Phonon redshift and Hubble friction in an expanding BEC

We revisit the theoretical analysis of an expanding ring-shaped Bose-Einstein condensate. Starting from the action and integrating over dimensions orthogonal to the phonon’s direction of travel, we derive an effective one-dimensional wave equation for azimuthally-travelling phonons. This wave equation shows that expansion redshifts the phonon frequency at a rate determined by the effective azimuthal sound speed, and damps the amplitude of the phonons at a rate given by \dot{\mathcal{V}}/{\mathcal{V}}𝒱̇/𝒱, where \mathcal{V}𝒱 is the volume of the background condensate. This behavior is analogous to the redshifting and ``Hubble friction’’ for quantum fields in the expanding universe and, given the scalings with radius determined by the shape of the ring potential, is consistent with recent experimental and theoretical results. The action-based dimensional reduction methods used here should be applicable in a variety of settings, and are well suited for systematic perturbation expansions.

Steady states of thin film droplets on chemically heterogeneous substrates

We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.

The strange instability of the equatorial Kelvin wave

<p>The Kelvin wave is perhaps the most important of the equatorially trapped waves in the terrestrial atmosphere and ocean, and plays a role in various phenomena such as tropical convection and El Nino. Theoretically, it can be understood from the linear dynamics of a stratified fluid on an equatorial β-plane, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave (but not the other equatorial waves) becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear <em>λ</em>, and the growth rate is proportional to exp(-1/λ^2) as λ → 0. This in contrast to most hydrodynamic instabilities, in which the growth rate typically scales as a positive power of λ-λ<sub>c</sub> as the control parameter λ passes through a critical value λ<sub>c</sub>.</p><p>This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause by analysing a quantum harmonic oscillator perturbed by a potential with a remote pole. Here we show how the growth rate and full spatial structure of the Kelvin wave instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with confluent hypergeometric functions in the exponentially-decaying tails of the Kelvin waves, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging, since special high-precision solutions of the governing ordinary differential equations are required even when the nondimensional shear is not that small (circa 0.5). </p>

Evaporation of a sessile droplet on a slope

We theoretically examine the drying of a stationary liquid droplet on an inclined surface. Both analytical and numerical approaches are considered, while assuming that the evaporation results from the purely diffusive transport of liquid vapor and that the contact line is a pinned circle. For the purposes of the analytical calculations, we suppose that the effect of gravity relative to the surface tension is weak, i.e. the Bond number (Bo) is small. Then, we express the shape of the drop and the vapor concentration field as perturbation expansions in terms of Bo. When the Bond number is zero, the droplet is unperturbed by the effect of gravity and takes the form of a spherical cap, for which the vapor concentration field is already known. Here, the Young-Laplace equation is solved analytically to calculate the first-order correction to the shape of the drop. Knowing the first-order perturbation to the drop geometry and the zeroth-order distribution of vapor concentration, we obtain the leading-order contribution of gravity to the rate of droplet evaporation by utilizing Green’s second identity. The analytical results are supplemented by numerical calculations, where the droplet shape is first determined by minimizing the Helmholtz free energy and then the evaporation rate is computed by solving Laplace’s equation for the vapor concentration field via a finite-volume method. Perhaps counter-intuitively, we find that even when the droplet deforms noticeably under the influence of gravity, the rate of evaporation remains almost unchanged, as if no gravitational effect is present. Furthermore, comparison between analytical and numerical calculations reveals that considering only the leading-order corrections to the shape of the droplet and vapor concentration distribution provides estimates that are valid well beyond their intended limit of very small Bo.

New model and dynamics of higher-dimensional nonlinear Rossby waves

In this work, the propagation of higher-dimensional nonlinear Rossby waves under the generalized beta effect is considered. Using the methods of weak nonlinear perturbation expansions and the multiple scales, we obtain a new (2 + 1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation for the first time. Furthermore, a new dispersion relation for the linear Rossby waves is given corresponding to the linearized Boussinesq equation. More importantly, based on the methods of the traveling wave setting and the Jacobi elliptic function expansions, several kinds of exact traveling wave solutions for the higher-dimensional nonlinear Rossby waves, including the periodic solutions, solitary solutions and others are obtained. Finally, we simulate the solitary solutions obtained by using the method of the Jacobi elliptic function. The numerical results show that the amplitude of the Rossby solitary waves is decreasing with the increase of generalized beta effect.