Point torque representations of ciliary flows

Ciliary flows are generated by a vast array of eukaryotic organisms, from unicellular algae to mammals, and occur in a range of different geometrical configurations. We employ a point torque—or “rotlet”—model to capture the time-averaged ciliary flow above a planar rigid wall. We demonstrate the advantages (i.e., accuracy and computational efficiency) of using this, arguably simpler, approach compared to other singularity-based models in Stokes flows. Then, to model ciliary flows in confined spaces, we extend the point torque solution to a bounded domain between two plane parallel no-slip walls. The flow field is resolved using the method of images and Fourier transforms, and we analyze the role of confinement by comparing the resultant fluid velocity to that of a rotlet near a single wall. Our results suggest that the flow field of a single cilium is not changed significantly by the confinement, even when the distance between the walls is commensurate with the cilium's length.

Figure 1

Schematic illustration of models used for capturing time-averaged flow generated by an individual cilium. (a) Colloidal rotor mimicking the motion of the ciliary or flagellar tip. (b) The rotlet model; (c) single-Stokeslet model; (d) two-Stokeslet model; and (e) four-Stokeslet model. The black arrows in the Stokeslet- and rotlet-based models denote the direction of $q\mathrm{th}$ point force ${\mathit{f}}_q$ and point torque $\mathit{\Omega }$ applied on the fluid domain, respectively, while $d$ and $e$ govern the placement of the singularities with respect to the boundary.

Figure 2

The heat map of the mean relative difference $\langle \text{RD}\rangle$ (a) in $(I_{\mathrm{\Omega }}/{\mathrm{\Omega }}_r,I_d/l_c)$ plane for the rotlet model, (b) in $(I_d/l_c,I_f/f_r)$ plane for the single-Stokeslet model, (c) in $(I_d/l_c,I_f/f_r)$ plane at $e/l_c=0.059$ for the two-Stokeslet model, and (d) in $(I_d/l_c,I_f/f_r)$ plane at $e/l_c=0.054$ for the four-Stokeslet model. The minimum value in the heat map is indicated using the black cross. The map is generated using $n_{\mathrm{\Omega }}=100$, $n_f=100$, $n_d=100$, and $n_e=50$ points on the intervals $I_{\mathrm{\Omega }}/{\mathrm{\Omega }}_r$, $I_f/f_r$, $I_d/l_c$ shown in panels (a)–(d), and $I_e/l_c=[0,0.25]$, respectively.

Figure 3

(a) The mean relative difference $\langle \text{RD}\rangle$ as a function of ${\text{log}}_{10}{n}_d$ for different steady singularity models. (b) The ratio of computational times taken to compute the time-averaged flow field using the steady singularity models indicated with the legend $\left(t_{\text{fit}}\right)$ and the colloidal rotor model $\left(t_{\text{min}}\right)$ as a function of ${\text{log}}_{10}\left(m\right)$, where $m$ is the number of computational grid points.

Figure 4

The relative difference ($\text{RD}$) obtained using (a) the rotlet model, (b) the single-Stokeslet model, (c) the two-Stokeslet model, and (d) the four-Stokeslet model in the window of length $L/l_c=10$ and height $H/l_c=10$ at optimal parameters values listed in Table 1. The red shaded semicircle in panels (a–d) corresponds to the region $\mathrm{\Delta }$ excluded from the calculations of RD.

Figure 5

Streamlines and velocity magnitude associated with the (a) time-averaged colloidal rotor, (b) the rotlet model, (c) the single-Stokeslet model, (d) the two-Stokeslet model, and (e) the four-Stokeslet model, all shown in the window of length $L/l_c=10$ and height $H/l_c=10$ at optimal parameters values listed in Table 1. Streamlines are shown with lines that have arrows to indicate the flow direction.

Figure 6

(a) From Ref. [18]: a sequence of instantaneous shapes of an isolated eukaryotic flagellum held by a micropipette (blue waveform); the resulting trajectories of passive microspheres are shown in black. (b) Trajectories of passive tracers, initially positioned at $x_0/l_c=-1.26$ and $z_0/l_c=0.25,2,3,4,5$, respectively, obtained using the different singularity-based models in a window of length $L/l_c=2.5$ and height $H/l_c=7$. As discussed in Sec. 2d, the model parameters are chosen for comparison with Ref. [18].

Figure 7

Schematic diagram showing the point torque, $\mathrm{\Omega }$, situated between the two parallel walls at $z=0$ and $z=H$, and its image system. The flow field is derived for arbitrary orientation of the point torque.

Figure 8

Heat map of the fluid velocity magnitude $\left|\mathit{u}\right|/U_t$ due to a point torque positioned at a distance $d=0.629l_c$ above the bottom wall. The top and bottom walls are separated by a distance of (a) $H=5l_c$, (b) $H=4l_c$, (c) $H=3l_c$, (d) $H=2l_c$, and (e) $H=1.26l_c$. The point torque is oriented parallel to both walls ($\mathrm{\Omega }=\left|\mathrm{\Omega }\right|{\widehat{\mathbf{e}}}_y$). The remaining parameters are listed in Table 1. Streamlines are shown with lines that have arrows to indicate the flow direction.

Figure 9

Variation of the scaled $x$ component of the velocity vector, i.e., $u_1/U_t$, with scaled $z/l_c$ in the $y=0$ plane for various distances between parallel walls calculated at (a) $x=0.1l_c$, (b) $x=0.5l_c$, and (c) $x=0.75l_c$ from the position of the point torque at $x_r=y_r=0$ and $z_r=d/l_c$ (where $d=0.629l_c$) oriented parallel to walls ($\mathrm{\Omega }=\left|\mathrm{\Omega }\right|{\widehat{\mathbf{e}}}_y$). Model parameters are listed in Table 1.

Figure 10

The same profiles as in Fig. 9, but for the $z$ component of the fluid velocity.

Figure 11

Heat map of the fluid velocity magnitude $\left|\mathit{u}\right|/U_t$ and streamlines due to a point torque oriented perpendicular to walls ($\mathrm{\Omega }=\left|\mathrm{\Omega }\right|{\widehat{\mathbf{e}}}_z$) separated by a distance of $H=1.26l_c$ at (a) $z=d-l_c/4$, (b) $z=d,$ and (c) $z=d+l_c/4$ (where $d=0.629l_c$). The remaining parameters are listed in Table 1.

Figure 12

Variation of the scaled $x$ component of the velocity vector, i.e., $u_1/U_t$, with scaled $z/l_c$ in the $y=0.005l_c$ plane for various distances between parallel walls calculated at (a) $x=0.1l_c$, (b) $x=0.5l_c$, and (c) $x=0.75l_c$ from the position of the point torque at $x_r=y_r=0$ and $z_r=d/l_c$ (where $d=0.629l_c$) oriented perpendicular to walls ($\mathrm{\Omega }=\left|\mathrm{\Omega }\right|{\widehat{\mathbf{e}}}_z$). Model parameters are listed in Table 1.

Figure 13

Heat map of the fluid velocity magnitude $\left|\mathit{u}\right|/U_t$ due to the rotlet above the wall lying along $z=0$. The associated contours compare the velocity magnitudes obtained using Eq. (B4) [black solid line] and Eq. (7) [blue dashed line] using the model parameters listed in Table 1. The displayed contour lines follow the sequence ${\displaystyle \left|\mathit{u}\right|/U_t:=〈1/[1+100(\ell -1)]:\ell \in \mathbb{N}〉}$, with velocity decreasing away from the point torque.