Reduced-order model for inertial locomotion of a slender swimmer

The inertial locomotion of an elongated model swimmer in a Newtonian fluid is quantified, wherein self-propulsion is achieved via steady tangential surface treadmilling. The swimmer has a length $2l$ and a circular cross section of longitudinal profile $aR\left(z\right)$, where $a$ is the characteristic width of the cross section, $R\left(z\right)$ is a dimensionless shape function, and $z$ is a dimensionless coordinate, normalized by $l$, along the centerline of the body. It is assumed that the swimmer is slender, $\epsilon =a/l\ll 1$. Hence, we utilize slender-body theory to analyze the Navier-Stokes equations that describe the flow around the swimmer. Therefrom, we compute an asymptotic approximation to the swimming speed, $U$, as $U/u_s=1-\beta [V\left(\mathrm{Re}\right)-\frac{1}{2}{\int }_{-1}^{1}zlnR\left(z\right)dz]/ln(1/\epsilon )+O[1/{ln}^2(1/\epsilon )]$, where $u_s$ is the characteristic speed of the surface treadmilling, $\mathrm{Re}$ is the Reynolds number based on the body length, and $\beta$ is a dimensionless parameter that differentiates between “pusher” (propelled from the rear, $\beta <0$) and “puller” (propelled from the front, $\beta >0$) -type swimmers. The function $V\left(\mathrm{Re}\right)$ increases monotonically with increasing $\mathrm{Re}$; hence, fluid inertia causes an increase (decrease) in the swimming speed of a pusher (puller). Next, we demonstrate that the power expenditure of the swimmer increases monotonically with increasing $\mathrm{Re}$. Further, the power expenditures of a puller and pusher with the same value of $\left|\beta \right|$ are equal. Therefore, pushers are superior in inertial locomotion as compared to pullers, in that they achieve a faster swimming speed for the same power expended. Finally, it is demonstrated that the flow structure predicted from our reduced-order model is consistent with that from direct numerical simulation of swimmers at intermediate $\mathrm{Re}$.

Figure 1

Illustration of a slender squirmer-type swimmer in a comoving frame. The various symbols in this sketch are described in the main text. The curved arrows at the body surface depict the surface treadmilling stroke around a puller $(\beta >0)$ and a pusher $(\beta <0)$. For a puller (pusher), the surface velocity ${\mathit{u}}_s$ is greatest at the front (rear) as thrust is generated from the front (rear). The dots near the ends of the body signify that ${\mathit{u}}_s$ vanishes at the ends.

Figure 2

Plots of $J_1$ (a) and $J_2$ (b) versus $z$ for various Re. The functions $J_1\left(z\right)$ and $J_2\left(z\right)$ are defined in (3.22) and (3.23), respectively, and appear in the $O[1/{ln}^2(1/\epsilon )]$ contribution to the force per unit length $f_2$ (3.21).

Figure 3

Plot of $V$ versus Re. The function $V\left(\mathrm{Re}\right)$ is defined in (3.25) and represents the inertial contribution to the $O[1/ln(1/\epsilon \left)\right]$ swimming speed $U_2$.

Figure 4

Distribution of inertial thrust per unit length $T_I$ at various Re. The thin vertical line at $z=0$ is to help the reader visualize the increasing left-right asymmetry in $T_I$ with increasing Re.

Figure 5

Mechanism of inertial thrust generation for a pusher. (a) At $\mathrm{Re}=0$ the pusher contribution to the stroke, $u_{sz}+1=-\beta z$ (shown by curved arrows on the body surface), results in a fore-aft antisymmetric vorticity distribution, as depicted by the oppositely signed, counter-rotating vortices. In particular, the magnitude of the vorticity is zero at the midpoint of the body, as indicated by the dashed vertical line where $\omega =0$. (b) At $\mathrm{Re}>0$ vorticity is advected past the swimmer by inertial forces, such that the fore-aft antisymmetry is lost. Hence, as explained in the main text, an inertial thrust $T_I$ is generated that leads to an increase in the swimming speed. An equivalent argument can readily be made to explain the decrease in the swimming speed of a puller with increasing Re.

Figure 6

Plot of $W$ versus Re. The function $W\left(\mathrm{Re}\right)$ is defined in (4.7) and represents the inertial contribution to the $O[1/{ln}^2(1/\epsilon )]$ power expenditure $p_2$.