Surfactant spreading in a two-dimensional cavity and emergent contact-line singularities

We model the advective Marangoni spreading of insoluble surfactant at the free surface of a viscous fluid that is confined within a two-dimensional rectangular cavity. Interfacial deflections are assumed small, with contact lines pinned to the walls of the cavity, and inertia is neglected. Linearising the surfactant transport equation about the equilibrium state allows a modal decomposition of the dynamics, with eigenvalues corresponding to decay rates of perturbations. Computation of the family of mutually orthogonal two-dimensional eigenfunctions reveals singular flow structures near each contact line, resulting in spatially oscillatory patterns of shear stress and a pressure field that diverges logarithmically. These singularities at a stationary contact line are associated with dynamic compression of the surfactant monolayer. We show how they can be regularised by weak surface diffusion. Their existence highlights the need for careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.

Boundary electromagnetic duality from homological edge modes

Recent years have seen a renewed interest in using ‘edge modes’ to extend the pre-symplectic structure of gauge theory on manifolds with boundaries. Here we further the investigation undertaken in [1] by using the formalism of homotopy pullback and Deligne- Beilinson cohomology to describe an electromagnetic (EM) duality on the boundary of M = B3 × ℝ. Upon breaking a generalized global symmetry, the duality is implemented by a BF-like topological boundary term. We then introduce Wilson line singularities on ∂M and show that these induce the existence of dual edge modes, which we identify as connections over a (−1)-gerbe. We derive the pre-symplectic structure that yields the central charge in [1] and show that the central charge is related to a non-trivial class of the (−1)-gerbe.

On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities

We present new families of weighted homogeneous and Newton nondegenerate line singularities that satisfy the Zariski multiplicity conjecture.

Contact-line singularities resolved exclusively by the Kelvin effect: volatile liquids in air

The contact line of a volatile liquid on a flat substrate is studied theoretically. We show that a remarkable result obtained for a pure-vapour atmosphere (Phys. Rev. E, vol. 87, 2013, 010401) also holds for an isothermal diffusion-limited vapour exchange with air. Namely, for both zero and finite Young’s angles, the motion- and phase-change-related contact-line singularities can in principle be regularised solely by the Kelvin effect (curvature dependence of saturation conditions). The latter prevents the curvature from diverging and rather leads to its versatile self-adjustment. To illustrate the point, the problem is resolved for a distinguished vicinity of the contact line (‘microregion’) in a ‘minimalist’ way, i.e. without any disjoining pressure, precursor film, Navier slip or any other microphysics. This also leads to the determination of the ‘Kelvin-only’ evaporation- and motion-induced apparent contact angles. With the Kelvin-only microscales actually turning out to be quite nanoscopic, other microphysics effects may nonetheless interfere too in reality. The Kelvin-only results will then yield a limiting case within such a more general formulation.