Irreducible Representations of Oscillatory and Swirling Flows in Active Soft Matter

Recent experiments imaging fluid flow around swimming microorganisms have revealed complex time-dependent velocity fields that differ qualitatively from the stresslet flow commonly employed in theoretical descriptions of active matter. Here we obtain the most general flow around a finite sized active particle by expanding the surface stress in irreducible Cartesian tensors. This expansion, whose first term is the stresslet, must include, respectively, third-rank polar and axial tensors to minimally capture crucial features of the active oscillatory flow around translating Chlamydomonas and the active swirling flow around rotating Volvox. The representation provides explicit expressions for the irreducible symmetric, antisymmetric, and isotropic parts of the continuum active stress. Antisymmetric active stresses do not conserve orbital angular momentum and our work thus shows that spin angular momentum is necessary to restore angular momentum conservation in continuum hydrodynamic descriptions of active soft matter.

Figure 1

Cross sections of complex time-dependent flows around a spherical active particle generated by a simple linear combination of $\mathbf{S}(t)$, $\mathbf{d}(t)$, and $\mathbf{\Gamma }(t)$ that capture essential features of flow fields around swimming Chlamydomonas [8]. Streamlines show the direction while the background color represents the natural logarithm of the strength of the flow. The red dot indicates the approximate position of the stagnation point. The inset shows the overall (blue solid line) and instantaneous (red dot) variation of $d_0(t)$ against $S_0(t)$.

Figure 2

Power dissipation and swimming efficiency of Chlamydomonas computed using a simple linear combination of $\mathbf{S}(t)$, $\mathbf{d}(t)$, and $\mathbf{\Gamma }(t)$. (a) Variation of $d_0(t)$ against $S_0(t)$, the former estimated from particle image velocimetry data of a swimming Chlamydomonas [8]. (b) Time variation of dissipated power $\stackrel{.}{W}(t)$. (c) The relative efficiency is maximum near the middle and end of the cycle. (d) Variation of the power $\stackrel{.}{W}$ against translational velocity $v_{\mathrm{CM}}$.