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.

Close

PrevNext

View figure in article

• 
• 
• 
• 
• 

• Locomotion
• Microfluidics
• Swimming