The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.

Vesicles are important surrogate structures made up of multiple phospholipids and cholesterol distributed in the form of a lipid bilayer. Tubular vesicles can undergo pearling – i.e. formation of beads on the liquid thread akin to the Rayleigh–Plateau instability. Previous studies have inspected the effects of surface tension on the pearling instabilities of single-component vesicles. In this study, we perform a linear stability analysis on a multicomponent cylindrical vesicle. We solve the Stokes equations along with the Cahn–Hilliard equation to develop the linearized dynamic equations governing the vesicle shape and surface concentration fields. This helps us to show that multicomponent vesicles can undergo pearling, buckling and wrinkling even in the absence of surface tension, which is a significantly different result from studies on single-component vesicles. This behaviour arises due to the competition between the free energies of phase separation, line tension and bending for this multi-phospholipid system. We determine the conditions under which axisymmetric and non-axisymmetric modes are dominant, and supplement our results with an energy analysis that shows the sources for these instabilities. Lastly, we delve into a weakly nonlinear analysis where we solve the nonlinear Cahn–Hilliard equation in the weak deformation limit to understand how mode-mixing alters the late time dynamics of coarsening. We show that in many situations, the trends from our simulations qualitatively match recent experiments (Yanagisawa et al., Phys. Rev. E, vol. 82, 2010, p. 051928).

The ‘Viroporin’ family comprises a number of mostly small-sized, integral membrane proteins encoded by animal and plant viruses. Despite their sequence and structural diversity, viroporins share a common functional trend: their capacity to assemble transmembrane channels during the replication cycle of the virus. Their selectivity spectrum ranges from low-pH-activated, unidirectional proton transporters, to size-limited permeating pores allowing passive diffusion of metabolites. Through mechanisms not fully understood, expression of viroporins facilitates virion assembly/release from infected cells, and subverts the cell physiology, contributing to cytopathogenicity. Compounds that interact with viroporins and interfere with their membrane-permeabilizing activity in vitro, are known to inhibit virus production. Moreover, viroporin-defective viruses comprise a source of live attenuated vaccines that prevent infection by notorious human and livestock pathogens. This review dives into the origin and evolution of the viroporin concept, summarizes some of the methodologies used to characterize the structure–function relationships of these important virulence factors, and attempts to classify them on biophysical grounds attending to their mechanisms of ion/solute transport across membranes.

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.

Hollow fibre membrane bioreactors provide a fast and efficient method for engineering functional tissue for use in medical treatments. Flow is utilised to overcome mass transport limitations by perfusing a nutrient-rich culture medium through the fibre lumen, which can then transport along the fibre lumen or across the porous membrane wall. Cells seeded at the outer membrane wall consume the nutrient and subsequently produce waste metabolites, which are transported away through an external extra-capillary space (ECS) along with excess nutrient. We present and investigate a two-dimensional axisymmetric model for fluid flow and solute transport through a single-fibre bioreactor configuration, with cells seeded to the external fibre wall. Fluid flow is modelled by steady lubrication and Darcy equations, which are coupled to the solute transport problem modelled by a system of advection–diffusion equations, supplemented with a reaction term to model the cell layer. Our model analysis reveals how spatially varying wall permeability distributions can be utilised to provide uniform nutrient delivery to a spatially uniform, homogeneous cell population. We also reveal how maximising the transmural pressure drop across the membrane wall is the dominant mechanism for waste removal rather than traditional experimental methods of flushing the ECS.

While the problem governing Stokes flow about a single particle that is subject to an external force is ill posed in two dimensions (the ‘Stokes paradox’), the related problem of two mutually repellent particles is well posed. Motivated by self-assembly phenomena in thin viscous membranes, we consider this problem in the limit of remote particles. Such limits are typically handled in the literature using reflection techniques, which provide successive approximations to the mutual hydrodynamic interactions. Since their starting point is a single particle in an unbounded fluid domain, these techniques are futile in the present two-dimensional problem. We show how this apparent contradiction is resolved via use of singular perturbations. We obtain a two-term approximation for the velocity acquired by circular disks, considering both rigid and free particle surfaces. We also illustrate our perturbation scheme for elliptic disks, deriving a renormalised single-particle velocity. The utility of our asymptotic scheme is illustrated in the general problem of hydrodynamic interaction between a cluster of remote disks.

We develop a mean-field model to examine the stability of a ‘quasi-2-D suspension’ of elongated particles embedded within a viscous membrane. This geometry represents several biological and synthetic settings, and we reveal mechanisms by which the anisotropic mobility of particles interacts with long-ranged viscous membrane hydrodynamics. We first show that a system of slender rod-like particles driven by a constant force is unstable to perturbations in concentration – much like sedimentation in analogous 3-D suspensions – so long as membrane viscous stresses dominate. However, increasing the contribution of viscous stresses from the surrounding 3-D fluid(s) suppresses such an instability. We then tie this result to the hydrodynamic disturbances generated by each particle in the plane of the membrane and show that enhancing subphase viscous contributions generates extensional fields that orient neighbouring particles in a manner that draws them apart. The balance of flux of particles aggregating versus separating then leads to a wave number selection in the mean-field model.

Many of the cell membrane's vital functions are achieved by the self-organization of the proteins and biopolymers embedded in it. The protein dynamics is in part determined by its drag. A large number of these proteins can polymerize to form filaments. In vitro studies of protein–membrane interactions often involve using rigid beads coated with lipid bilayers, as a model for the cell membrane. Motivated by this, we use slender-body theory to compute the translational and rotational resistance of a single filamentous protein embedded in the outer layer of a supported bilayer membrane and surrounded on the exterior by a Newtonian fluid. We first consider the regime where the two layers are strongly coupled through their inter-leaflet friction. We find that the drag along the parallel direction grows linearly with the filament's length and quadratically with the length for the perpendicular and rotational drag coefficients. These findings are explained using scaling arguments and by analysing the velocity fields around the moving filament. We then present and discuss the qualitative differences between the drag of a filament moving in a freely suspended bilayer and a supported membrane as a function of the membrane's inter-leaflet friction. Finally, we briefly discuss how these findings can be used in experiments to determine membrane rheology. In summary, we present a formulation that allows computation of the effects of membrane properties (its curvature, viscosity and inter-leaflet friction), and the exterior and interior three-dimensional fluids’ depth and viscosity on the drag of a rod-like/filamentous protein, all in a unified theoretical framework.

Janus membranes, thin permeable structures with chemical and geometrical asymmetric properties, show great potential in industrial separation processes. Yet the link between the micro- and macro-scale behaviours of these membranes needs to be established rigorously. Here, we develop interface conditions to describe the solvent–solute flow across Janus membranes within a homogenization-based framework. Upstream and downstream spatial averages are introduced to account for discontinuities induced by the microstructure. The homogenized model quantifies the macroscopic jump, across the membrane, in the solvent velocity and stresses, and in the solute concentration and fluxes through coefficients obtained via closure problems at the micro-scale. The model paves the way towards a better understanding of fundamental interface phenomena such as osmosis and phoresis via homogenization.

Coupling surface deformations with active stresses in two-dimensional nematic liquid crystal films leads to a rich area of investigation, particularly in biological fluid mechanics across multiple scales from tissue mechanics to cell membrane mechanics. In Al-Izzi & Morris (J. Fluid Mech., vol. 957, 2023, A4), the authors derive the complete set of governing equations for such systems. Their results provide an extended theoretical framework with which active nematic fluid films with in-plane flow and out-of-plane deformation can be analysed. To illustrate the potential applications of this framework, a few specific biologically inspired examples are discussed.

Morphodynamic equations governing the behaviour of active nematic fluids on deformable curved surfaces are constructed in the large deformation limit. Emphasis is placed on the formulation of objective rates that account for normal deformations whilst ensuring that tangential flows are Eulerian, and the use of the surface derivative (rather than the covariant derivative) in the nematic free energy, which elastically couples local order to out-of-plane bending of the surface. Focusing on surface geometry and its dynamical interplay with the hydrodynamics, several illustrative instabilities are then characterised. These include cases where the role of the Scriven–Love number and its nematic analogue are non-negligible, and where the active nematic forcing can be characterised by an analogue of the Föppl–von Kármán number. For the former, flows and changes to the nematic texture are coupled to surface geometry by viscous dissipation. This is shown to result in non-trivial relaxation dynamics for a nematic tube. For the latter, the nematic active forcing couples to the surface bending terms of the nematic free energy, resulting in extensile (active ruffling) and contractile (active pearling) instabilities in the tube shape, as well as active bend instabilities in the nematic texture. In comparison to the flat case, such bend instabilities now have a threshold set by the extrinsic curvature of the tube. Finally, we examine a topological defect located on an almost flat surface, and show that there exists a steady state where a combination of defect elasticity, activity and non-negligible spin connection drive a shape change in the surface.

Elastocapillarity has attracted increasing interest in recent years due to its important roles in many industrial applications. In this work, we derive a thermodynamically consistent continuum model for the dynamics of two immiscible fluids on a thin and inextensible elastic sheet in two dimensions. With the sheet being modelled by a deformable curve with the Wilmore energy and local inextensibility constraint, we derive a two-phase hydrodynamics model with the interfacial and boundary conditions consistent with the second law of thermodynamics. In particular, the boundary conditions on the sheet and at the moving contact line take the form of force balances involving the fluid stress, surface tensions, the sheet bending force and sheet tension, as well as friction forces arising from the slip of fluids on the sheet. The resulting model obeys an energy dissipation law. To demonstrate its capability of modelling complex elastocapillary interactions, we consider two applications: (1) the relaxation dynamics of a droplet on an elastic sheet and (2) the transport of a droplet driven by bendotaxis in a channel bounded by elastic sheets. Numerical solutions for the coupled fluid–sheet dynamics are obtained using the finite element method. The detailed information provided by the full hydrodynamics model allows us to better understand the dynamical processes as compared to other simplified models that were used in previous work.

Piezoelectric macro-fibre composite (MFC) actuators are employed onto the flexible leeward surface of an airfoil for active control. Time-resolved aerodynamic forces, membrane deformations and flow fields are synchronously measured at low Reynolds number (Re = 6 × 104). Mean aerodynamics show that the actively controlled airfoil can achieve lift-enhancement and drag-reduction simultaneously in the angle of attack range of 10° ≤ α ≤ 14°, where the rigid airfoil encounters stall. The maximum increments of lift and lift-to-drag ratio are 27.1 % and 126 % at the reduced actuation frequency of ${f^ + } = 3.52$. The unsteady coupling features are further analysed at α = 12°, where the maximum lift-enhancement occurs. It is newly discovered that the membrane vibrations and flow fields are locked into half of the actuation frequency when ${f^ + } > 3$. The shift of the dominant vibration mode from bending to inclining is the reason for the novel ‘half-frequency lock-in’ phenomenon. To the fluid–structure interaction, there are three characteristic frequencies for the actively controlled airfoil: $S{t_1} = 0.5{f^ + }$, $S{t_2} = {f^ + }$, and $S{t_3} = 1.5{f^ + }$. Here, St1 and its harmonics (St2, St3) are coupled with the natural frequencies of the leading-edge shear layer, resulting in the generation of multi-scale flow structures and suppression of flow separation. The lift presents comparable dominant frequencies between St1 and St3, which means the instantaneous lift is determined by the flow structures of St1 and St3. The local membrane bulge and dent affect the instantaneous swirl strength of flow structures near the maximum vibration amplitude location, which is the main reason for the variation of instantaneous lift.

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.

We study the kinematics, dynamics and flow fields generated by an oscillating, compliant membrane hydrofoil extracting energy from a uniform water stream at a chord-based Reynolds number $Re \approx 3 \times 10^4$. Hydrodynamic forces during the foil's motion cause the membrane to dynamically morph its shape, effectively increasing the camber during the oscillation cycle. The membrane's deflection is modelled using the Young–Laplace equation, with pressure term approximated from thin-airfoil theory. Simultaneous tracking of the membrane deformation and the surrounding flow field using laser profiling and particle image velocimetry, respectively, reveals the role of dynamic cambering in stabilizing the leading-edge vortices on the membrane. In this regime of operation, we obtain up to 160 % higher power extraction when compared to a rigid, symmetric hydrofoil. The present work provides a demonstration of how passive compliance of soft materials interacting with fluids may be exploited in tidal and fluvial energy extraction.

We develop a new model, to our knowledge, for the many-body hydrodynamics of amphiphilic Janus particles suspended in a viscous background flow. The Janus particles interact through a hydrophobic attraction potential that leads to self-assembly into bilayer structures. We adopt an efficient integral equation method for solving the screened Laplace equation for hydrophobic attraction and for solving the mobility problem for hydrodynamic interactions. The integral equation formulation accurately captures both interactions for near touched boundaries. Under a linear shear flow, we observe the tank-treading deformation in a two-dimensional vesicle made of Janus particles. The results yield measurements of intermonolayer friction, membrane permeability and, at large shear rates, membrane rupture. The simulation studies include a Janus particle vesicle in both linear and parabolic shear flows, and interactions between two Janus particle vesicles in shear and extensional flows. The hydrodynamics of the Janus particle vesicle is similar to the behaviour of an inextensible, elastic vesicle membrane with permeability.

Biological membranes are host to proteins and molecules which may form domain-like structures resulting in spatially varying material properties. Vesicles with such heterogeneous membranes can exhibit intricate shapes at equilibrium and rich dynamics when placed into a flow. Under the assumption of small deformations and a two-dimensional system, we develop a reduced-order model to describe the fluid-structure interaction between a viscous background shear flow and an inextensible membrane with spatially varying bending stiffness and spontaneous curvature. Material property variations of a critical magnitude, relative to the flow rate and internal/external viscosity contrast, can set off a qualitative change in the vesicle dynamics. A membrane of nearly constant bending stiffness or spontaneous curvature undergoes a small amplitude swinging motion (which includes tangential tank-treading), while for large enough material variations the dynamics pass through a regime featuring tumbling and periodic phase-lagging of the membrane material, and ultimately for very large material variation to a rigid-body tumbling behaviour. Distinct differences are found for even and odd spatial modes of domain distribution. Full numerical simulations are used to probe the theoretical predictions, which appear valid even when studying substantially deformed membranes.

In this paper, we study the fluid–structure interaction of a three-dimensional (3-D) flexible membrane immersed in an unsteady separated flow at moderate Reynolds numbers. We employ a body-conforming variational fluid–structure interaction solver based on the recently developed partitioned iterative scheme for the coupling of turbulent fluid flow with nonlinear structural dynamics. Of particular interest is to understand the flow-excited instability of a 3-D flexible membrane as a function of the non-dimensional mass ratio ($m^{*}$), Reynolds number ($Re$) and aeroelastic number ($Ae$). For a wide range of parameters, we examine two distinct stability regimes of the fluid–membrane interaction: deformed steady state (DSS) and dynamic balance state (DBS). We propose stability phase diagrams to demarcate the DSS and DBS regimes for the parameter space of mass ratio versus Reynolds number ($m^{*}$-$Re$) and mass ratio versus aeroelastic number ($m^{*}$-$Ae$). With the aid of the global Fourier mode decomposition technique, the distinct dominant vibrational modes are identified from the intertwined membrane responses in the parameter space of $m^{*}$-$Re$ and $m^{*}$-$Ae$. Compared to the deformed steady membrane, the flow-excited vibration produces relatively longer attached leading-edge vortices which improve the aerodynamic performance when the coupled system is near the flow-excited instability boundary. The optimal aerodynamic performance is achieved for lighter membranes with higher $Re$ and larger flexibility. Based on the global aeroelastic mode analysis, we observe a frequency lock-in phenomenon between the vortex-shedding frequency and the membrane vibration frequency causing self-sustained vibrations in the dynamic balance state. To characterize the origin of the frequency lock-in, we propose an approximate analytical formula for the nonlinear natural frequency by considering the added mass effect and employing a large deflection theory for a simply supported rectangular membrane. Through our systematic high-fidelity numerical investigation, we find that the onset of the membrane vibration and the mode transition has a direct dependence on the frequency lock-in between the natural frequency of the tensioned membrane and the vortex-shedding frequency or its harmonics. These findings on the fluid-elastic instability of membranes have implications for the design and development of control strategies for membrane wing-based unmanned systems and drones.