# Simulation of deformable objects transported in fluid flow

Abstract: Deformable particles suspended in a viscous fluid can be found in many industrial and biological applications. In this thesis, two different numerical tools have been developed to simulate suspensions of capsules, thin membranes enclosing a second fluid and a rigid nucleus so to work as model for ”Eukaryotic” cells, and flexible slender bodies known as filaments/fibres. Both tools use a semi-implicit fluid flow solver with different approaches for the deformable structure. The capsule membrane is modelled as a thin hyperelastic material and the elasticity equations are solved with an accurate spectral representation of the capsule shape as a truncated number of spherical harmonics. The filaments are considered as one dimensional inextensible slender bodies obeying Euler-Bernoulli beam equations which is solved by a two-step method using finite difference discretisation. The immersed boundary method is exploited to couple the fluid and solid motion using different versions for the two different objects considered. The nucleus inside the capsules is modelled either as a second stiffer capsule or as a rigid particle. In order to avoid membrane-membrane, membrane-wall and membrane-nucleus overlapping, a short range repulsive force is implemented in terms of a potential function of the distance between the approaching objects. For the short range interactions between the filaments, both lubrication correction and collision forces are considered and it is found that the inclusion of the lubrication correction has significant effect on the rheology in shear flow. Both codes are validated against the numerical and experimental data in the literature. We study the capsule behaviour in a simple shear flow created by with two walls moving in opposite directions. The membrane obeys the Neo-Hookean constitutive equations and, in the simulations with a rigid nucleus, its radius is fixed to half the capsule initial radius. The filaments, on the other hand, are studied in 4 different flow configurations: shear flow, channel flow, settling in quiescent fluid and homogeneous isotropic turbulence. The results indicate that for single capsule, the nucleus reduces the membrane deformation significantly and changes the deformed shaped when there is negligible bending resistance of the membrane. The rheological properties of nucleated capsule suspensions result from the competition between the capsule deformation and their orientation angle and similarly to the case of single capsules, the nucleus reduces the mean deformation. By increasing the capsule volume fraction, the relative viscosity increases and capsules become more oriented in the mean flow direction. Filament suspensions in shear flow exhibit shear thinning behaviour with respect to deformability; inertia has a significant effect on the rheological properties of the suspensions as documented here. For the case of settling fibres, we document the formation of columnar structures with higher settling velocity known as streamers, which are more pronounced at higher volume fractions and for flexible fibres. For a single filament in homogeneous isotropic turbulence, two distinct regimes for the filament motion are identified with a sharp transition from one to another at a critical bending stiffness. In turbulent channel flow, we demonstrate how finite-size filaments cause considerable drag reduction, of the order of 30% for volume fractions of the order of 1.5%, and that the main averaged quantities are almost independent of the filament flexibility for the bending rigidities studied here.

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