We computationally study the motion of an initially spherical capsule flowing through a straight channel with an orthogonal lateral branch, using a three-dimensional immersed-boundary lattice-Boltzmann method. The capsule is enclosed by a strain-hardening membrane and contains an internal fluid of the same viscosity as the fluid in which it is suspended. Our primary focus is to study the influence of the geometry of the side branch on the capsule path selection. Specifically, we consider the case where the side branch cross-section is half that of the straight channel and study various bifurcation configurations, where the branch is rectangular or square, centred or not on the straight channel axis. The capsule is initially centred on the axis of the straight channel. We impose the flow rate split ratio between the two downstream branches of the bifurcation. We summarise the results obtained for different capsule-to-channel size ratios, flow Reynolds number $Re$ (based on the parent channel size and average flow speed) and capsule mechanical deformability (as measured by the capillary number) in phase diagrams giving the critical flow rate split ratio above which the capsule flows into the side branch. A major finding is that, at equal flow rate split between the two downstream branches, the capsule will enter a branch which is narrow in the spanwise direction, but will not enter a branch which is narrow in the flow direction. For $Re\leqslant 5$, this novel intriguing phenomenon primarily results from the background flow, which is strongly influenced by the side branch geometry. For higher values of $Re$, the capsule relative size and deformability also play specific roles in the path selection. The capsule trajectory does not always obey the classical Fung’s bifurcation law, which stipulates that a particle (in Fung’s case, a red blood cell) enters the bifurcation branch with the highest flow rate. We also consider the same branched channels operating under constant pressure drop conditions and show that such systems are difficult to control due to the transient additional pressure drop caused by the capsule. The present results obtained for dilute systems open new perspectives on the design of microfluidic systems, with optimal channel geometries and flow conditions to enrich cell and particle suspensions.

The motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear softening constitutive law. The three-dimensional fluid–structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyse experimental data on the deformation of artificial capsules in such channels.

The objective of this study is to investigate the motion of an ellipsoidal capsule in a simple shear flow when its revolution axis is initially placed off the shear plane. We consider prolate capsules with an aspect ratio of two or three enclosed by a membrane, which is either strain-hardening or strain-softening. We seek the equilibrium motion of the capsule as we increase the capillary number $\mathit{Ca}$, which measures the ratio between the viscous and elastic forces. The three-dimensional fluid–structure interaction problem is solved numerically by coupling a boundary integral method (for the internal and external flows) with a finite element method (for the wall deformation). For any initial inclination with the flow vorticity axis, a given capsule converges towards a unique equilibrium configuration which depends on $\mathit{Ca}$. At low capillary number, the stable equilibrium motion is the rolling regime: the capsule aligns its long axis with the vorticity axis, while the membrane tank-treads. As $\mathit{Ca}$ increases, the capsule takes a complex wobbling motion at equilibrium, precessing around the vorticity axis. As $\mathit{Ca}$ is further increased, the capsule long axis oscillates about the shear plane, while the membrane rotates around a capsule cross-section that also oscillates over time (oscillating–swinging regime). The amplitude of the oscillations about the shear plane decreases as $\mathit{Ca}$ increases and the capsule finally takes a swinging motion in the shear plane. It is found that the transitions from rolling to wobbling and from wobbling to oscillating–swinging depend on the mean energy stored in the membrane.

The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite-element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law and ratio of shear to area dilatation moduli) on the capsule behaviour is investigated. Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary numbers, the capsule tumbles, behaving mostly like a solid particle. At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.

The motion and deformation of a spherical elastic capsule freely suspended in a simple shear flow is studied numerically, focusing on the effect of the internal-to-external viscosity ratio. The three-dimensional fluid–structure interactions are modelled coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). For low viscosity ratios, the internal viscosity affect the capsule deformation. Conversely, for large viscosity ratios, the slowing effect of the internal motion lowers the overall capsule deformation; the deformation is asymptotically independent of the flow strength and membrane behaviour. An important result is that increasing the internal viscosity leads to membrane compression and possibly buckling. Above a critical value of the viscosity ratio, compression zones are found on the capsule membrane for all flow strengths. This shows that very viscous capsules tend to buckle easily.

We computationally study the transient motion of an initially spherical capsule flowing through a right-angled tube bifurcation, composed of tubes having the same diameter. The capsule motion and deformation is simulated using a three-dimensional immersed-boundary lattice Boltzmann method. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane following the Skalak’s law (Skalak et al., Biophys. J., vol. 13(3), 1973, pp. 245–264). The fluids inside and outside the capsule are assumed to have identical viscosity and density. We mainly focus on path selection of the capsule at the bifurcation as a function of the parameters of the problem: the flow split ratio, the background flow Reynolds number $Re$, the capsule-to-tube size ratio $a/R$ and the capillary number $Ca$, which compares the viscous fluid force acting on the capsule to the membrane elastic force. For fixed physical properties of the capsule and of the tube flow, the ratio $Ca/Re$ is constant. Two size ratios are considered: $a/R=0.2$ and 0.4. At low $Re$, the capsule favours the branch which receives most flow. Inertia significantly affects the background flow in the branched tube. As a consequence, at equal flow split, a capsule tends to flow straight into the main branch as $Re$ is increased. Under significant inertial effects, the capsule can flow into the downstream main tube even when it receives much less flow than the side branch. Increasing $Ca$ promotes cross-stream migration of the capsule towards the side branch. The results are summarized in a phase diagram, showing the critical flow split ratio for which the capsule flows into the side branch as a function of size ratio, $Re$ and $Ca/Re$. We also provide a simplified model of the path selection of a slightly deformed capsule and explore its limits of validity. We finally discuss the experimental feasibility of the flow system and its applicability to capsule sorting.

The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio $a/b$, viscosity ratio ${\it\lambda}$ between the internal and external fluids and the capillary number $Ca$ which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For $a/b=0.5$, only four regimes are stable when ${\it\lambda}=1$: tumbling and swinging in the low and medium $Ca$ range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When ${\it\lambda}$ is increased, the tumbling-to-swinging transition occurs for higher $Ca$; the wobbling regime takes place at lower $Ca$ values and within a narrower $Ca$ range. For ${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the $Ca$–${\it\lambda}$ phase diagram is qualitatively similar for higher aspect ratio. Only the $Ca$-range over which wobbling is stable increases with $a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.