The Quarterly Journal of Mechanics and Applied Mathematics - current issue

Resonances of an elastic plate coupled with a compressible confined flow

Bonnet-Ben Dhia, A.-S., Mercier, J.-F. — 2009-04-13

A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid–structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations (k) = k², where is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically.

Nonaxisymmetric stokes flow between concentric cones

Hall, O., Hills, C. P., Gilbert, A. D. — 2009-04-13

We study the fully three-dimensional Stokes flow within a geometry consisting of two infinite cones with coincident apices. The Stokes approximation is valid near the apex and we consider the dominant flow features as it is approached. The cones are assumed to be stationary and the flow to be driven by an arbitrary far-field disturbance. We express the flow quantities in terms of eigenfunction expansions and allow for the first time for nonaxisymmetric flow regimes through an azimuthal wave number. The eigenvalue problem is solved numerically for successive wave numbers. Both real and complex sequences of eigenvalues are found, their relative dominance affecting the flow features observed. The implications for the presence of eddy-like structures (analogous to those found in other corner geometries) are discussed and we find that these flow features depend not only upon the internal angles of the two cones but also upon the symmetry of the driving mechanism. For an arbitrary disturbance, the dominant flow mode is not axisymmetric but rather is associated with wave number one and, by breaking axisymmetry, eddies can be avoided in this geometry.

On obtaining effective orthotropic elasticity tensors

Kochetov, M., Slawinski, M. A. — 2009-04-13

We consider the problem of obtaining the effective orthotropic tensor that corresponds to a given generally anisotropic one; herein, by ‘effective’, we mean the closest in the sense of the Frobenius norm, without a priori assuming the orientation of the orthotropic tensor. It is difficult to find the absolute minimum of the distance function since the minimization process is nonlinear, exhibiting several local minima. To find the effective orthotropic tensor, the minimization process must be performed on a three-dimensional manifold SO(3). In the case of monoclinic and transversely isotropic tensors, it can be performed on a two-dimensional sphere, which lends itself to an insightful plot that allows us to guide a numerical method. We use the orientation of the symmetry-plane normal of the effective monoclinic tensor to guide the method and obtain the effective orthotropic tensor—a two-step process.

Circular map for supercavitating flow in a multiply connected domain

Antipov, Y. A., Silvestrov, V. V. — 2009-04-13

A nonlinear free boundary-value problem of supercavitating flow past n + 1 hydrofoils is analyzed. To describe the cavities’ closure mechanism, the Tulin–Terent'ev single-spiral-vortex model is employed. The flow domain is considered as the image of an (n + 1)-connected circular domain. The conformal map is constructed in terms of the solutions to two Riemann–Hilbert problems of the theory of symmetric automorphic functions. One of the problems is homogeneous and its coefficients are continuous functions while the second problem is inhomogeneous and has discontinuous coefficients. The exact solutions to the problems are found by using quasiautomorphic and quasimultiplicative analogs of the Cauchy kernel. The case of a single plate is considered in detail and the numerical results are reported.

Interaction of elastic plate weakened by multiple holes with a set of patches

Zemlyanova, A. Y., Silvestrov, V. V. — 2009-04-13

The problem of interaction of a thin elastic infinite plate weakened by several holes with a system of superimposed patches is considered. Each patch covers completely one or several holes or is located in the exterior of all holes. Some holes may not be covered by patches. The patches are attached to the plate continuously along their boundaries. The patches and the holes can be of arbitrary shape and size. The plate is loaded with given in-plane stresses at infinity and along the boundaries of the holes. All parts of the construction are assumed to be in a generalised plane stressed state. Using special integral representations of complex potentials, the problem is reduced to a system of singular integral equations. Numerical examples on reinforcement of a plate with two or three holes covered by one or two patches are given.