Eur. Phys. J. E (2020) 43: 58
https://doi.org/10.1140/epje/i2020-11980-9

Regular Article

Towards an analytical description of active microswimmers in clean and in surfactant-covered drops

¹ Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225, Düsseldorf, Germany
² School of Mechanical Engineering, Purdue University, 47907, West Lafayette, IN, USA
³ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warsaw, Poland
⁴ Department of Bioengineering, Stanford University, 443 Via Ortega, 94305, Stanford, CA, USA
⁵ Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, 19104, Philadelphia, PA, USA
⁶ Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Av. Manuel Montt 367, Providencia, Santiago de Chile, Chile
⁷ Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

^* e-mail: sprenger@thphy.uni-duesseldorf.de
^** e-mail: ider@thphy.uni-duesseldorf.de

Received: 29 May 2020
Accepted: 10 August 2020
Published online: 11 September 2020

Abstract

Geometric confinements are frequently encountered in the biological world and strongly affect the stability, topology, and transport properties of active suspensions in viscous flow. Based on a far-field analytical model, the low-Reynolds-number locomotion of a self-propelled microswimmer moving inside a clean viscous drop or a drop covered with a homogeneously distributed surfactant, is theoretically examined. The interfacial viscous stresses induced by the surfactant are described by the well-established Boussinesq-Scriven constitutive rheological model. Moreover, the active agent is represented by a force dipole and the resulting fluid-mediated hydrodynamic couplings between the swimmer and the confining drop are investigated. We find that the presence of the surfactant significantly alters the dynamics of the encapsulated swimmer by enhancing its reorientation. Exact solutions for the velocity images for the Stokeslet and dipolar flow singularities inside the drop are introduced and expressed in terms of infinite series of harmonic components. Our results offer useful insights into guiding principles for the control of confined active matter systems and support the objective of utilizing synthetic microswimmers to drive drops for targeted drug delivery applications.