We present a fluid-structure adaptive solver designed to compute the dynamics of microcapsules flowing in confined geometries.

The aim of our work is to study the motion of deformable capsules in microfluidic devices. We simulate a capsule of radius a flowing under an average inlet flow velocity U in a square cross-section channel of half-width l. Its motion and deformation are governed by the following dimensionless numbers: the aspect ratio a/l, the Reynolds number Re=ρUl/µ and the viscoelastic capillary number Ca=µU/Gs, where ρ and µ are the density and viscosity of the suspending fluid, and Gs is the surface shear modulus of the capsule membrane.

The fluid-structure solver is based on two independent existing fluid and solid solvers. The solid solver, which is part of Caps3D [1, 2] solves for the capsule membrane equilibrium using a membrane Finite Element method over a Lagrangian grid. The fluid solver is Basilisk [3], a Finite Volume open source solver for computing the Navier-Stokes equations over a Eulerian grid. The two solvers are coupled using an Immersed Boundary Method [4].

At each time step Caps3D uses the position of the Lagrangian capsule nodes to compute the membrane surface elastic forces, which are then converted into volume forces by the Immersed Boundary Method; in a second step Basilisk uses the volume forces as the source term for the fluid resolution. A new fluid velocity field is obtained and used to update the position of the capsule nodes. The integrated adaptive quadtree mesh tool of Basilisk is used along with specific immersed boundary filters that we have specially developed to account for confined particle flows. They are crucial for the computation of the thin layer between the capsule and the fluid field boundary.

We propose a validation and a convergence study for different grid mesh sizes Δx and integration time steps Δt. The aspect ratio is set to a/l = 0.85 and the capillary number to Ca=0.05 and 0.1. The simulations are run both under Stokes flow conditions and solving for the Navier-Stokes equations at Re=1.

The steady-state capsule shape and velocity are compared to existing numerical results [5]. A good consistency has been found. For simulations up to t=10 in advective time units, we find that the volume variation of the capsule is less than 0.1%, which is much better than results found by recent studies (e.g. [6]).

The novelty is that the fluid-structure code can resolve the flow of capsules of aspect ratios greater than 1 at non-zero Reynolds numbers, which are required to study capsules in microsystems. The coupling between the fluid dynamics and capsule deformation is also studied considering the streamline changes under each capillary number.

Such simulations are a useful complement to experimental measurements, as they provide local field quantities (fluid pressure and velocity, membrane tensions, ...), which cannot easily be evaluated experimentally, as well as information on topographic changes in the streamlines. They thus provide useful additional information for the study of capsules in microsystems.

References:

[1] J. Walter, PhD thesis, Université de Technologie de Compiègne, 2009.
[2] J. Walter, A.-V. Salsac, D. Barthès-Biesel and P. Le Tallec, Int. J. Numer. Meth. Engng. 2010, 83:829–850., 2010
[3] S. Popinet, SIAM conference on Parallel Processing for Scientific Computing, April 12-15 2016, Paris, 2016.
[4] Charles S. Peskin, Acta Numerica 11:479-517, January 2002.
[5] X.-Q. Hu, B. Sévénié, A.-V. Salsac, E. Leclerc, and D. Barthès-Biesel, Phys. Rev. E 87, 063008 – 13 June 2013
[6] R. M. Carroll and N. R. Gupta, International Journal of Multiphase Flows, 87:114-123., 2016