Elastic capsules in shear flow: Analytical solutions for constant and time-dependent shear rates

TL;DR

This paper derives analytical solutions for microcapsule dynamics in shear flow, revealing how time-dependent shear can induce tumbling motion and resonance effects, advancing understanding of capsule behavior under varying flow conditions.

Contribution

It provides the first analytical expressions for capsule dynamics in time-dependent shear flow using matched asymptotic expansions, extending previous numerical studies.

Findings

Analytical formulas for mean tumbling rate in variable shear flow.

Resonance peaks in capsule response depend on modulation frequency.

Tumbling can be induced by shear modulation even in swinging regimes.

Abstract

We investigate the dynamics of microcapsules in linear shear flow within a reduced model with two degrees of freedom. In previous work for steady shear flow, the dynamic phases of this model, i.e. swinging, tumbling and intermittent behaviour, have been identified using numerical methods. In this paper, we integrate the equations of motion in the quasi-spherical limit analytically for time-constant and time-dependent shear flow using matched asymptotic expansions. Using this method, we find analytical expressions for the mean tumbling rate in general time-dependent shear flow. The capsule dynamics is studied in more detail when the inverse shear rate is harmonically modulated around a constant mean value for which a dynamic phase diagram is constructed. By a judicious choice of both modulation frequency and phase, tumbling motion can be induced even if the mean shear rate corresponds to the swinging regime. We derive expressions for the amplitude and width of the resonance peaks as a function of the modulation frequency.