The coupled motion between shallow-water sloshing in a moving vehicle and the vehicle dynamics is considered, with the vehicle dynamics restricted to horizontal motion. The paper is motivated by Cooker's experiments and theory for water waves in a suspended container. A new derivation of the coupled problem in the Eulerian fluid representation is given. However, it is found that transformation to a Lagrangian representation leads to a formulation which has nice properties for numerical simulation. In the Lagrangian representation, a simple and fast numerical algorithm with excellent energy conservation over long times, based on the Störmer–Verlet method, is implemented. Numerical simulations of the coupled dynamics in both the linear and non-linear case are presented.

We study the coupled surface and grain boundary motion in bi- and tricrystals in three-space dimensions, building on previous work by the authors on the simplified two-dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple-junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical finite-element approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric finite-element approximation we show well posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time-step size. Moreover, the induced discrete equations are linear and easy to solve. A generalisation to anisotropic surface energies is straightforward. Several numerical results in two- and three-space dimensions are presented, including simulations for thermal grooving and sintering. Three-dimensional simulations featuring quadruple junction points, non-standard boundary contact angles and fully anisotropic surface energies are also presented.

This paper is devoted to the study of the asymptotic behaviour of the high-frequency spectrum of the wave equation with periodic coefficients in a ‘thin’ elastic strip Ση=(0, 1)×(−η/2, η/2), η > 0. The main geometric assumption is that the structure period is of the order of magnitude of the strip thickness η and is chosen in such a way that η−1 is a positive large integer. On the boundary ∂Ση, we set Dirichlet (clamped) or Neumann (traction-free) boundary conditions. Aiming to describe sequences of eigenvalues of order η−2 in the above problem, which correspond to oscillations of high frequencies of order η−1, we study an appropriately rescaled limit of the spectrum. Using a suitable notion of two-scale convergence for bounded operators acting on two-scale spaces, we show that the limiting spectrum consists of two parts: the Bloch (or band) spectrum and the ‘boundary’ spectrum. The latter corresponds to sequences of eigenvectors concentrating on the vertical boundaries of Ση, and is characterised by a problem set in a semi-infinite periodic strip with either clamped or stress-free boundary conditions. Based on the observation that some of the related eigenvalues can be found by solving an appropriate periodic-cell problem, we use modal methods to investigate finite-thickness semi-infinite waveguides. We compare our results with those for finite-thickness infinite waveguides given in Adams et al. (Proc. R. Soc. Lond. A, vol. 464, 2008, pp. 2669–2692). We also study infinite-thickness semi-infinite waveguides in order to gain insight into the finite-height analogue. We develop an asymptotic algorithm making use of the unimodular property of the modal method to demonstrate that in the weak contrast limit, and when wavenumber across the guide is fixed, there is at most one surface wave per gap in the spectrum. Using the monomode property of the waveguide we can consider the gap structure for the nth mode, when doing so, for traction-free boundaries, we find exactly one surface wave in each n-band gap.