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We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Here, we observe and model mathematically an unexpected fundamental mechanism by which particles suspended within stratification may self-assemble and form large aggregates without adhesion. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles at the same heights. We observe that many particles demonstrate a collective motion revealing a system which appears to solve jigsaw-like puzzles on its way to organizing into a large-scale disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments isolate the individual dynamics, which are quantitatively predicted by simulations. Numerical force calculations with two spheres are used to build many-body simulations which capture observed features of self-assembly.

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross‐sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle and localized, cross‐wise uniform initial distributions, short‐time skewness is calculated exactly to be positive, while long‐time asymptotics shows it to be negative. Monte Carlo simulations confirm these predictions, and document the timescale for sign change. The equilateral triangle appears to be the only regular polygon with this property—all others possess positive skewness at all times. Alternatively, closed‐form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how both nonzero short‐time skewness and the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (more mass to the right of the mean) or back (more mass to the left of the mean) “loading” properties of the evolving tracer distribution along the pipe. Short‐ and long‐time asymptotics confirm this condition, and Monte Carlo simulations verify this at all times. The simulations are also used to examine the role of corners and boundaries on the distribution for short‐time evolution ofpoint source, as opposed to cross‐wise uniform, initial data.