Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

The melting of wall-mounted ice deep inside a water layer flow is investigated, the ice being initially in the form of a slender hump or step-up and the oncoming water upstream near the wall being warmer than the ice. The wall is at the same temperature as the oncoming water except beneath the ice where the wall temperature is the same as that of the ice. The unsteady interaction of the flow, heat transfer and phase change is studied analytically and computationally for a basic pure-ice model in two spatial dimensions at high flow rates. In-flow predictions of ice-shape evolution are presented along with the complete melting times and the vanishing points where melting is completed on the solid surface. This is for a range of initial conditions and background heat transfers. The unsteady movement of the contact points at the edges of the ice formation is determined explicitly for the cases of humps and steps, together with the ultimate behaviours of both the ice humps and the ice steps. Effects of reducing the sub-ice wall temperature and accretion leading to flow separation are also discussed.

Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under ${\ell }_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for ${\ell }_1$-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian Spike-and-slab Gaussian Lasso prior. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochastic Gradient Langevin Dynamics (SGLD) which gives a computationally tractable way of Bayesian posterior sampling within our framework. Bayesian SINDy autoencoder achieves better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification suggested by the experimental studies. The Bayesian SINDy autoencoder can be applied to real video data, withaccurate physics discovery which correctly identifies the governing equation and provides a close estimate for standard physics constants like gravity $g$, for example, in videos of a pendulum.

The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.

Approximate streamsurfaces of a three-dimensional velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in three-dimensional flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for three-dimensional flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex) and a hollow cylinder (Taylor–Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows) and Rayleigh–Bénard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh–Bénard convection.

Pancake-like vortices are often generated by turbulence in geophysical flows. Here, we study the inertia-gravity oscillations that can exist within such geophysical vortices, due to the combined action of rotation and gravity. We consider a fluid enclosed within a triaxial ellipsoid, which is stratified in density with a constant Brunt–Väisälä frequency (using the Boussinesq approximation) and uniformly rotating along a (possibly) tilted axis with respect to gravity. The wave problem is then governed by a mixed hyperbolic-elliptic equation for the velocity. As in the rotating non-stratified case considered by Vantieghem (2014, Proc. R. Soc. A, 470, 20140093. (doi:10.1098/rspa.2014.0093)), we find that the spectrum is pure point in ellipsoids (i.e. only consists of eigenvalues) with polynomial eigenvectors. Then, we characterize the spectrum using numerical computations (obtained with a bespoke Galerkin method) and asymptotic spectral theory. Finally, the results are discussed in light of natural applications (e.g. for Mediterranean eddies or Jupiter’s vortices).

We present mathematical theory for self-similarity induced spectral gaps in the spectra of systems generated by generalised Fibonacci tilings. Our results characterise super band gaps, which are spectral gaps that exist for all sufficiently large periodic systems in a Fibonacci-generated sequence. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We apply our analytic results to characterise spectra in three different settings: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. The theory is shown to give accurate predictions of the super band gaps, with minimal computational cost and significantly greater precision than previous estimates. It also provides a mathematical foundation for using periodic approximants (supercells) to predict the transmission gaps of quasicrystalline samples, as we verify numerically.

A quasi steady-state model (QSM) for accurately predicting the detailed diffusion-dominated dissolution process of polydisperse spheroidal (prolate, oblate and spherical) particle systems with a broad range of distributions of particle size and aspect ratio has been developed. A rigorous, mathematics-based QSM of the dissolution of single spheroidal particles has been incorporated into the well-established framework of polydisperse dissolution models based on the assumption of uniform bulk concentration. Validation against experimental results shows that this model can accurately predict the increase in bulk concentration of polydisperse systems with various particle sizes and shape parameters. A series of representative instances involving the dissolution of polydisperse felodipine particles at various concentration ratios is used to demonstrate the model's effectiveness, rendering it a valuable tool for understanding and managing complex systems with diverse particle characteristics.

We consider the propagation of a harmonic elastic wave in a composite inclusion–matrix structure subjected to plane deformation. The interface between the inclusion and matrix is described by the complete Gurtin-Murdoch model with non-vanishing interface tension and interface stretching rigidity. We consider an inclusion of general shape and formulate the corresponding boundary value problem for the wave functions in the inclusion and matrix when a harmonic compressional or shear wave is incident on the edge of the matrix. The problem is then solved by series expansion methods for the case of a circular inclusion embedded in an infinite matrix. The series solution obtained is validated by checking its convergence and via comparisons with existing static and dynamic solutions for certain reduced cases. Numerical examples are presented for the case of a small-sized circular hole embedded in a soft matrix demonstrating the influence of surface tension on the incident wave-induced dynamic stress concentration in the matrix. We find that the presence of surface tension relieves the peak stress around the circular hole when the frequency of the incident wave is below a certain critical value, while it tends to intensify the peak stress for a high-frequency incident wave.

This paper presents a coupled model that considers the nonlinear compressibility effect in fluid–structure interaction (FSI) during air-blast loading on flexible structures. In this coupled model, structural behaviour is idealized as a linear single-degree-of-freedom mass-spring-damper system whereas nonlinear fluid compressibility is considered by applying Rankine–Hugoniot jump conditions across a moving plate. The surrounding fluid medium is modelled with an ideal gas equation and hence, this model can be applied for FSI analysis with relatively strong shocks (reflection coefficient of up to 8). The nonlinear compressibility of the fluid medium at the backside of the plate is also considered in this coupled formulation and its effects on the structural responses are examined. Moreover, the negative/underpressure phase of the reflected wave profile, which is typically neglected in a decoupled model, is also considered in the proposed model and its influence on the structural response is also investigated. The study reveals that the nonlinear compressibility of fluid medium significantly influences the coupled FSI phenomena, especially in flexible lightweight structures. Numerical examples are presented to highlight the implications of the nonlinear compressibility effect in FSI on the reflected pressure profile and the response of flexible structures. Parametric dependencies of response on structural mass and natural frequency are examined thoroughly and a response spectrum is obtained. It is envisaged that the lightweight protective structure design under higher blast intensity may benefit from this study.

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form$$\frac{\mathrm{d}{x}_i}{\mathrm{d}t}=x_i{\varphi }_i(x_1,\dots ,x_N),$$where $N$ represents the number of species and $x_i$, the abundance of species $i$. Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to$${\varphi }_i(x_1,\dots ,x_N)=r_i-x_i+{(\Gamma \mathbf{\text{x}})}_i,$$play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, $r_i$ is the intrinsic growth of species $i$ and $\Gamma$ stands for the interaction matrix: ${\Gamma }_{ij}$ represents the effect of species $j$ over species $i$. For large $N$, estimating matrix $\Gamma$ is often an overwhelming task and an alternative is to draw $\Gamma$ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.