PyStokes: phoresis and Stokesian hydrodynamics in Python

PyStokes is a Python library for studying phoretic and hydrodynamic interactions between spherical particles when these interactions can be described by the solutions of, respectively, the Laplace and Stokes equations. The library has been specifically designed for studying these interactions in suspensions of active particles, which are distinguished by their ability to produce flow, and thus motion, in the absence of external forces or torques. Such particles are endowed with a mechanism to produce hydrodynamic flow in a thin interfacial layer, which may be due to the motion of cilia, as in microorganisms (Brennen & Winet, 1977) or osmotic flows of various kinds in response to spontaneously generated gradients of phoretic fields (Ebbens & Howse, 2010). The latter, often called autophoresis, is a generalisation of well- known phoretic phenomena including, inter alia, electrophoresis (electric field), diffusiophoresis (chemical field) and thermophoresis (temperature field) that occur in response to externally imposed gradients of phoretic fields (Anderson, 1989).

Hydrodynamic and phoretic interactions between active particles in a viscous fluid are central to the understanding of their collective dynamics. Under experimentally relevant conditions, the motion of the fluid is governed by the Stokes equation and that of the phoretic field, if one is present, by the Laplace equation. The “activity” appears in these equations as boundary conditions on the particle surfaces that prescribe the slip velocity in the Stokes equation and flux of the phoretic field in the Laplace equation. The slip velocity and the 

phoretic flux are related by a linear constitutive law that can be derived from a detailed analysis of the boundary layer physics (Anderson, 1989). The Stokes and Laplace equations are coupled by this linear constitutive law only at the particle boundaries. The linearity of the governing equations and the coupling boundary conditions allows for a formally exact solution of the problem of determining the force per unit area on the particle surfaces. This formally exact solution can be approximated to any desired degree of accuracy by a truncated series expansion in a complete basis of functions on the particle boundaries. This, in turn, leads to an efficient and accurate numerical method for computing hydrodynamic and phoretic interactions between active particles.

In addition to the joint computation of phoretic and hydrodynamic interactions, the PyStokes library can be used to compute the hydrodynamically interacting motion of squirming particles where the slip is specified independently of a phoretic field, or the dynamics of passive sus- pensions where the slip vanishes and forces and torques are prescribed. The PyStokes library can also compute hydrodynamically correlated Brownian motion, and thus, allows the study of the interplay between passive, active, and Brownian contributions to motion.

The PyStokes library has been used to model suspensions of microorganisms (Bolitho, Singh, & Adhikari, 2020; Singh & Adhikari, 2016), synthetic autophoretic particles (Singh & Adhikari, 2016; R. Singh et al., 2019) and self-propelling droplets (Thutupalli, Geyer, Singh, Adhikari, & Stone, 2018). Our software implementation uses a polyglot programming approach that combines the readability of Python with the speed of Cython and retains the advantages of a high-level, dynamically typed, interpreted language without sacrificing performance.