Interactions between hyperelastic bio-membranes and fluid play a crucial role in the flight (or swimming) motion of many creatures, such as bats, flying squirrels and lemurs. Bio-membranes are characterised by high stretchability and micro-bending stiffness, leading to unique fluid–solid coupling properties (Mathai et al., 2023, Phys. Rev. Lett., vol. 131, 114003). This study presents a high-fidelity numerical exploration of the hyperelastic characteristics of a pitching foil inspired by bio-membranes in fluid within a low Reynolds number regime. The focus is on the effect of foil compliance on its self-propulsion performance, mimicking natural propulsion mechanisms, with the foil free to move in the horizontal direction. We find that with certain compliance, the foil may experience a velocity crisis, meaning that its propulsive capability is completely lost. This phenomenon is caused by the loss of beat speed when the foil’s passive deformation is out of phase with the pitching motion. By contrast, the two motions can be in phase at proper compliance, leading to an increased beat speed. This will significantly enhance propulsive velocity up to $33\,\%$ compared with the rigid case. The results demonstrate the feasibility of compliance tuning to circumvent the velocity crisis and improve the propulsive speed, which are helpful in the design of micro aerial robots using biomimetic membranes.

A new arbitrary Lagrangian–Eulerian (ALE) formulation for Navier–Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterisation that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier–Stokes equations and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.

Based on present knowledge of atmospheric composition, a mechanism for the natural formation of vesicles in the lakes of Titan is proposed. It involves precipitation-induced spray droplets coated by a monolayer of amphiphiles. On interaction with the monolayer on the lake’s surface, bilayer membranes are being formed that encapsulate the liquid phase of the original droplet. The resulting vesicles develop thermodynamic stability by continuous compositional selection of various types of amphiphiles in a dynamic equilibrium, leading to an optimized vesicle stability. Different populations of stable vesicles may compete, initiating a long-term evolution process that could eventually result in primitive protocells. The existence of any type of vesicles on Titan would prove that early steps towards increasing order and complexity have taken place, which represent the necessary precondition for abiogenesis. A valid analytical approach could involve a laser device with combined light scattering analysis and surface enhanced Raman spectroscopy. It would allow for very sensitive detection of amphiphiles as well as for the observation of dispersed vesicles.

Transport in nanofluidic devices is often characterized by complex electrohydrodynamic coupling. Electro-osmotic flow (EOF), i.e. the motion of fluid due to an external electric field, is one of the most common electrohydrodynamic phenomena. However, the classical continuum description of EOF cannot be directly applied at the nanoscale, and no generic experimental techniques exist to measure EOF for nanopores just a few nanometres in size. This led to the development of approximate approaches to express EOF through experimentally accessible quantities. The most popular one, derived by Gu et al. in 2003, employs nanopore selectivity measured via reversal potential experiments and expresses EOF as the sum of water molecules dragged by each ion moving through the pore. Here, combining theoretical arguments, continuum electrohydrodynamic and molecular dynamics simulations, we discuss the limitations of these approximations. Our results indicate that, although some approximate expressions contradict basic fluid dynamics scaling arguments, they still capture the order of magnitude of EOF for very narrow biological nanopores such as MspA, CytK and CsgG. Finally, we highlight some caveats of the method, particularly when dealing with non-cylindrical biological pores and the effects of localized alterations of the pore surface charge, such as point mutations commonly employed in nanopore sensing technology.

Capsules are widely used in bioengineering, chemical engineering and industry. The development of drug delivery systems using deformable capsules is progressing, yet the regulation of drug release within a capsule remains a challenge. Meanwhile, a microswimmer enclosed in a capsule can generate a large lubrication force on the capsule membrane, which could result in deformation and mechanical damage to the membrane. In this study, we numerically investigate how a capsule can be damaged by an enclosed microswimmer. The capsule membrane is modelled as a two-dimensional neo-Hookean material, with its deformability parametrised by capillary number. An isotropic brittle damage model is applied to express membrane rupture, with the Lighthill–Blake squirmer serving as the microswimmer model. In a sufficiently small capillary number regime, pusher-type squirmers exhibit stable swimming along the capsule membrane, while neutral-type and puller-type squirmers exhibit swimming towards the membrane and remain stationary. As capillary number increases, the damage to the membrane increases and rupture occurs in all swimming modes. For pusher-type squirmers, the critical capillary number leading to rupture is dependent on the initial incidence angle, whereas neutral-type and puller-type squirmers are independent of the initial value. Furthermore, we present methods for controlling membrane damage by magnetically orienting the microswimmer. The findings reveal that a static magnetic field can orient the microswimmer, leading to membrane damage and rupture even for a capsule that cannot be damaged by free swimming, while controlling the swimming path with a rotating magnetic field enables soft membranes to maintain deformation without rupture.

A spherical capsule (radius $R$) is suspended in a viscous liquid (viscosity $\mu$) and exposed to a uniaxial extensional flow of strain rate $E$. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus $G$ and an area dilatation modulus $K$. Dimensional arguments imply that the slenderness $\epsilon$ of the deformed capsule depends only upon $K/G$ and the elastic capillary number ${Ca}=\mu R E/G$. We address the coupled flow–deformation problem in the limit of strong flow, ${Ca}\gg 1$, where large deformation allows for the use of approximation methods in the limit $\epsilon \ll 1$. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that $\epsilon$ is proportional to ${Ca}^{-2/3}$, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon $\ln \epsilon$. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio $K/G$; in particular, we find the approximation $\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$ for the relation between $\epsilon$ and $Ca$. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models.

Porous membranes, like nets or filters, are thin structures that allow fluid to flow through their pores. Homogenisation can be used to rigorously link the flow velocity with the stresses on the membrane via several coefficients (e.g. permeability and slip) stemming from the solution of Stokes problems at the pore level. For a periodic microstructure, the geometry of a single pore determines these coefficients for the whole membrane. However, many biological membranes are not periodic, and the porous microstructure of industrial membranes can be modified to address specific needs, resulting in non-periodic patterns of solid inclusions and pores. In this case, multiple microscopic calculations are needed to retrieve the local non-periodic membrane properties, negatively affecting the efficiency of the homogenised model. In this paper, we introduce an adjoint-based procedure that drastically reduces the computational cost of these operations by computing the pore-scale solution’s first- and second-order sensitivities to geometric modifications. This adjoint-based technique only requires the solution of a few problems on a reference geometry and allows us to find the homogenised solution on any number of modified geometries. This new adjoint-based homogenisation procedure predicts the macroscopic flow around a thin aperiodic porous membrane at a fraction of the computational cost of classical approaches while maintaining comparable accuracy.

The hydrodynamic analysis of motion of small particles (e.g. proteins) within lipid bilayers appears to be naturally suitable for the framework of two-dimensional Stokes flow. Given the Stokes paradox, the problem in an unbounded domain is ill-posed. In his classical paper, Saffman (J. Fluid Mech., vol. 73, 1976, pp. 593–602) proposed several possible remedies, one of them based upon the finite extent of the membrane. Considering a circular boundary, that regularisation was briefly addressed by Saffman in the isotropic configuration, where the particle is concentrically positioned in the membrane. We investigate here the hydrodynamic problem in bounded membranes for the general case of eccentric particle position and a rectilinear motion in an arbitrary direction. Symmetry arguments provide a representation of the hydrodynamic drag in terms of ‘radial’ and ‘transverse’ coefficients, which depend upon two parameters: the ratio $\lambda$ of particle to membrane radii and the eccentricity $\beta$. Using matched asymptotic expansions we obtain closed-form approximations for these coefficients in the limit where $\lambda$ is small. In the isotropic case ($\beta = 0$) we find that the drag coefficient is $4\pi /(\ln ({1}/{\lambda })- {1})$, contradicting the value $4\pi /(\ln ({1}/{\lambda })- {1}/{2})$ obtained by Saffman. We explain the oversight in Saffman’s argument.

We present numerical analysis of the lateral movement of a spherical capsule in the steady and pulsatile channel flow of a Newtonian fluid for a wide range of oscillatory frequencies. Each capsule membrane satisfying strain-hardening characteristics is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$. Our numerical results showed that capsules with high $Ca$ exhibit axial focusing at finite $Re$ similarly to the inertialess case. We observe that the speed of the axial focusing can be substantially accelerated by making the driving pressure gradient oscillate in time. We also confirm the existence of an optimal frequency that maximises the speed of axial focusing, which remains the same found in the absence of inertia. For relatively low $Ca$, however, the capsule exhibits off-centre focusing, resulting in various equilibrium radial positions depending on $Re$. Our numerical results further clarify the existence of a specific $Re$ for which the effect of the flow pulsation to the equilibrium radial position is maximum. The roles of channel size on the lateral movements of the capsule are also addressed. Throughout our analyses, we have quantified the radial position of the capsule in a tube based on an empirical expression. Given that the speed of inertial focusing can be controlled by the oscillatory frequency, the results obtained here can be used for label-free cell alignment/sorting/separation techniques, e.g. for circulating tumour cells in cancer patients or precious haematopoietic cells such as colony-forming cells.

Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue. We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface; and in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic $Q$-tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.

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.

Viruses are highly dynamic macromolecular assemblies. They undergo large-scale changes in structure and organization at nearly every stage of their infectious cycles from virion assembly to maturation, receptor docking, cell entry, uncoating and genome delivery. Understanding structural transformations and dynamics across the virus infectious cycle is an expansive area for research that that can also provide insight into mechanisms for blocking infection, replication, and transmission. Additionally, the processes viruses carry out serve as excellent model systems for analogous cellular processes, but in more accessible form. Capturing and analyzing these dynamic events poses a major challenge for many structural biological approaches due to the size and complexity of the assemblies and the heterogeneity and transience of the functional states that are populated. Here we examine the process of protein-mediated membrane fusion, which is carried out by specialized machinery on enveloped virus surfaces leading to delivery of the viral genome. Application of two complementary methods, cryo-electron tomography and structural mass spectrometry enable dynamic intermediate states in intact fusion systems to be imaged and probed, providing a new understanding of the mechanisms and machinery that drive this fundamental biological process.

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.