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The dynamics of thin, non-circular droplets evaporating in the diffusion-limited regime are examined. The challenging non-rectilinear mixed-boundary problem this poses is solved using a novel asymptotic approach and an asymptotic expansion for the evaporative flux from the free surface of the droplet is found. While theoretically valid only for droplets that are close to circular, it is demonstrated that the methodology can successfully be applied to droplets with a wide variety of footprint shapes, including polygons and highly non-convex domains. While the applications of this are numerous, the analytically-tractable case of deposition of solute from large droplets is examined in detail, including a matched asymptotic analysis to resolve the pressure, streamlines and deposition up to second order., Comment: 21 pages, 8 figures

The error incurred in the representation of the contact pressure at the edges of incomplete contacts by first order asymptotes is treated, and the maximum value of the relative error found for a range of geometries, both symmetric and non-symmetric. Shear tractions are excited by both the application of a shear force and the application of bulk tension in one body. An asymptotic representation of the shear traction distribution under conditions of full stick is presented.

We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup., Comment: Lemma 4.3 is incorrect and cannot be corrected by minor revision. The main algorithm requires the states described in its conclusion, and so must be either reworked or removed

We consider the effect of droplet geometry on the early-stages of coffee ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small-capillary number, large-solutal Peclet number limit for two evaporative models: a kinetic model, in which the evaporative flux is constant across the droplet, and a diffusive model, in which the evaporative flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow while, local to the contact line, solute diffusion counters advection. The resulting interplay leads to the formation of the nascent coffee ring. By exploiting a coordinate system embedded in the contact line, we solve explicitly the local leading-order problem, deriving a similarity profile (in the form of a gamma distribution) that describes the coffee ring profile in its early stages. Notably, the ring characteristics change due to the concomitant asymmetry in the shape of the droplet free surface, the evaporative flux (in the diffusive evaporative regime) and the mass flux into the contact line. We utilize the asymptotic model to determine the effects of geometry on the growth of the coffee ring for a droplet with an elliptical contact set. Our results offer mechanistic insight into the effect of contact-line curvature on the development of the coffee-ring from deposition up to jamming of the solute; moreover our model predicts when finite concentration effects become relevant.

Methods for finding the optimal choices of the applied remote loads -- the applied normal force, moment, shear force and remote bulk stresses -- needed to solve frictional contact problems in partial-slip using half-plane theory are derived by using data from contacts analysed by the finite element method. While the normal and shear forces and moment are readily found from equilibrium considerations, in order to determine the bulk stresses we must exploit details of the traction ratio and the direct strain within the contact, both of which are readily extracted from simulations. These contact loads enable the formulation of an equivalent half-plane problem for the contact, which can be used to determine much more precise estimates of the slip-zone sizes than are obtainable from direct use of frictional finite element analysis, as aggregated data from the finite element output is employed, and the half-plane analysis will add precision in terms of satisfaction of the laws of frictional slip and stick., Comment: 22 pages, 10 figures

This contribution presents an asymptotic formulation for the stick-slip behaviour of incomplete contacts under oscillatory variation of normal load, moment, shear load and differential bulk tension. The asymptotic description allows us not only to approximate the size of the slip zones during the steady-state of a cyclic problem without knowledge of the geometry or contact law, but provides a solution when all known analytical solutions for incomplete contacts reach their limitations, that is, in the presence of a varying moment and a differential bulk tension large enough to reverse the direction of slip at one end of the contact. An insightful comparison between the mathematically explicit analytical solution and the asymptotic approach is drawn using the example geometry of a shallow wedge., Comment: 21 pages, 11 figures

In this analysis, we consider the effects of non-quiescent initial conditions driven by pre-impact air-water interactions on the classical Wagner model of impact theory. We consider the problem of a rigid, solid impactor moving vertically towards a liquid pool. Prior to impact, viscous forces in the air act to deform the liquid free surface, inducing a flow in the pool. These interactions are then incorporated as initial conditions in the post-impact analysis. We derive expressions for the size of the effective contact set, the leading-order pressure and force on the impactor, and the speed and thickness of the jet at its base. In all cases, we show that the effect of the pre-impact behaviour is to cushion the impactor, reducing the size of the effective contact set and, hence, the force on the impactor. Small- and large-time asymptotic solutions are derived for general power-law impactors, and we show that the effects of the air die away as the impact progresses, so that we approach the classical Wagner solution.