# Numerical simulations of elastic capsules with nucleus in shear flow

## TL;DR

This paper presents a numerical method to simulate the deformation of elastic capsules with a nucleus in shear flow, revealing how nuclear properties affect cell deformation and flexibility.

## Contribution

The study introduces a spectral method for simulating elastic capsules with a nucleus, analyzing the impact of various parameters on cell deformation in shear flow.

## Key findings

- Eukaryotic cells deform less than prokaryotic cells under shear.
- Deformation decreases with higher membrane bending stiffness.
- Cells with a nucleus are less flexible passing through narrow capillaries.

## Abstract

The shear-induced deformation of a capsule with a stiff nucleus, a model of eukaryotic cells, is studied numerically. The membrane of the cell and of its nucleus are modelled as a thin and impermeable elastic material obeying a Neo-Hookean constitutive law. The membranes are discretised by a Lagrangian mesh and their governing equations are solved in spectral space using spherical harmonics, while the fluid equations are solved on a staggered grid using a second-order finite differences scheme. The fluid-structure coupling is obtained using an immersed boundary method. The numerical approach is presented and validated for the case of a single capsule in a shear flow. The variations induced by the presence of the nucleus on the cell deformation are investigated when varying the viscosity ratio between the inner and outer fluids, the membrane elasticity and its bending stiffness. The deformation of the eukaryotic cell is smaller than that of the prokaryotic one. The reduction in deformation increases for larger values of the capillary number. The eukaryotic cell remains thicker in its middle part compared to the prokaryotic one, thus making it less flexible to pass through narrow capillaries. For a viscosity ratio of 5, the deformation of the cell is smaller than in the case of uniform viscosity. In addition, for non-zero bending stiffness of the membrane, the deformation decreases and the shape is closer to an ellipsoid. Finally, we compare the results obtained modeling the nucleus as an inner stiffer membrane with those obtained using a rigid particle.