Demetrios T. Papageorgiou

A weakly nonlinear analysis is conducted on a two-ferrofluid system in a rod-annular geometry, where a thin ferrofluid film coats a current-carrying wire, and a second ferrofluid of different magnetic susceptibility fills the region between the film and the annulus. An azimuthal field generated by the current and an applied axial field produce a magnetic forcing, interacting with surface tension at the interface of the fluids. A nonlocal PDE is derived, governing the interface evolution of the initially stationary system, where numerical solutions reveal the complexity of the interplay between the magnetic and capillary forcing. The magnetic field can further enhance the development of lobe and collar structures, features that arise in the hydrodynamic case, thereby increasing the surface area of the interface. Under certain settings, it can provide partial stabilisation and even a flat stable interface. The interface morphology is readily influenced by the field strengths, the domain size, and the radius of the annulus, offering precise control over interfacial profiles and surface area. Such manipulability may have important implications for applications in coating, printing, heat transfer, and microfluidic transport.

This study investigates the effect of a tangential electric field on the curvature singularity encountered by evolving vortex sheets. Weakly nonlinear models are derived by using asymptotic expansions based on the Dirichlet–Neumann operators. A fully nonlinear boundary integral method based on Cauchy’s integral formula and an angle –arclength derivative formulation in a Lagrangian framework is proposed. Linear stability analysis reveals a transition from the unstable regime to a neutrally stable regime at a critical electric field strength. However, both weakly nonlinear models and fully nonlinear simulations suggest that the curvature singularity persists regardless of the value of the electric field strength. The analysis demonstrates that the singularity time obeys two different asymptotic rules. For subcritical values of the tangential electric field and small initial amplitudes, the singularity time scales with the logarithm of the inverse of the amplitude, and monotonically increases with the electric field strength. For supercritical values of the tangential electric field the singular time is asymptotically larger, follows an inverse power law with respect to the initial amplitude and reaches local maxima as the electric field strength varies. Fully nonlinear simulations of the underlying electrodynamic Euler equations confirm that the curvature singularity can be significantly delayed when the electric field strength takes some special values. In the limit of large electric field strengths the numerical simulations indicate that the time to singularity formation is proportional to the inverse square root of the electric field strength.

We develop a model for steady, laminar boundary layers over small-scale textured surfaces. Although the texture is small relative to the boundary-layer thickness, it modifies the flow via a slip length. We use matched asymptotic expansions to simplify the problem, dividing the flow into outer, boundary-layer and inner regions. The far-field behaviour of the inner problem yields a slip boundary condition for the boundary layer. We derive an asymptotic solution valid when the slip length is small, and for arbitrary slip lengths, we develop a numerical method combining Chebyshev collocation and finite differences. We apply this framework to canonical small-scale textured surfaces, including superhydrophobic surfaces and riblets, and utilise existing analytical slip formulae. However, the framework is expected to extend to liquid-infused, porous, compliant or deformable surfaces with a variety of regular or random textures. We demonstrate how slip modifies the boundary layer’s velocity field, wall shear stress and displacement thickness across a range of surface configurations, and examine the linear stability of the resulting slip-modified boundary layers. Our approach enables computationally inexpensive modelling of a wide range of small-scale textured surfaces within laminar boundary-layer flows, providing predictive capability for drag, boundary-layer growth and transition across applications ranging from microfluidics to turbo-machinery and marine transport.

This paper investigates the stability of interfacial long waves in two-layer plane Couette flow using a nonlinear, non-local asymptotic model derived from the Navier–Stokes equations and valid for thin upper layers. Non-locality enters through a coupling of the thin and main layers, and crucial inertial effects are retained. The models generically support bistability phenomena observed in experiments where two stable travelling waves, one unimodal and the other bimodal, are recorded at the same lid velocity. In direct comparisons with experiments, the models show remarkable agreement, both qualitatively and quantitatively. The two stable travelling waves are identified and their basins of attraction characterised via large-time computations for different initial conditions. We also identify a new symmetry-breaking travelling-wave branch bifurcating from the bimodal family, compute higher-wavenumber travelling-wave branches and present time-periodic orbits arising via Hopf bifurcation. A symmetry is also presented that links solutions for thin upper layers to those corresponding to thin lower layers. The instability of the two solutions is shown to be identical.

This study utilises chromocapillary stresses induced by light-actuated photosurfactants to demonstrate theoretically that a stable uniform liquid layer wetting a substrate can be sculpted and stirred on the microscale. A mathematical model is presented for two photosurfactant species that can switch from trans to cis states. Switching takes place in the bulk and on the interface, and convection–diffusion–reaction equations describe the local concentrations there. Under uniform light illumination (e.g. blue light) the equilibrium concentrations of trans and cis are non-uniform with layer depth, and a quiescent state with a flat interface exists. A non-uniform light intensity along the layer is superimposed to drive the system out of equilibrium, and induce interfacial deformations and flow in the bulk. This is carried out asymptotically for small-intensity non-uniformities and the first-order non-uniform solutions are found in semi-analytic form. The solutions show that a local increase in intensity increases the surface tension locally by sweeping surfactant off the interface to generate an inward trapping flow (known as a ‘Marangoni tweezer’ in experiments). Light intensities with a sinusoidal variation along the interface are also considered to show that vortical mixing motions are set up. Additionally, the liquid sculpting problem is analysed and a class of inverse problems are solved to predict the distribution of the light intensity required to produce a desired target interfacial shape. Finally, a parametric study is carried out to evaluate the effect of Biot, Damköhler and Marangoni numbers on the maximum light-induced interfacial velocity.

We develop a model for steady, laminar boundary layers over small-scale textured surfaces. Although the texture is small relative to the boundary-layer thickness, it modifies the flow via a slip length. We use matched asymptotic expansions to simplify the problem, dividing the flow into outer, boundary-layer and inner regions. The far-field behaviour of the inner problem yields a slip boundary condition for the boundary layer. We derive an asymptotic solution valid when the slip length is small. For arbitrary slip lengths, we develop a numerical method combining Chebyshev collocation and finite differences. We apply this framework to canonical small-scale textured surfaces, including superhydrophobic surfaces and riblets, and utilise existing analytical slip formulae. We demonstrate how slip can effect the boundary layer’s velocity field, wall shear stress and displacement thickness. Our approach enables computationally inexpensive modelling of small-scale textured surfaces in applications ranging from microfluidics to marine transport.

A steady equilibrium solution is presented showing how a stagnant cap of insoluble surfactant is affected by surface diffusion in the limit of low Reynolds and capillary numbers. Exploiting a recently discovered connection between the complex Burgers equation and Marangoni flows in a half-plane of viscous fluid with surfactants, it is shown how, in the steady case, a linearizable Riccati-type ordinary differential equation (ODE) governs the system. This allows the equilibrium solution, including surface diffusion effects, to be expressed in terms of parabolic cylinder functions. The stagnant cap equilibrium is recovered using a Liouville–Green approximation, in the limit of vanishing surface diffusion, through a modification of large-parameter expansions of the parabolic cylinder function. Moreover, connection formulae for such functions can be used to show how simple poles of the solution in the non-physical upper half-plane coalesce, as surface diffusion lessens, to select a choice of branch cut joining square root branch points at the two edges of a limiting stagnant cap. This singularity structure, which is expected to be generic for such Marangoni systems, is discussed in the context of the Stokes phenomenon for solutions of nonlinear ODEs.

This paper investigates the effects of a light-actuated photosurfactant on the canonical problem of the linear stability of a viscous thread surrounded by a dynamically passive fluid. A model consisting of the Navier–Stokes equations and a set of molar concentration equations is presented that capture light-induced switching between two stable surfactant isomer states, trans and cis. These two states display significantly different interfacial properties, allowing for some external control of the stability behaviour of the thread via incident light. Normal modes are used to generate a generalized eigenvalue problem for the growth rate which is solved with a hybrid analytical and numerical method. The results are validated with appropriate analytical solutions of increasing complexity, beginning with a solution to a clean interface, then analytical solutions for one insoluble surfactant, one soluble surfactant and a special case of two photosurfactants with a spatially uniform undisturbed state. Presenting each of these cases allows for a holistic discussion of the effect of surfactants in general on the stability of a liquid thread. Finally, the numerical solutions in the presence of two photosurfactants that display radially non-uniform undisturbed states are presented, and details of the impact of the illumination on the linear stability of the thread are discussed.