Collinear swimmer propelling a cargo sphere at low Reynolds number

The swimming velocity and rate of dissipation of a linear chain consisting of two or three little spheres and a big sphere is studied on the basis of low Reynolds number hydrodynamics. The big sphere is treated as a passive cargo, driven by the tail of little spheres via hydrodynamic and direct elastic interaction. The fundamental solution of Stokes equations in the presence of a sphere with a no-slip boundary condition, as derived by Oseen, is used to model the hydrodynamic interactions between the big sphere and the little spheres.

Figure 1

Schematic shape of longitudinal swimmer consisting of three beads and a cargo sphere.

Figure 2

Plot of the reduced efficiency $(b^3/a^3)E_T^C$ of the 3C-chain with longitudinal linear motion as a function of the stiffness parameter $\sigma$ in the asymptotic limit $a\ll d\ll b$ (solid curve) and for $d=5a,b=10a$ (dashed curve).

Figure 3

Plot of the reduced efficiency $(b^3/a^3)E_T^C$ of the 3C-chain with longitudinal linear motion as a function of the ratio $b/a$ for $d=5a$ and $\sigma =10$.

Figure 4

Plot of the reduced efficiency $(b^3/a^3)E_{T\text{max}}^C$ of the 4C-chain with longitudinal linear motion as a function of the stiffness parameter $\sigma$ in the asymptotic limit $a\ll d\ll b$ (solid curve) and for $d=5a,b=10a$ (dashed curve).

Figure 5

Plot of the reduced efficiency $(b^3/a^3)E_{T\text{max}}^C$ of the 4C-chain with longitudinal linear motion as a function of the ratio $b/a$ for $d=5a$ and $\sigma =10$.

Figure 6

Plot of the motion of a 4C-chain with $d=5a$, $b=10a$, and $\sigma =6.271$ in the $r_1{r}_2$ plane, where $r_1=x_2-x_1,r_2=x_3-x_2$, for ten periods with actuating forces as given by Eq. (5.3) with $\varepsilon =100$.

Figure 7

Plot of the positions $x_1\left(t\right),x_2\left(t\right),x_3\left(t\right)$ for the 4C-chain during the last period of Fig. 5.

Figure 8

Plot of the mean rate of dissipation of a 4C-chain with $d=5a$, $b=10a$, and $\sigma =6.271$ as a function of the amplitude factor $\varepsilon$.

Figure 9

Plot of the efficiency $E_T$ of a 4C-chain with $d=5a$, $b=10a$, and $\sigma =6.271$ as a function of the amplitude factor $\varepsilon$.