Summary We derive analytical solutions based on singular Green’s functions, which enable efficient computations of scattering simulations or Floquet–Bloch dispersion relations for waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales; we cross-verify against finite element numerical simulations to gain further confidence in our approach, which relies upon the hypothesis of Euler–Bernoulli beam theory considerably simplifying continuity conditions such that each beam can be replaced by point forces and moments applied to the neutral plane of the plate. The representations of Green’s functions by Fourier series or Fourier transforms readily follows, yielding rapid and accurate analytical schemes. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect. Topologically protected states are produced and coexist along: interfaces between adjoining chiral-mirrored bulk media, and edges between one such chiral bulk and the surrounding bare elastic plate, allowing topological circuits to be designed with robust waveguiding. Our topologically protected interfacial states correspond to zero-line modes, and our topological edgestates are produced in accordance with the bulk-edge correspondence. These topologically non-trivial states exist within near flexural resonances of the constituent beams of the phononic crystal and hence can be tuned into a deep-subwavelength regime.

Summary In this article, we prove that for isotropic functions that depend on $P$ vectors, $N$ symmetric tensors and $M$ non-symmetric tensors (a) the minimal number of irreducible invariants for a scalar-valued isotropic function is $3P+9M+6N-3,$ (b) the minimal number of irreducible vectors for a vector-valued isotropic function is $3$ and (c) the minimal number of irreducible tensors for a tensor-valued isotropic function is at most $9$. The minimal irreducible numbers given in (a), (b) and (c) are, in general, much lower than the irreducible numbers obtained in the literature. This significant reduction in the numbers of irreducible isotropic functions has the potential to substantially reduce modelling complexity.

SummaryWe investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.

SummaryWhile several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a non-constant vorticity affects the flow structure, such as the appearance of stagnation points. In order to shed light on this topic, we investigate in detail the flow beneath solitary waves propagating on an exponentially decaying sheared current. Our focus is to analyse numerically the emergence of stagnation points. For this purpose, we approximate the velocity field within the fluid bulk through the classical Korteweg-de Vries asymptotic expansion and use the Matlab language to evaluate the resulting stream function. Our findings suggest that the flow beneath the waves can have 0, 1 or 2 stagnation points in the fluid body. We also study the bifurcation between these flows. Our simulations indicate that the stagnation points emerge from a streamline with a sharp corner.

SummaryA system of nonlinear stochastic equations excited by a Gaussian random term leading to a statistically stationary solution is considered. The Carleman linearization is used to handle the nonlinearity and the statistical characterization of the solution is formulated in terms of a sequence of correlations of increasing order. Truncated in a finite but arbitrary order, the problem leads to a linear system of equations yielding directly the correlations. It is demonstrated in two examples that, if the excitation is not too strong, the solution converges as a function of the truncation order and provides an alternative to the Monte Carlo approach consisting in ensemble averaging a large number of time-dependent random solutions. For a low-dimensional system, it is shown that replacing the tensor indexing by a numbering accounting for redundancies makes it possible to keep the total problem dimension within reasonable limits even if relatively high-order correlations are accounted for. A model reduction based on the singular value decomposition of the second-order correlation matrix is tested with success for a case of a partial differential equation (Burgers’ equation), showing that the method can be potentially applied even to high-dimensional systems originating in partial differential equations.

SummaryWe consider the long-term evolution of an axisymmetric bubble and explore the ways in which it may develop. Linearised inviscid analysis is used to predict the stability of the bubble with a small disturbance while a nonlinear inviscid extension shows that the growth of unstable modes is ultimately limited by the formation of axisymmetric curvature singularities. The addition of surface tension is shown to delay, but not entirely prevent, these singularities. Our results are found to agree well with a viscous Boussinesq theory at least to early times. The inclusion of viscosity means that the development of the bubble structure is not limited by the creation of singularities, and the bubble may ultimately adopt one of a wide range of possible large-scale deformations. Among these, perhaps the most exotic are jet-like structures which can pinch off and break into several distinct parts. Spectral methods are employed to solve the inviscid and Boussinesq models while the linearised inviscid model admits a closed-form series solution.

SummaryWe consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of $u$. We conclude the article by validating this theory against full numerical computations for two specific examples.

Summary We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates.

Summary Theoretical evidence is given that it is possible for superhydrophobicity to enhance steady laminar convective heat transfer in pressure-driven flow along a circular pipe or tube with constant heat flux. Superhydrophobicity here refers to the presence of adiabatic no-shear zones in an otherwise solid no-slip boundary. Adding such adiabatic no-shear zones reduces not only hydrodynamic friction, leading to greater fluid volume fluxes for a given pressure gradient, but also reduces the solid surface area through which heat enters the fluid. This leads to a delicate trade-off between competing mechanisms so that the net effect on convective heat transfer along the pipe, as typically measured by a Nusselt number, is not obvious. Existing evidence in the literature suggests that superhydrophobicity always decreases the Nusselt number, and therefore compromises the net heat transfer. In this theoretical study, we confirm this to be generally true but, significantly, we identify a situation where the opposite occurs and the Nusselt number increases thereby enhancing convective heat transfer along the pipe.

Summary This article analyzes the axisymmetric contact problem of two elastic inhomogeneous bodies whose Young moduli are power functions of depth and the exponents are not necessarily the same. It is shown that the model problem is equivalent to an integral equation with respect to the pressure distribution whose kernel is a linear combination of two Weber–Schafheitlin integrals. The pressure is expanded in terms of the Jacobi polynomials, and the expansion coefficients are recovered by solving an infinite system of linear algebraic equations of the second kind. The coefficients of the system are represented through Mellin convolution integrals and computed explicitly. The Hertzian and Johnson–Kendall–Robertson adhesive models are employed to determine the contact radius, the displacement of distant points of the contacting bodies, the pressure distribution and the elastic normal displacement of surface points outside the contact circular zone. The effects of the exponents of the Young moduli and the surface energy density on the pressure distribution and the displacements are numerically analyzed.