For nonlinear hyperbolic equations of a gas dynamic type with flow acceleration and friction terms, the classification of a special class of periodic travelling waves, which are known as roll waves, is given. As an illustration, the shallow water equations for the inclined channels of an arbitrary cross-section are considered. The analysis shows that the flow patterns depend upon the sign of the second derivative of the pressure function. The roll waves in regular channels with the convex pressure have the same structure as the waves described in Dressler [6]. For a nonconvex pressure function, the multi-jump configuration of roll waves is found.

Mathematical models for the growth of nutrient-rich tissue are presented and a number of properties of the resulting models outlined. The focus is on obtaining broadly applicable results for the simplest appropriate formulations by using matched-asymptotic, moving-boundary and thin-film approaches; the relevance of the results to a variety of specific biological applications will be addressed elsewhere, as will the inclusion of additional physical phenomena.

We consider a Stefan problem with a kinetic condition at the free boundary and prove the convergence of the solution as $t$ tends to infinity either to a travelling wave solution or to a self-similar solution. The key idea is to transform this problem into a problem for a single nonlocal parabolic equation which admits a comparison principle.

The problem of a bubble rising due to buoyancy in a Hele–Shaw cell filled with a viscous fluid is a classical free-boundary problem first posed and solved by Saffman & Taylor [11]. In fact, due to linearity of the flow equations the problem is reduced to that of a bubble transported by uniform fluid flow. Saffman and Taylor provided explicit expressions for the bubble shape. Steady propagation of bubbles and fingers in a Hele–Shaw cell filled with a nonlinearly-viscous fluid was studied by Alexandrou & Entov [1]. In Alexandrou & Entov [1], it was shown that for a nonlinearly viscous fluid the problem of a rising bubble cannot be reduced to that of a steadily transported bubble, and should be treated separately. This note presents a solution of the problem following the general framework suggested in Alexandrou & Entov [1]. The hodograph transform is used in combination with finite-difference and collocation techniques to solve the problem. Results are presented for the cases of a Bingham and power-law fluids.

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

The aim of this paper is to develop a theory for the growth of multiple crystals in a polymer melt. This leads to nonlinear moving boundary problems for the temperature, with normal growth speed of the crystal boundaries determined by a nonlinear Gibbs-Thompson relation. Particular attention is paid to the effect of impingement, i.e., the event of two crystals hitting each other, which stops the growth on the contact interface. In one space dimension, the well-posedness of the growth model coupled to the heat equation is shown for an arbitrary number of crystals, both in a quasi-stationary and in an instationary situation. The resulting evolution of a fixed crystal in presence of other crystals is compared to the pure single crystal case. Finally, some basic features of the model in higher spatial dimensions and the main problems encountered in the attempt to prove a general well-posedness result are discussed.

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.