The red blood cell (RBC) membrane is composed of a lipid bilayer and a cytoskeleton interconnected by protein junction complexes, allowing for potential sliding between the lipid bilayer and the cytoskeleton. Despite this biological reality, it is most often modelled as a single-layer model, a hyperelastic capsule or a fluid vesicle. Another approach involves incorporating the membrane's composite structure using double layers, where one layer represents the lipid bilayer and the other represents the cytoskeleton. In this paper, we computationally assess the various modelling strategies by analysing RBC behaviour in extensional flow and four distinct regimes that simulate RBC dynamics in shear flow. The proposed double-layer strategies, such as the vesicle–capsule and capsule–capsule models, account for the fluidity and surface incompressibility of the lipid bilayer in different ways. Our findings demonstrate that introducing sliding between the layers offers the cytoskeleton a considerable degree of freedom to alleviate its elastic stresses, resulting in a significant increase in RBC elongation. Surprisingly, our study reveals that the membrane modelling strategy for RBCs holds greater importance than the choice of the cytoskeleton's reference shape. These results highlight the inadequacy of considering mechanical properties alone and emphasise the need for careful integration of these properties. Furthermore, our findings fortuitously uncover a novel indicator for determining the appropriate stress-free shape of the cytoskeleton.

Three-dimensional simulations of the ordering of elastic capsule suspensions within planar Poiseuille flow channels are reported. The simulations utilize the immersed boundary method coupled with the lattice Boltzmann method to capture the complex flow-induced capsule deformations and hydrodynamic interactions within the suspensions. A parametric study is presented whereby the confinement ratio and the particle deformability are varied independently within a two-dimensional range relevant to this ordering phenomenon. The initial distribution of capsules is random, and the simulations evolve the system from a disordered state to an ordered one, while an order parameter that quantifies the fraction of capsules belonging to one-dimensional train assemblies is computed throughout time. A monotonic increase in ordering is observed with increasing deformability. However, an optimal confinement ratio is identified corresponding to a peak in the order parameter. This peak is attributed to the competition between increasing long-range capsule attractions and decreasing in-plane capsule density (with fixed volume fraction) as the confinement ratio increases. Simulations are also performed to understand how dispersity in capsule size and deformability impact the degree of ordering. It is shown that ordering is quite sensitive to dispersity in capsule size, and much less sensitive to dispersity in deformability. Overall, the results provide important insights for the design of microfluidic devices.

A computational model is developed to study the time-averaged mean dynamics of red blood cells (RBCs) driven by the time-averaged mean stress generated by two phase-shifted orthogonal ultrasonic standing waves in a viscous fluid. The cell is modelled as an ellipsoidal viscoelastic membrane enclosing the viscous fluid cytoplasm, the motion of which is described by the inclination angle of the ellipsoidal cell shape and the phase angle of the potential membrane cycle. Based on the acoustic perturbation method, the acoustic field and acoustic streaming field are solved to obtain the time-averaged mean stress, and then the temporal evolution equations of the inclination and phase angles of the cell are determined considering the torque balance and energy conservation. At a small acoustic pressure amplitude, this model reproduces the experimentally observed features of cell motion in orthogonal standing waves: the transition from steady stationary orientation to unsteady tumbling with the increase of the phase difference between the two standing waves. By turning up the acoustic pressure amplitude above a critical value, it is further predicted that the previously observed motions can be accompanied by the membrane tank-treading rotation. Observations of these motions, combined with the present computational model, can help to evaluate the mechanical properties of RBC membranes in an automated and high-throughput manner by acoustic methods.

Membrane viscosity is known to play a central role in the transient dynamics of isolated viscoelastic capsules by decreasing their deformation, inducing shape oscillations and reducing the loading time, that is, the time required to reach the steady-state deformation. However, for dense suspensions of capsules, our understanding of the influence of the membrane viscosity is minimal. In this work, we perform a systematic numerical investigation based on coupled immersed boundary–lattice Boltzmann (IB-LB) simulations of viscoelastic spherical capsule suspensions in the non-inertial regime. We show the effect of the membrane viscosity on the transient dynamics as a function of volume fraction and capillary number. Our results indicate that the influence of membrane viscosity on both deformation and loading time strongly depends on the volume fraction in a non-trivial manner: dense suspensions with large surface viscosity are more resistant to deformation but attain loading times that are characteristic of capsules with no surface viscosity, thus opening the possibility to obtain richer combinations of mechanical features.

This work investigates the time-averaged shape dynamics of a soft elastic capsule in two phase-shifted orthogonal ultrasonic standing waves. The capsule consists of an elastic membrane that encloses a viscous fluid and is immersed in another viscous fluid. Combining the acoustic perturbation theory of fluid dynamics with the thin-shell mechanics of capsule membrane deformation, two sets of equations are established to govern the ultrasonic propagation and the time-averaged response of the fluid–capsule system, respectively. These governing equations are solved numerically based on the finite element method. Numerical simulations show that the ultrasonic standing waves have pure elongation and pure rotation effects on the initially circular capsule when the phase difference is 0 and ${\rm \pi}/2$, respectively. By setting the phase difference between 0 and ${\rm \pi}/2$, it is found that the initially circular capsule exhibits a tank-treading motion due to the combined effect of the elongation and rotation. The capsule membrane elasticity and internal fluid viscosity have significant effects on the tank-treading behaviour of the initially circular capsule, including elongation deformation, inclination angle and tank-treading velocity. For the initially non-circular capsule, three types of dynamical states, including stable orientation, swinging and tumbling, are predicted by varying the phase difference and intensity of the ultrasonic standing waves, as well as the initial shape, membrane elasticity and internal fluid viscosity of the initially non-circular capsule. This work enriches the cell manipulation capabilities of ultrasonic standing wave micro-acoustofluidics and may inspire new biological applications.

With an immersed-boundary lattice-Boltzmann method, we consider the transit of a three-dimensional initially spherical capsule with a viscoelastic membrane through a cross-slot microchannel. The capsule is released with a small initial off-centre distance in the feeding channel, to mimic experiments where capsules or cells are not perfectly aligned with the centreline. Our main objective is to establish the phase diagram of the capsule's deformation modes as a function of the flow inertia and capsule membrane viscosity. We mainly find three deformation modes in the channel cross-slot. For a capsule with low membrane viscosity, a quasi-steady mode occurs at low Reynolds numbers ($Re$), in which the capsule can reach and maintain a steady ellipsoidal shape near the stagnation point, for a considerable time period. With $Re$ increasing to 20, an overshoot–retract mode is observed. The capsule deformation oscillates on an inertial–elastic time scale, suggesting that the dynamics is mainly driven by the balance of the inertial and membrane elastic forces. The membrane viscosity slows down the capsule deformation and suppresses the overshoot–retract mode. A capsule with high membrane viscosity undergoes a continuous-elongation mode, in which its deformation keeps increasing during most of its journey in the channel cross-slot. We summarise the results in phase diagrams, and propose a scaling model which can predict the deformation modes of a viscoelastic capsule in the inertial flow regime. We also discuss implications of the present findings for practical experiments for mechanical characterisation of capsules or cells.

In this paper, we present a generic approach of a dynamical data-driven model-order reduction technique for three-dimensional fluid–structure interaction problems. A low-order continuous linear differential system is identified from snapshot solutions of a high-fidelity solver. The reduced-order model uses different ingredients, such as proper orthogonal decomposition, dynamic mode decomposition and Tikhonov-based robust identification techniques. An interpolation method is used to predict the capsule dynamics for any values of the governing non-dimensional parameters that are not in the training database. Then a dynamical system is built from the predicted solution. Numerical evidence shows the ability of the reduced model to predict the time evolution of the capsule deformation from its initial state, whatever the parameter values. Accuracy and stability properties of the resulting low-order dynamical system are analysed numerically. The numerical experiments show very good agreement, measured in terms of modified Hausdorff distance between capsule solutions of the full-order and low-order models, in the case of both confined and unconfined flows. This work is a first milestone to move towards real-time simulation of fluid–structure problems, which can be extended to nonlinear low-order systems to account for strong material and flow nonlinearities. It is a valuable innovation tool for rapid design and for the development of innovative devices.

We present a numerical analysis of the lateral movement and equilibrium radial positions of red blood cells (RBCs) with major diameter 8 $\mathrm {\mu }$m under a Newtonian fluid in a circular channel with 50 $\mathrm {\mu }$m diameter. Each RBC, modelled as a biconcave capsule whose membrane satisfies strain-hardening characteristics, is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$, the latter of which indicates the ratio of the fluid viscous force to the membrane elastic force. The effects of initial orientation angles and positions on the equilibrium radial position of an RBC centroid are also investigated. The numerical results show that depending on their initial orientations, RBCs have bistable flow modes, so-called rolling and tumbling motions. Most RBCs have a rolling motion. These stable modes are accompanied by different equilibrium radial positions, where tumbling RBCs are further away from the channel axis than rolling ones. The inertial migration of RBCs is achieved by alternating orientation angles, which are affected primarily by the initial orientation angles. Then the RBCs assume the aforementioned bistable modes during the migration, followed by further migration to the equilibrium radial position at much longer time periods. The power (or energy dissipation) associated with membrane deformations is introduced to quantify the state of membrane loads. The energy expenditures rely on stable flow modes, the equilibrium radial positions of RBC centroids, and the viscosity ratio between the internal and external fluids.

This paper examines the shape dynamics of deformable elastic and viscoelastic particles in an ambient Newtonian fluid subjected to simple shear. The particles are allowed to undergo large deformation, with the elastic stress determined using the neo-Hookean constitutive relation. We first present a method to determine the shape dynamics of initially ellipsoidal particles that is an extension of the method of Roscoe (J. Fluid Mech., vol. 28, issue 2, 1967, pp. 273–293), originally used to determine the shape at steady state of an initially spherical particle. We show that our method recovers earlier results for the in-plane trembling and tumbling dynamics of initially prolate spheroids in simple shear flow, obtained by a different approach. We then examine the in-plane dynamics of oblate spheroids and triaxial ellipsoids in simple shear flow, and show that they too, like prolate spheroids, exhibit time-periodic tumbling or trembling dynamics, depending on the initial aspect ratios of the particle and the elastic capillary number $G \equiv \mu \dot {\gamma }/\eta$, where $\mu$ is the viscosity of the fluid, $\eta$ is the elastic shear modulus of the particle and $\dot {\gamma }$ is the shear rate. In addition, we find a novel state wherein the particle extends indefinitely in time and asymptotically aligns with the flow axis. We demarcate all the dynamical regimes in the parameter space comprising $G$ and the initial particle aspect ratios. When the particles are viscoelastic, damped oscillatory dynamics is observed for initially spherical particles, and the tumbling–trembling boundary is altered for initially prolate spheroids so as to favour tumbling.

We investigate the formation and stability of a pair of identical soft capsules in channel flow under mild inertia. We employ a combination of the lattice Boltzmann, finite element and immersed boundary methods to simulate the elastic particles in flow. Validation tests show excellent agreement with numerical results obtained by other research groups. Our results reveal new trajectory types that have not been observed for pairs of rigid particles. While particle softness increases the likelihood of a stable pair forming, the pair stability is determined by the lateral position of the particles. A key finding is that stabilisation of the axial distance occurs after lateral migration of the particles. During the later phase of pair formation, particles undergo damped oscillations that are independent of initial conditions. These damped oscillations are driven by a strong hydrodynamic coupling of the particle dynamics, particle inertia and viscous dissipation. While the frequency and damping coefficient of the oscillations depend on particle softness, the pair formation time is largely determined by the initial particle positions: the time to form a stable pair grows exponentially with the initial axial distance. Our results demonstrate that particle softness has a strong impact on the behaviour of particle pairs. The findings could have significant ramifications for microfluidic applications where a constant and reliable axial distance between particles is required, such as flow cytometry.

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.

Although the behaviour of fluid-filled vesicles in steady flows has been extensively studied, far less is understood regarding the shape dynamics of vesicles in time-dependent oscillatory flows. Here, we investigate the nonlinear dynamics of vesicles in large amplitude oscillatory extensional (LAOE) flows using both experiments and boundary integral (BI) simulations. Our results characterize the transient membrane deformations, dynamical regimes and stress response of vesicles in LAOE in terms of reduced volume (vesicle asphericity), capillary number (${Ca}$, dimensionless flow strength) and Deborah number (${De}$, dimensionless flow frequency). Results from single vesicle experiments are found to be in good agreement with BI simulations across a wide range of parameters. Our results reveal three distinct dynamical regimes based on vesicle deformation: pulsating, reorienting and symmetrical regimes. We construct phase diagrams characterizing the transition of vesicle shapes between pulsating, reorienting and symmetrical regimes within the two-dimensional Pipkin space defined by ${De}$ and ${Ca}$. Contrary to observations on clean Newtonian droplets, vesicles do not reach a maximum length twice per strain rate cycle in the reorienting and pulsating regimes. The distinct dynamics observed in each regime result from a competition between the flow frequency, flow time scale and membrane deformation time scale. By calculating the particle stresslet, we quantify the nonlinear relationship between average vesicle stress and strain rate. Additionally, we present results on tubular vesicles that undergo shape transformation over several strain cycles. Broadly, our work provides new information regarding the transient dynamics of vesicles in time-dependent flows that directly informs bulk suspension rheology.

A computational study is presented on cross-stream migration and focusing of deformable capsules in curved microchannels of square and rectangular sections under inertial and non-inertial regimes. The numerical methodology is based on immersed boundary methods for fluid–structure coupling, a finite-volume-based flow solver and finite-element method for capsule deformation. Different focusing behaviours in the two regimes are predicted that arise due to the interplay of inertia, deformation, altered shear gradient, streamline curvature effect and secondary flow. In the non-inertial regime, a single-point focusing occurs on the central plane, and at a radial location between the interior face (i.e. face with highest curvature) of the channel and the location of zero shear. The focusing position is nearly independent of capsule deformability (represented by the capillary number, $Ca$). A two-step migration is observed that is comprised of a faster radial migration, followed by a slower migration toward the centre plane. The focusing location progressively moves further toward the interior face with increasing curvature and width, but decreasing height. In the inertial regime, single-point focusing is observed near the interior face for channel Reynolds number $Re_{C}\sim {O}(1)$, that is also highly sensitive to $Re_{C}$ and $Ca$, and moves progressively toward the exterior face with increasing $Re_{C}$ but decreasing $Ca$. As $Re_{C}$ increases by an order, secondary flow becomes stronger, and two focusing locations appear close to the centres of the Dean vortices. This location becomes practically independent of $Ca$ at even higher inertia. The inertial focusing positions move progressively toward the exterior face with increasing channel width and decreasing height. For wider channels, the equilibrium location is further toward the exterior face than the vortex centre.

We present a microfluidic method to measure the elastic properties of a population of microcapsules (liquid drops enclosed by a thin hyperelastic membrane). The method is based on the observation of flowing capsules in a cylindrical capillary tube and an automatic inverse analysis of the deformed profiles. The latter requires results from a full numerical model of the fluid–structure interaction accounting for nonlinear membrane elastic properties. For ease of use, we provide them under the form of databases, when the initially spherical capsule has a membrane governed by a neo-Hookean or a general Hooke's law with different surface Poisson ratios. Ultimately, the microfluidic method yields information on the type of elastic constitutive law that governs the capsule wall material together with the value of the elastic parameters. The method is applied to a population of ovalbumin microcapsules and is validated by means of independent experiments of the same capsules subjected to a different flow in a microrheological device. This is of great interest for quality control purposes, as small samples of capsule suspensions can be diverted to a measuring test section and mechanically tested with a 10 % precision using an automated process.

Microfluidic systems consisting of a square microchannel with an orthogonal side branch are promising tools to enrich or sort suspensions of deformable capsules. To allow their operating control, we numerically consider a train of initially spherical identical capsules, equally spaced along the axis of the feeding channel. The capsules have a strain-hardening membrane, an internal fluid viscosity identical to that of the external fluid and a size comparable to that of the channel. We study the influence of the interspacing on the capsule path selection at the channel bifurcation using a three-dimensional immersed boundary–lattice Boltzmann method. Our objectives are to establish a phase diagram and identify the critical interspacing above which hydrodynamic interaction between capsules no longer affects their path selection. We find two main regimes. At low interspacing, strong capsule interaction leads to an unsteady regime for which the capsule path selection follows either a periodic or a disordered state. Above a critical initial interspacing $d_{ct}$, a steady regime is achieved where interaction between capsules is too weak to affect their path selection. The capsules then follow an identical steady trajectory. We find that the dependence of the interspacing $d_{ct}$, normalised by the capsule radius, on the flow split ratio falls onto a universal curve regardless of the flow strength, capsule size and membrane shear elasticity. We also compare the path selection of a capsule train with that of a two-capsule system, and discuss applications of the present results in controlling capsule trains in microfluidic suspension enrichment devices.

Capsules, composed of a liquid core protected by a thin deformable membrane, offer high-potential applications in many fields of industry such as bioengineering. One of their limitations comes from the absence of models of capsule damage and/or rupture when they are subjected to an external flow. To assess when rupture is initiated, we develop a fluid–structure interaction (FSI) numerical model of a capsule in Stokes flow that accounts for potential damage of the capsule membrane. We consider the framework of continuum damage mechanics and model the membrane with an isotropic brittle damage model, in which the membrane damage state depends on the history of loading. The FSI problem is solved by coupling the finite element method, to solve for the membrane deformation, with the boundary integral method, to solve for the inner and outer fluid flows. The model is applied to an initially spherical capsule subjected to a simple shear flow. Damage initiates at a critical value of the capillary number, ratio of the fluid viscous forces to the membrane elastic forces and rupture occurs at a higher capillary number, when it reaches a threshold value. The material parameters introduced in the damage model do not influence the mode of damage but only the values of the critical and threshold capillary numbers. When the capillary number is larger than the critical value, damage develops in the two symmetric central regions containing the vorticity axis. It is indeed in these regions that the internal tensions are the highest on the membrane.

We investigate the rheology of strain-hardening spherical capsules, from the dilute to the concentrated regime under a confined shear flow using three-dimensional numerical simulations. We consider the effect of capillary number, volume fraction and membrane inextensibility on the particle deformation and on the effective suspension viscosity and normal stress differences of the suspension. The suspension displays a shear-thinning behaviour that is a characteristic of soft particles such as emulsion droplets, vesicles, strain-softening capsules and red blood cells. We find that the membrane inextensibility plays a significant role on the rheology and can almost suppress the shear-thinning. For concentrated suspensions a non-monotonic dependence of the normal stress differences on the membrane inextensibility is observed, reflecting a similar behaviour in the particle shape. The effective suspension viscosity, instead, grows and eventually saturates, for very large inextensibilities, approaching the solid particle limit. In essence, our results reveal that strain-hardening capsules share rheological features with both soft and solid particles depending on the ratio of the area dilatation to shear elastic modulus. Furthermore, the suspension viscosity exhibits a universal behaviour for the parameter space defined by the capillary number and the membrane inextensibility, when introducing the particle geometrical changes at the steady state in the definition of the volume fraction.

Cylindrical vesicle and cell membranes under tension can undergo a Rayleigh–Plateau instability leading to break-up. In Part 1 (Graessel et al., J. Fluid Mech., vol. xxx, 2021, Ax) we showed that anisotropic tension, created by active biological processes underneath the cell membrane, can significantly influence this process for a liquid–liquid interface. Here, we study the combined influence of anisotropic tension and membrane elasticity on the Rayleigh–Plateau instability. We analytically derive the dispersion relation for an interface endowed with bending and/or shear elasticity considering explicitly the dynamics of the suspending fluid. We find that the combination of bending elasticity and tension anisotropy leads to three qualitatively different regimes for the Rayleigh–Plateau scenario: (i) the classical regime in which short wavelengths are stable and long wavelengths are unstable, (ii) the suppressed regime in which the system is stable against all perturbation wavelengths and (iii) the restricted regime, in which a stable region at short and another one at long wavelengths are separated by a range of unstable modes centred around the dimensionless wavenumber $kR_0=1$. The width of this unstable range as well as the dominant wavelength of the instability depend on the bending modulus and tension anisotropy. For shear elasticity and area dilatation, on the other hand, only the classical and the suppressed regimes are observed, with the transition between them being independent of the tension anisotropy.

Numerous experiments and theoretical calculations have shown that cylindrical vesicles can undergo a pearling instability similar to the Rayleigh–Plateau instability of a liquid jet when they are subjected to external tension. In a living cell, a Rayleigh–Plateau-like instability could be triggered by internal tension generated in the cell cortex. This mechanism has been suggested to play an essential role in biological processes such as cell morphogenesis. In contrast to the simple, passive and isotropic membrane of vesicles, the cortical tensions generated by biological cells are often strongly anisotropic. Here, we theoretically investigate how this anisotropy affects the Rayleigh–Plateau instability mechanism. We do so in the limit of both low and high Reynolds numbers and accordingly cover cell behaviour under anisotropic cortical tension as well as fast liquid jets with anisotropic surface tension. Combining analytical linear stability analysis with numerical simulations we report a strong influence of the anisotropy on the dominant wavelength of the instability: increasing azimuthal with respect to axial tension leads to destabilisation and to a shorter break-up wavelength. In addition, compared to the classical isotropic Rayleigh–Plateau situation, the range of unstable modes grows or shrinks when the azimuthal tension is higher or lower than the axial tension, respectively. We explore nonlinear effects like an altered break-up time and formation of satellite droplets under anisotropic tension. In Part 2 (Bächer et al. J. Fluid Mech., vol. xxx, 2021, Ax) of this series we continue our analysis by analytically investigating the influence of bending and shear elasticity, usually present in vesicles and cells, on this anisotropic Rayleigh–Plateau instability.

Cross-streamline migration of deformable capsules is studied in three-dimensional curved vessels using a numerical model. Two geometries are chosen: torus vessels to study the effect of constant vessel curvature, and U-shaped vessels to study the effect of change in curvature. A wide range of inertia is considered to include a broad range of applications, from the microcirculation to inertial microfluidics. Vessel curvature and change in curvature are shown to affect capsule migration because of the way they affect the undisturbed flow. In toroidal vessels at negligible inertia, no secondary flow exists and the axial fluid velocity is shifted toward the inner surface of the vessel. In this limit, capsules settle at a location that is away from the vessel centreline, and between the location of the maximum fluid velocity and the innermost surface. Increasing the vessel curvature results in the equilibrium position being increasingly closer to the inner surface. The results suggest the presence of a curvature-induced migration that drives the capsule continuously toward the region of higher streamline curvature. The equilibrium locations are on the symmetry plane of the torus, and are stable equilibria when inertia is negligible. At finite inertia, capsules released on the symmetry plane first settle at an equilibrium located on this plane, which may either be stable or unstable. This equilibrium location depends on both capsule deformability (characterised by capillary number, $Ca$) and the tube Reynolds number $R{e_t}$. It is closer to the inner edge of the torus for smaller $R{e_t}$ and larger $Ca$, and toward the outer edge for larger $R{e_t}$ and smaller $Ca$. This dependence of the equilibria arises due to the secondary flow (or Dean's vortex), which opposes the curvature-induced inward migration. The equilibrium locations on the symmetry plane are unstable if the secondary flow is strong, in which case capsules depart the symmetry plane, and set on to a spiralling motion, eventually settling near the centres of the Dean's vortices. For the U-shaped vessels, the curvature change is shown to create a radial velocity component, even at small inertia, that can be of comparable magnitude to the axial velocity, and that increases with increasing curvature. This radial velocity causes a large and abrupt change in capsule trajectory toward the inner side of the vessel where the curvature increases, but toward the outer side where the curvature decreases.