Partial drift volume due to a self-propelled swimmer

We assess the ability of a self-propelled swimmer to displace a volume of fluid that is large compared to its own volume via the mechanism of partial drift. The swimmer performs rectilinear locomotion in an incompressible, unbounded Newtonian fluid. The partial drift volume $D$ is the volume of fluid enclosed between the initial and final profiles of an initially flat circular disk of marked fluid elements; the disk is initially aligned perpendicular to the direction of locomotion and subsequently distorted due to the passage of the swimmer, which travels a finite distance. To focus on the possibility of large-scale drift, we model the swimmer simply as a force dipole aligned with the swimming direction. At zero Reynolds number ($\mathrm{Re}=0$), we demonstrate that $D$ grows without limit as the radius of the marked fluid disk $h$ is made large, indicating that a swimmer at $\mathrm{Re}=0$ can generate a partial drift volume much larger than its own volume. Next, we consider a steady swimmer at small $\mathrm{Re}$, which is modeled as the force-dipole solution to Oseen's equation. Here, we find that $D$ no longer diverges with $h$, which is due to inertial screening of viscous forces, and is effectively proportional to the magnitude of the force dipole exerted by the swimmer. The validity of this result is extended to $\mathrm{Re}\ge O\left(1\right)$—the realm of intermediate-$\mathrm{Re}$ swimmers such as copepods—by taking advantage of the fact that, in this case, the flow is also described by Oseen's equations at distances much larger than the characteristic linear dimension of the swimmer. Next, we utilize an integral momentum balance to demonstrate that our analysis for a steady inertial swimmer also holds, in a time-averaged sense, for an unsteady swimmer that does not experience a net acceleration over a stroke cycle. Finally, we use experimental data to estimate $D$ for a few real swimmers. Interestingly, we find that $D$ depends heavily on the kinematics of swimming, and, in certain cases, $D$ can be significantly greater than the volume of the swimmer at $\mathrm{Re}\ge O\left(1\right)$. Our work also highlights that $D$ due to a self-propelled body is fundamentally different than that due to a body towed by an external force. In particular, predictions of $D$ in the latter case cannot be utilized to estimate $D$ for a self-propelled swimmer.

Figure 1

The drift volume induced by the swimmer is the (shaded) region between the initial and final profiles of a marked plane of fluid.

Figure 2

Pusher (a) and puller (b) type swimmers translate at velocity $\mathit{U}$. Pullers ($b<0$) are propelled from the front, and hence the center of thrust is in front of the center of drag. Pushers ($b>0$) are propelled from the rear, where the opposite is true. Note that $F_{\text{thrust}}$ and $F_{\text{drag}}$, which are equal and opposite, represent the force exerted by the swimmer on the fluid, and hence $F_{\text{thrust}}$ is shown to be directed opposite to the swimming direction.

Figure 3

The lines correspond to streamlines of the far-field flow disturbance (in the laboratory frame), generated by a steady, pusher-type ($b>0$) swimmer translating at unit speed ($U=1$) in the ${\mathit{e}}_x$ direction at $\mathrm{Re}\ll 1$. Arrows indicate the flow direction. Dashed lines indicate positive isocontours of ${\psi }_{\mathit{v}}$, while solid lines indicate negative isocontours. (a) The flow pattern at distances $r>O(1/\mathrm{Re})$ from the swimmer, in the Oseen region, is affected appreciably by inertia. The initial position of the dyed sheet of fluid is illustrated to the right of the swimmer. A viscous wake of height $w$ grows parabolically with distance downstream of the swimmer. (b) At distances $r\ll 1/\mathrm{Re}$, in the Stokes region, inertial effects are relatively weak and the flow pattern approaches the $\mathrm{Re}=0$ stresslet. Equivalent plots for a puller ($b<0$) may be visualized from the above illustration by reversing the streamline directions and the sign of ${\psi }_{\mathit{v}}$ (but not the swimming direction).

Figure 4

The (partial) drift volume $D$ is normalized by $bh$ and plotted versus $\tau$. Here, we have taken ${\tau }_i\to -\infty$ (and $\tau ={\tau }_f$). The dashed line represents ${\mathrm{Re}}_h=0$ (fluid inertia is completely neglected), while the solid lines represent ${\mathrm{Re}}_h=\{0.1,1,10,100\}$ as the curves respectively descend. The drift volume for arbitrary values of ${\tau }_i$ and ${\tau }_f$ may be obtained by subtracting$D\left({\tau }_i\right)/bh$ from $D\left({\tau }_f\right)/bh$ on a chosen ${\mathrm{Re}}_h$ curve.

Figure 5

A control volume $V\left(t\right)$ surrounds an unsteady, self-propelled swimmer. The reference frame translates at the constant, time-averaged velocity $\overline{\mathit{U}}$ of the swimmer over a periodic swimming stroke cycle. In this frame, the swimmer periodically fluctuates about a central position at velocity ${\mathit{U}}^{\prime }\left(t\right)$, and exerts the instantaneous force $\mathit{F}\left(t\right)$ on $V$ as it accelerates and decelerates. The outer bounding surface $A$ is fixed and assumed to be far enough from the swimmer such that the fluid velocity beyond $A$ is only disturbed slightly from free stream ($\mathit{v}\ll \overline{\mathit{U}}$). The inner boundary $S\left(t\right)$ coincides with the moving surface of the swimmer's body.