Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a numerical method for two-fluid systems subject to surface tension and depletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressible two-phase Navier-Stokes equations, and uses a mass-conserving level set method to capture the fluid interface. The four novel ingredients proposed here are, firstly, an interface-correction level set (ICLS) method; global mass conservation is achieved by performing an additional advection near the interface, with a correction velocity obtained by locally solving an algebraic equation, which is easy to implement in both 2D and 3D. Secondly, we report a second-order accurate geometric estimation of the curvature at the interface and, thirdly, the combination of the ghost fluid method with the fast pressurecorrection approach enabling an accurate and fast computation even for large density contrasts. Finally, we derive a hydrodynamic model for the interaction forces induced by depletion of surfactant micelles and combine it with a multiple level set approach to study short-range interactions among droplets in the presence of attracting forces.
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
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