface, determined by the dynamic boundary condition. A numerical procedure, used to solve the obtained system of equations, and numerical results concerning the effect of surface tension on the time evolution of the free boundary, are discussed.

We study the propagation of a crack in critical equilibrium for a brittle material in a Mode III field. The energy variations for small virtual extensions of the crack are handled in a novel way: the amount of energy released is written as a functional over a family of univalent functions on the upper half plane. Classical techniques developed in connection to the Bieberbach Conjecture are used to quantify the energy-shape relationship. By means of a special family of trial paths generated by the so-called Löwner equation we impose a stability condition on the field which derives in a local crack propagation criterion. We called this the anti-symmetry principle, being closely related to the well known symmetry principle for the in-plane fields.