We consider the thermal insulation property of homogeneous anisotropically heat-conducting bodies, i.e. those whose thermal tensor (matrix) A is constant throughout the body but not generally a constant times the identity. This anisotropy is a common feature of nano-composite materials. We propose using the principal Dirichlet eigenvalue λ of the associated elliptic differential operator −∇ ⋅ A∇ as a simple measurement for the insulating ability of the material, because the time scale of thermal flow is of the order of 1/λ. 1/λ is a generalization of the ‘R-value’ used in engineering practice as a measure of the insulating ability of isotropic conductors. If the thermal tensor A depends on parameters, e.g. inherited from nanostructure, then so does λ. It is important to know how λ depends on the parameters. For the material to be a good insulator, λ should be suppressed. But calculation or estimation of this principal elliptic eigenvalue, particularly over ranges of parameter values, is not a simple task. The focus of this paper is estimation – by fitted formulas and new exact bounds – of the principal elliptic Dirichlet eigenvalue of ellipses (2D) and ellipsoids (3D) using only simple expressions in the eigenvalues of the matrix A. Our simplest approximations and bounds avoid even the calculation of matrix eigenvalues and use in 2D merely the trace and determinant of A, and in 3D the trace and determinant of A and the trace of A2. The new bounds are shown to imply an extremal property in homogenization theory and a new condition for enhancement in Taylor dispersion. Recently, Zheng, Forest, Lipton, Zhou and Wang published formulas for the thermal tensor A in the case of strong extensional fiber-type flow of a polymer with dilute rod-like nano-inclusions, including explicitly the influence of the probability distribution of inclusion orientation. Our results are combined with these formulas to quantify the effect of variations of the probability distribution on the insulating ability of the composite.

In this paper, we consider the asymptotic behaviour for the non-local parabolic problem with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is non-increasing. It is found that (a) for 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 < p < 2, u(x, t) is globally bounded for any λ > 0; (c) for p = 2, if 0 < λ < 2|∂Ω|2, then u(x, t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution and u(x, t) is a global solution and u(x, t) → ∞ as t → ∞ for all x ∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution and u(x, t) blows up in finite time for all x ∈ Ω; (d) for p > 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* and u0(x) sufficiently large, u(x, t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour of u(x, t) as it blows up are obtained for p ≥ 2.

Motivated by a recent investigation of Millar and McKay [Director orientation of a twisted nematic under the influence of an in-plane magnetic field. Mol. Cryst. Liq. Cryst435, 277/[937]–286/[946] (2005)], we study the magnetic field twist-Fréedericksz transition for a nematic liquid crystal of positive diamagnetic anisotropy with strong anchoring and pre-twist boundary conditions. Despite the pre-twist, the system still possesses ℤ2 symmetry and a symmetry-breaking pitchfork bifurcation, which occurs at a critical magnetic-field strength that, as we prove, is above the threshold for the classical twist-Fréedericksz transition (which has no pre-twist). It was observed numerically by Millar and McKay that this instability occurs precisely at the point at which the ground-state solution loses its monotonicity (with respect to the position coordinate across the cell gap). We explain this surprising observation using a rigorous phase-space analysis.

We study the stability and dynamics of melting icebergs. Specifically, we address the ‘toppling’ or ‘rollover’ observed for floating icebergs. The rollover is thought to occur because the ocean melts the iceberg from below, causing its overall mass and mass distribution to change with time. We model the evolution of equilibrium positions for a general homogeneous body afloat in an ideal fluid, as this homogeneous body is subjected to ‘melting’, i.e. a slow removal of material from the submerged part. If this process is the dominating melting mechanism, can the likelihood of a toppling be inferred from observing only the above-surface part? We show here that some information about the evolution of stability due to melting can be inferred from the surface geometry of the iceberg.

In this paper, we provide the mathematical basis for three different magneto-acoustic imaging approaches (vibration potential tomography, magneto-acoustic tomography with magnetic induction and magneto-acoustic current imaging) and propose new algorithms for solving the inverse problem for each of them.