Effective viscosity of a suspension of flagellar-beating microswimmers: Three-dimensional modeling

Micro-organisms usually can swim in their liquid environment by flagellar or ciliary beating. In this numerical work, we analyze the influence of flagellar beating on the orbits of a swimming cell in a shear flow. We also calculate the effect of the flagellar beating on the rheology of a dilute suspension of microswimmers. A three-dimensional model is proposed for Chlamydomonas Reinhardtii swimming with a breaststroke-like beating of two anterior flagella modeled by two counter-rotating fore beads. The active swimmer model reveals unusual angular orbits in a linear shear flow. Namely, the swimmer sustains orientations transiently across the flow. Such behavior is a result of the interplay between shear flow and the swimmer's periodic beating motion of flagella, which exert internal torques on the cell body. This peculiar behavior has some significant consequences on the rheological properties of the suspension. We calculate Einstein's viscosity of the suspension composed of such isolated modeled microswimmers (dilute case) in a shear flow. We use numerical simulations based on a Rotne-Prager-like approximation for hydrodynamic interaction between simplified flagella and the cell body. The results show an increased intrinsic viscosity for active swimmer suspensions in comparison to nonactive ones as well as a shear thinning behavior in accordance with previous experimental measurements [Phys. Rev. Lett. 104, 098102 (2010)].

Figure 1

The active swimmer-model consists of three beads where the left ($L$) and the right ($R$) satellite bead of radius $R^{\prime }$ are connected to the body-bead of radius $R$ and centered in $C$ by frictionless Hookean springs, $S_{sa,sb,m}^L$ or $S_{sa,sb,m}^R$, respectively. $R/R^{\prime }=3$ and the equilibrium length of the springs can be chosen time-dependent or stationary. The unit vector $\widehat{\mathbf{n}}$ defines the mean swimming direction and defines together with the unit vector $\widehat{\mathbf{m}}$ the mean flagellar beating plane.

Figure 2

The velocity field and streamlines around the model swimmer in a fluid at rest (in the laboratory reference frame) and averaged over one period of the flagellar beating. The arrow inside the main sphere indicates the motion of the swimmer. The closed dashed curves represent the orbits of each satellite bead with respect to the central bead. The swimmer moves back and forth during one period but on average it moves forward with a velocity $v\sim 54\mu \mathrm{m}/\mathrm{s}$ obtained for chosen parameters (see the text and Table 1). The flow lines as well as the velocity field compare very well with the experimental results of Drescher et al. [27].

Figure 3

Temporal orientation angles of active (dashed lines) or inactive microswimmers (solid line) in a linear shear flow for three different swimmer orientations. (a) $\widehat{\mathbf{n}}$ (swimming direction) and $\widehat{\mathbf{m}}$ are both parallel to the shear ($xy$) plane. (b) $\widehat{\mathbf{n}}$ parallel to the $xy$ plane and $\widehat{\mathbf{m}}\parallel \widehat{\mathbf{z}}$. (c) $\widehat{\mathbf{m}}\parallel$ to the shear plane and $\widehat{\mathbf{n}}\parallel \widehat{\mathbf{z}}$. In parts (a) and (b), $\theta$ is the angle between $\widehat{\mathbf{n}}$ and $\widehat{\mathbf{x}}$, while in part (c), $\theta$ is the angle between $\widehat{\mathbf{m}}$ and $\widehat{\mathbf{x}}$. The time dependence is plotted in units of the shear-rate-dependent tumbling period $T$, which is different for each curve (see Table 2).

Figure 4

The top part, I, is for the swimming direction perpendicular to the flow direction, i.e., $\widehat{\mathbf{n}}\parallel y$ and $\widehat{\mathbf{m}}\parallel z$. The bottom part, II, is for the swimming direction parallel to the flow direction, i.e., $\widehat{\mathbf{n}}\parallel x$ and $\widehat{\mathbf{m}}\parallel z$. In I and II, part (a) shows the time dependence of the actual length $l\left(t\right)$ of the main springs in units of $R$ (solid curve) and the time-dependent equilibrium length $l_m\left(t\right)$ of the main springs (dashed curve) during one period $T_b$ of the satellite bead motion. Part (b) shows in I and II the dimensionless torque originating from the main springs (dashed curve), the supporting springs (dotted curve), and all springs together (dashed-dotted curve). The solid curve shows the total torque exerted by the satellite beads on the body. Part (c) shows the orientation angle $\theta \left(t\right)$ of the swimmer in the shear flow.

Figure 5

The intrinsic viscosity as a function of the shear rate. The solid line (squares) corresponds to inactive swimmers and the dashed line (circles) corresponds to active swimmers.