Stability of dancing Volvox

Biflagellate algal cells of the genus Volvox form spherical colonies that propel themselves, vertically upwards in still fluid, by the coordinated beating of thousands of flagella, that also cause the colonies to rotate about their vertical axes. When they are swimming in a chamber of finite depth, pairs (or more) of Volvox carteri colonies were observed by Drescher et al. ( Phys. Rev. Lett. , vol. 102, 2009, 168101) to exhibit hydrodynamic bound states when they are close to a rigid horizontal boundary. When the boundary is above, the colonies are attracted to each other and orbit around each other in a 'waltz'; when the boundary is below they perform more complex 'minuet' motions. These dances are simulated in the present paper, using a novel 'spherical squirmer' model of a colony in which, instead of a time-independent but <span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" &#x03B8; "> θ -dependent tangential velocity being imposed on the spherical surface (radius <span id="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" a "> a ; <span id="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" &#x03B8; "> θ is the polar angle), a time-independent and uniform tangential shear stress is applied to the fluid on a sphere of radius <span id="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" ( 1 + &#x03F5; ) a , &#x03F5; &#x226A; 1 "> ( 1 + ϵ ) a , ϵ ≪ 1 , where <span id="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" &#x03F5; a "> ϵ a represents the length of the flagella. The fluid must satisfy the no-slip condition on the sphere at radius <span id="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" a "> a . In addition to the shear stress, the motions depend on two dimensionless parameters that describe the effect of gravity on a colony: <span id="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" F g "> F g , proportional to the ratio of the sedimentation speed of a non-swimming colony to its swimming speed, and <span id="MathJax-Element-8-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" G b h "> G b h , that represents the fact that colonies are bottom heavy; <span id="MathJax-Element-9-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" G b h "> G b h is the ratio of the time scale to swim a distance equal to the radius, to the time scale for gravitational reorientation of the colony's axis to the vertical when it is disturbed. In addition to reproducing both of the dancing modes, the simulations are able to determine values of <span id="MathJax-Element-10-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" F g "> F g and <span id="MathJax-Element-11-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 18px; vertical-align: baseline; display: inline-table; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0" role="presentation" data-mathml=" G b h "> G b h for which they are stable (or not); there is reasonable agreement with the experiments. A far-field model for the minuet motions is also shown to have qualitative agreement, but does not describe some features that are reproduced in the full simulations.