Simulating squirmers with volumetric solvers
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TECHNICAL PAPER
Simulating squirmers with volumetric solvers Stevens Paz1 · Gustavo C. Buscaglia1 Received: 1 May 2020 / Accepted: 3 September 2020 / Published online: 25 September 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract Squirmers are models of a class of microswimmers that self-propel in fluids without significant deformation of their body shape, such as ciliated organisms and phoretic particles. Available techniques for their simulation are based on the boundary element method and do not contemplate nonlinearities such as those arising from the inertia of the fluid or non-Newtonian rheology. This article describes a methodology to simulate squirmers that overcomes these limitations by using volumetric numerical methods, such as finite elements or finite volumes. It deals with interface conditions at the surface of the squirmer that generalize those in the published literature, which are generally restricted to the imposition of slip velocities. The actual procedures to be performed on a fluid solver to implement the proposed methodology are provided, including the treatment of metachronal surface waves. Among the several numerical examples, a two-dimensional simulation is shown of the hydrodynamic interaction of two individuals of Opalina ranarum. Keywords Squirmer model · Numerical microfluidics · Ciliated organisms · Phoretic particles · Fluid–solid interaction · Finite element/volume methods
1 Introduction Microswimmers are organisms or particles with self-driven capacity of locomotion [41]. A large class of microswimmers is that of ciliated organisms, in which cilia act as oars that bend, stretch and rotate generating forces and displacements in the surrounding fluid [10, 39]. A squirmer, initially introduced by Lighthill [44], is a model of a microswimmer consisting of a deformable body that swims via small shape oscillations [14]. It was applied to ciliates by Blake [6] using the concept of ciliary envelope, in which the tips of the numerous cilia are treated as a deformable shell that covers the body. This model has been extensively used in the literature to study energy dissipation and swimming efficiency [40, 45, 50], nutrient uptake [31, 48, 49, 51, 52]
Technical Editor Daniel Onofre de Almeida Cruz, D.Sc. * Stevens Paz stevenspsanchez@gmail.com Gustavo C. Buscaglia gustavo.buscaglia@icmc.usp.br 1
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Paulo, Brazil
and the mechanical effect of the squirmer’s geometry while swimming [63]. Within the ciliary envelope model, the microswimmer has a smooth effective impermeable surface Γ through which it interacts with the surrounding fluid. We restrict here to the important class of tangential squirmers, in which only the tangential motions of the envelope are considered [22]. Since normal-to-the-surface deformations are neglected, tangential squirmers move as rigid bodies that exhibit a tangential slip velocity 𝐮s with respect to the adjacent fluid. To sustain the slippage bet
