The Quarterly Journal of Mechanics and Applied Mathematics - Advance Access

Quasi-Steady Spreading of a Thin Ridge of Fluid with Temperature-Dependent Surface Tension on a Heated or Cooled Substrate

Dunn, G. J., Duffy, B. R., Wilson, S. K., Holland, D. — 2009-06-26

We investigate theoretically the problem of the quasi-steady spreading or contraction of a thin two-dimensional sessile or pendent ridge of viscous fluid with temperature-dependent surface tension on a planar horizontal substrate that is uniformly heated or cooled relative to the atmosphere. We derive an implicit solution of the leading-order thin-film equation for the free-surface profile of the ridge and use this to examine the quasi-steady evolution of the ridge, the dynamics of the moving contact lines being modelled by a ‘Tanner law’ relating the velocity of the contact line to the contact angle; in particular, we obtain a complete description of the possible forms that the evolution may take. In both the case of a (sessile or pendent) ridge on a heated substrate and the case of a pendent ridge on a cooled substrate when gravitational effects are relatively weak, there is one stable final state to which the ridge may evolve. In the case of a pendent ridge on a cooled substrate when gravitational effects are stronger, there may be one or two stable final states; moreover, the contact angles may vary non-monotonically with time during the evolution to one of these states. In the case of a pendent ridge on a cooled substrate when gravitational effects are even stronger, there may be up to three stable final states with qualitatively different solutions; moreover, the ridge may evolve via an intermediate state from which quasi-steady motion cannot persist, and so there will be a transient non-quasi-steady adjustment (in which the contact angles change rapidly, with the positions of the contact lines unaffected), after which quasi-steady motion is resumed. Lastly, we consider the behaviour of the ridge in the asymptotic limits of strong heating or cooling of the substrate and of strong or weak gravitational effects.

An Analytical Model Of The Motion Of A Conformable Sheet Over A General Convex Surface In The Presence Of Frictional Coupling

Cottenden, D. J., Cottenden, A. M. — 2009-06-22

Friction is important across a wide range of applications. In particular, in health care, friction is thought to be the cause of some pressure ulcers in largely immobile patients, and abrasion due to friction contributes to the deterioration of skin health in incontinence pad users, especially in the presence of liquid. Some of these frictional forces are due to stress in materials wrapped around curved anatomical surfaces, which are often complicated shapes. The little work to date that has considered friction arising by this mechanism has assumed very simplified geometries (prisms, or even cylinders), which have enabled coefficients of friction to be extracted from laboratory tests on arms, but which are certainly not applicable to, for example, the diaper region. This work describes the development of a much more general mathematical model for friction between a draped, stressed sheet and the substrate, relating geometry, material mechanical properties and stress for essentially any convex surface. A general, wide, class of frictional interfaces is described (which includes those which obey Amontons’ law), and the model is presented in differential form for a generic member of this class. Finally, an analytical solution is developed for convex, instantaneously rigid substrates isomorphic to the plane draped with a low-density sheet exhibiting no Poisson contraction, a fair approximation to some anatomical situations. The solution is explicitly calculated for a general prism and a general cone, producing expressions consistent with previous published models and with limited new experimental data.

Axisymmetric Contact of a Rigid Inclusion Embedded at the Interface of a Piezoelectric Bimaterial

Eskandari, M., Moeini-Ardakani, S. S., Shodja, H. M. — 2009-06-03

The axisymmetric contact problem of a rigid inclusion embedded in the piezoelectric bimaterial frictionless interface subjected to simultaneous far-field compression and electric displacement is addressed. With the aid of a robust technique, the coupled governing integral equations of this mixed boundary-value problem are reduced to decoupled Fredholm integral equations with a constraint equation. A useful limiting case for the contact problem of transversely isotropic bimaterials is addressed. The present solution is analytically in agreement with the existing solution for an isotropic bimaterial. Selected numerical results of interest to engineering applications including the radius of separation zone, contact pressure and contact electric displacement are plotted to portray the effects of precompression, piezoelectric coupling and material properties.

EFFECTIVE CONDUCTIVITY OF UNIDIRECTIONAL CYLINDERS WITH INTERFACIAL RESISTANCE

Drygas, P., Mityushev, V. — 2009-05-15

We investigate the effective conductaivity tensor for unidirectionally arbitrary distributed cylinders when the contact between the components is not perfect. The corresponding boundary value problem is reduced to a system of functional-differential equations, which is solved using symbolic computations. The major advantage of the method presented here is to provide approximate analytical expressions for the local thermal flux and for the effective conductivity in a systematic manner in terms of the concentration of inclusions, of the contrast parameter, of the thermal resistance and of the location of the parallel fibres.

Eigenvectors of a Rotation Matrix

Ahmad, F., Ahmad Khan, R. — 2009-05-12

If a tensor is invariant under rotation about a fixed axis, the matrices representing the tensor and the rotation commute with each other. The two matrices have common eigenvectors, therefore a knowledge of eigenvectors of the rotation matrix provides us with some information about eigenvectors of the tensor. This result is applied to derive familiar representations of a transversely isotropic tensor of rank 2 and the elasticity tensor possessing tetragonal symmetry. Representation of the elasticity tensor belonging to a particular symmetry class can be achieved in an elegant manner.

Water-Wave Scattering by Submerged Elastic Plates

Mahmood-Ul-Hassan, , Meylan, M. H., Peter, M. A. — 2009-05-11

We present a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. The eigenfunction expansion depends on the solution of a special dispersion equation for a submerged elastic plate and this is discussed in detail. We show how the solution can be calculated for the case of normal incidence on a semi-infinite plate in two spatial dimensions and then extend this solution to obliquely incident waves, to a plate of finite length and to a circular finite plate in three dimensions. Numerical calculations showing various properties of the solutions are presented and a near-orthogonality relation for the eigenfunctions is used to derive an energy-balance relation.

On The Static Term for The Electric Field in Crystal Optics

Ortner, N., Wagner, P. — 2009-05-07

A short formula for the so-called static term E₁ in the fundamental matrix of the time-dependent system of crystal optics is given. This improves the correct, albeit very complicated expression for E₁ derived previously by Burridge and Qian. Furthermore, the nature of the singularity of E₁ at the origin is elucidated precisely.

ON THE EXISTENCE OF WAVES GUIDED BY A CAVITY IN AN ELASTIC FILM

Cherednichenko, K. D., Sabina, F. J. — 2009-04-03

In the regime where linearised elasticity is a suitable approximation to the behaviour of an elastic body, the existence of a wave guided along the cavity in a film of a homogeneous and isotropic elastic material is proved, at least for a certain range of frequencies. Using the theory of linear self-adjoint operators, it is shown that the associated eigenvalue problem has a nontrivial solution in an appropriate Sobolev class, with the corresponding eigenvalue lying below the continuous spectrum. We study the existence of localised modes of this kind in two particular cases: under the assumption that the cavity is sufficiently narrow in the direction transverse to the film and for a rectangular cavity of arbitrary size.

WAVE PROPAGATION THROUGH CASCADING SCREENS OF FINITE THICKNESS WITH PERIODIC DISTRIBUTION OF OPENINGS

Scarpetta, E., Tibullo, V. — 2009-04-03

We develop an analytical approach to study normal penetration of a scalar plane wave into an arbitrary number of parallel, equidistant screens of finite thickness; on each screen, there is a periodic distribution of rectangular openings. After reducing the problem to a system of integral equations and then to an algebraic linear system, suitable assumptions on the physical and geometrical parameters lead to explicit representations for the scattered wave field and the relevant parameters. Some figures are given to reflect the peculiar wave properties of the cascading structure under consideration.