An improved squirmer model for Volvox locomotion

We recently used the Lighthill-Blake envelope (or 'squirmer') model for ciliary propulsion to predict the mean swimming speed $U$ and angular velocity $\Omega$ of spherical $Volvox$ colonies [1]. Input was the measured flagellar beating patterns (a symplectic metachronal wave) of $Volvox~carteri$ colonies with different radii $a$ [2]. The predictions were compared with independent measurements of $U$ and $\Omega$ as functions of $a$, and proved to be substantial underestimates of both $U$ and $\Omega$, by about $80\%$, probably because the envelope model ignores the fact that, during the recovery stroke, most of a flagellum is much closer to the no-slip colony surface than during the power stroke. In consequence $U$ and $\Omega$ will be proportional to the beating amplitude $\epsilon$ not to $\epsilon^2$ as in the Lighthill-Blake theory. A new model is proposed, based on a shear-stress (not velocity) distribution (cf [4]) that is applied at a smaller radius in the recovery stroke than in the power stroke. Agreement with experiment is greatly improved\\ \\$[1]$ Pedley et al, JFM 798:165,2016. [2] Brumley et al, PRL 109:268102,2012. [3] Drescher et al,PRL 102:168101,2009. [4]Short et al, PNAS 103:8315,2006.