A highly accurate boundary integral equation method for surfactant-laden drops in 3D

Abstract

The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

A 3D boundary integral method for the electrohydrodynamics of surfactant-covered drops

“…where G and T have been defined in (8) and L l (x 0 , x) =x r 3 . This approach results to be cheaper compared to a standard approach as observed already in [35,40] thanks to the orthogonality and symmetry properties of the spherical harmonics [5]. Indeed the systems to solve (for computing the velocity field -eq.…”

“…We can do that in a number of aligned points and then perform a 1D Lagrange interpolation at the original target point. This procedure was firstly proposed by Ying et al [50] and then optimized and tailored to our specific setting in [40].…”

“…The system is evolved in time using the combination of the Midpoint Rule for the evolution of the drop with an Implicit-Explicit (IMEX) second-order Runge-Kutta scheme for the evolution of the surfactant concentration. To make the implicit part of the solver efficient also for large diffusion coefficients, a preconditioner is designed taking advantages of spherical harmonics eigenfunction properties [40]. The overall scheme is adaptive with respect to drop deformation and surfactant concentration; in [34] we showed the efficiency in minimizing the number of BIM computations while still meeting the set tolerance.…”

Scalable simulation of realistic volume fraction red blood cell flows through vascular networks

“…We use the improved algorithm in [28] to precompute the singular integration operator and substantially improve overall complexity. To compute u γ i (X ) for X close to, but not on γ i , we follow the approaches of [28,43], which use a variation of the high-order near-singular evaluation scheme of [58]. Rather than extrapolating the velocity from nearby check points as in Section 3, we use [48] to compute the velocity on surface, upsampled quadrature on γ i to compute the velocity at check points and interpolate the velocity between them to the desired location.…”

Regularized single and double layer integrals in 3D Stokes flow

“…Since the errors decay rapidly away from the surface, the values near the boundary were obtained by interpolating between the values at points on the surface and points that are sufficiently separated from the surface along the surface normals. This algorithm was adapted and optimized by Sorgentone and Tornberg [25] for close interactions of viscous drops with surface tension, where a spherical harmonics expansion was used to parameterize the surface. A quadrature by expansion method was developed by Klöckner et al [16] and Barnett [3] for evaluation of Laplace and Helmholtz potentials, through local expansions.…”