Effects of surface proximity and force orientation on the feeding flows of microorganisms on solid surfaces

Many aquatic microorganisms are attached to solid surfaces while creating feeding flows that bring prey particles to them. To explore the effects of surface proximity and orientation of the flow-generating force, we analyze the low-Reynolds-number flow due to a point force above a plane no-slip surface. The presence of the surface reduces the feeding flow relative to the unbounded situation. We show that the reduction of the flow rate through a circular clearance disk perpendicular to the force and centered at its position is twice as large when the force is perpendicular to the surface as when it is parallel. When the force is perpendicular to the surface, the flow forms a toroidal eddy with closed streamlines, and the resulting flow recirculation may lead to refiltration of water that has already been cleared for prey. We prove that due to the nature of the far-field flow, the shortest recirculation time along a streamline through a circular clearance disk is inversely proportional to the flow rate to the power four. Finally, we discuss the biological advantages and disadvantages of perpendicular and parallel force orientation and the effects of prey diffusion and ambient flow, and we argue that recirculation is irrelevant in the typical perpendicular feeding flow since the recirculation time is long compared to the biologically relevant timescales.

Figure 1

Examples of suspension feeding microorganisms on solid surfaces. (a) The choanoflagellate S. rosetta (scanning electron micrograph courtesy of Mark J. Dayel) [13], (b) the ciliate V. convallaria (image courtesy of Rachel E. Pepper), and (c) the ciliate E. vannus sitting on a lump of organic material. The equivalent spherical radius of the cell of S. rosetta is typically $2\mu \mathrm{m}$. The blue arrows indicate the flow directions. The choanoflagellate S. rosetta and the ciliate V. convallaria generate feeding flows away from and toward the surface, respectively, and the ciliate E. vannus generates a feeding flow roughly parallel to the surface.

Figure 2

Streamlines and clearance zones in the point force model. The point force (red arrow) is placed at the height $h$ above the plane no-slip surface, and it is oriented respectively (a) perpendicular and (b) parallel to the surface. Both plots show a circular clearance disk (green) that is centered at the position of the point force and oriented perpendicular to it.

Figure 3

Clearance rates for different force orientations. (a) The complete expression (10) for the perpendicular case (black solid line), the approximation (11) for the perpendicular case (red dots), the complete expression for the parallel case evaluated numerically (orange solid line), and the approximation (13) for the parallel case (blue dots). The clearance rate is equal to $90\%$ of the clearance rate for the Stokeslet in the unbounded domain when $h/a=3.8$ in the parallel case (gray dashed line) and $h/a=7.5$ in the perpendicular case (gray dotted line). (b) The clearance rate as function of the angle between the force and the inward surface normal when $h/a=3.8$ (black dashed line) and (magenta dots) and $h/a=7.5$ (black dotted line) and (green dots). The complete expressions are shown as lines and the approximations (14) as dots.

Figure 4

Optimal clearance zones and clearance rates when the force is parallel to the surface. (a) The boundaries of the optimal clearance disks are contour lines of $v_x$ (blue solid lines), and the position of the force is indicated by the red dot. (b) The flow rate, $Q$, for the optimal clearance disks as function of the effective radius of the clearance disk, $b$. The flow rate is normalized by the total flow rate defined in Eq. (15). The contour lines in (a) correspond to the points (blue dots) in (b), and the numbers indicate the values of $k=v_x/v_0$, where $v_0$ is defined in Eq. (6).

Figure 5

Recirculation times along the closed streamlines when the force is perpendicular to the wall. (a) The numerically determined recirculation times are shown for selected streamlines in units of the timescale $T_0$. (b) The recirculation time, $T$, as function of the stream function, $\mathrm{\Psi }$, normalized by $T_0$ and ${\mathrm{\Psi }}_0$, respectively. Numerical results for the complete flow solution (blue solid line) and the analytically determined power law in Eq. (23) (orange dots). The inset shows the numerical results for the complete flow solution (blue solid line) and the analytic approximation in Eq. (24) (red dots).

Figure 6

The flow rate, $Q$, through a circular clearance disk and the shortest recirculation time, $T$, along a streamline through the disk as functions of the radius of the disk, $a$, when the force is perpendicular to the surface. The clearance disk is positioned at the height $d=z_{\max }$ above the surface corresponding to the height of the center of the toroidal eddy. (a) The flow rate in Eq. (8) is normalized by the maximum flow rate, $Q_{\max }$, determined in Eq. (9), and (b) the numerically determined recirculation time is normalized by the timescale $T_0$.