The Quarterly Journal of Mechanics and Applied Mathematics - current issue

Surface tension effects on interaction between two fluids near a wall

Vanden-Broeck, J.-M., Smith, F. T. — 2008-04-24

Interaction between two fluids near a fixed solid surface is modelled, with surface tension acting as an important influence on the assumed planar motion. The two fluids are immiscible, incompressible and have small density and viscosity ratios; the heavier more viscous body of fluid is approaching the solid surface and the other fluid is lying as a thin layer in between. In the so-called supercritical range where, for both fluids, inviscid forces dominate over viscous ones, a pair of pressure–shape relations is found which leads to a nonlinear integro-differential equation for the unknown interface shape. Analysis, computation and comparisons are applied to the equation. Travelling-state solutions are found of periodic and non-periodic form, including interesting cases which exhibit parabolic growth of the layer thickness in the far field.

Nonlinear transversely isotropic elastic solids: an alternative representation

Shariff, M. H. B. M. — 2008-04-24

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis–Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.

Dynamics of a bridged crack in a discrete lattice

Mishuris, G. S., Movchan, A. B., Slepyan, L. I. — 2008-04-24

The paper addresses a problem of partial fracture of a lattice by a propagating fault modelling a crack bridged by elastic fibres. It is assumed that the strength of bonds within the lattice alternates periodically, so that during the dynamic crack propagation only weaker bonds break, whereas the stronger bonds remain intact. The mathematical problem is reduced to the functional equation of the Wiener–Hopf type, which is solved analytically. The load–crack speed dependence is presented, which also has implications on the stability analysis for the bridged crack propagating within the lattice. In particular, we address the evaluation of the dissipation rate, which is found to be strongly dependent on the crack speed. In this lattice model, our results also cover the case of the supercritical crack speed.

Exact solutions for the evolution of ellipsoidal inclusions in porous media

Buchak, P., Crowdy, D. — 2008-04-24

This paper derives exact mathematical solutions for the time-dependent evolution of a single ellipsoidal inclusion in a porous medium when a linear straining flow is active in the far field. This represents a two-phase free boundary problem. It is shown that the dynamics is such that an initially ellipsoidal inclusion remains ellipsoidal under evolution. The theory of ellipsoidal harmonics is used to determine the system of ordinary differential equations governing the geometrical parameters of the ellipsoidal inclusion.

Fourth-order cartesian tensors: old and new facts, notions and applications

Moakher, M. — 2008-04-24

Fourth-order tensors can be represented in many different ways. For instance, they can be represented as multilinear maps or multilinear forms. It is also possible to describe a fourth-order tensor in a given vector space by a second-order tensor but in another vector space with higher dimension. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. In this paper, we use these three descriptions to discuss the different symmetries of fourth-order tensors, to present the algebra of the space of fourth-order symmetric tensors and to describe different metrics on this space. Isotropic tensors and orientation tensors are presented using these different representations of fourth-order tensors. Applications to elasticity and high angular resolution diffusion imaging are discussed. Finally, we present a systematic and consistent approach to finding the tensor of an even order that best fits in some sense a tensor of higher even order.

Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions

Merkin, J. H. — 2008-04-24

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T)^p, where T is the local temperature and T is the ambient temperature, is considered when there are power-law variations in either the wall temperature or the wall heat flux which enables the equations to be reduced to similarity form. When the wall temperature is prescribed, solutions are found for p ≤ 2 and p ≥ p_c (p_c = 10.673) with a saddle-node bifurcation at p = p_c and two solution branches for p > p_c. When the wall heat flux is prescribed, solutions are found only for p < 2. The special case p = 2 is considered and the limiting forms as p -> 2 and p -> are obtained and compared with the solutions obtained from solving the similarity equations numerically

Electromagnetic fields in the presence of an infinite dielectric wedge: the phased line source excitation case

Salem, M. A., Kamel, A. H. — 2008-04-24

Electromagnetic fields, excited by an electric phased line source in the presence of an infinite dielectric wedge, are determined by application of the Kontorovich–Lebedev transform. The Maxwell's equations together with the conditions of continuity of the tangential field components at the material interfaces are formulated as a vector boundary-value problem. By representing the field components as Kontorovich–Lebedev integrals, the problem is reduced to a system of singular integral equations for the unknown spectral functions. We construct numerical solutions to those equations that permit fields evaluation for values of the wedge refractive index, not necessarily close to unity, and for arbitrary positioned source and observer. Numerical results showing the influence of a wedge presence on the directivity of a phased line source are presented and verified through finite-difference frequency-domain simulations.

Matrix Wiener-Hopf approximation for a partially clamped plate

Abrahams, I. D., Davis, A. M. J., Llewellyn Smith, S. G. — 2008-04-24

This article examines the classic problem of deflection of a thin elastic plate subjected to static or dynamic normal loading. The plate is infinite in extent in one coordinate direction and finite in the other. On one infinite edge, the plate is clamped, and on the other the plate has mixed boundary conditions, clamped on a semi-infinite part of the edge and free on the remaining half. The boundary-value problem is reduced to a Wiener–Hopf equation, but it is of matrix form belonging to a class for which no exact solution technique is known. An explicit approximate solution, in general accurate to any specified degree, is obtained by a recent method which employs Padé approximants. Numerical results are presented for the plate deflection, and these exhibit convergence to the exact solution as the order of the approximant is increased.

On the buckling of elastic plates

Balmforth, N. J., Craster, R. V., Slim, A. C. — 2008-04-24

The Föppl-von Kármán equations are used to explore the onset of linear instability and the subsequent nonlinear development of buckling patterns in a flat elastic plate due to an imposed shear or body force such as gravity. Experimental results are also presented for a clamped and sheared sheet of Neoprene rubber and these compare favourably with theory.