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Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory. 

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number,$${\textit {Bq}}\gg 1$$. A semianalytic solution of the dual integral equations governing the flow at arbitrary$${\textit {Bq}}$$was devised by Hugheset al.(J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit$${\textit {Bq}}\to 0$$, it produces the value$$8$$for the dimensionless translational drag, which is$$50\,\%$$larger than the classical$$16/3$$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit$${\textit {Bq}}\to 0$$from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction$$(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$$,$$\gamma _E$$being the Euler–Mascheroni constant. 

Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule or viscous drop in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations. 

A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $$\tau _{*c}=\lambda _{*} a_{*}/D_{*}$$ , where $$\lambda _{*}$$ is the Debye screening length, $$a_{*}$$ is the inclusion length scale and $$D_{*}$$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer. 

The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems – namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class – the rotation of an infinite circular cylinder whose boundary is periodically decorated by a finite number of infinite grooves – with the goal of calculating the rotational mobility (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters – namely, the number $$N$$ of grooves and the solid fraction $$\unicode[STIX]{x1D719}$$ . Using matched asymptotic expansions we analyse the large- $$N$$ limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation, $$\begin{eqnarray}\displaystyle 1+{\displaystyle \frac{2}{N}}\ln \csc {\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}}{2}}, & & \displaystyle \nonumber\end{eqnarray}$$ for the ratio of the enhanced mobility to the homogeneous-cylinder mobility. Making use of conformal-mapping techniques and inductive arguments we prove that the preceding approximation is actually exact for $$N=1,2,4,8,\ldots$$ . We conjecture that it is exact for all $$N$$ .