Effect of inertia on laminar swimming and flying of an assembly of rigid spheres in an incompressible viscous fluid

A mechanical model of swimming and flying in an incompressible viscous fluid in the absence of gravity is studied on the basis of assumed equations of motion. The system is modeled as an assembly of rigid spheres subject to elastic direct interactions and to periodic actuating forces which sum to zero. Hydrodynamic interactions are taken into account in the virtual mass matrix and in the friction matrix of the assembly. An equation of motion is derived for the velocity of the geometric center of the assembly. The mean power is calculated as the mean rate of dissipation. The full range of viscosity is covered, so that the theory can be applied to the flying of birds, as well as to the swimming of fish or bacteria. As an example a system of three equal spheres moving along a common axis is studied.

Figure 1

Plot of the eigenvalue ${\lambda }_{+}$ and the absolute value $|{\xi }_{+}|$ for the corresponding eigenvector ${\mathbf{\xi }}_{+}=(1,{\xi }_{+})$ of the three-sphere model with $d=5a$ as functions of the dimensionless viscosity ${\eta }_{*}$.

Figure 2

Snapshot of the spheres and their velocities in the rest frame at $t=0$ for the motion given by Eq. (8.2) with $\varepsilon =3,d=5a$, and ${\eta }_{*}=0.01$.

Figure 3

Plot of the elliptical orbit in the $r_1{r}_2$ plane corresponding to Eq. (8.2) with $\varepsilon =0.1,d=5a$, and ${\eta }_{*}=0.01$ (solid curve). We also plot the elliptical orbit for the high viscosity limit (dashed curve). The two curves cannot be distinguished on the scale of the figure.

Figure 4

Plot of the reduced mean swimming velocity $\overline{U}/\left({\varepsilon }^2\omega a\right)$ for $d=5a$ and ${\eta }_{*}=0.01$ as a function of the amplitude $\varepsilon$ as calculated from Eq. (3.8) for displacements given by Eq. (8.3).

Figure 5

Parametric plot of the reduced mean swimming power $\overline{\mathcal{P}}/\left({\varepsilon }^2\eta {\omega }^2{a}^3\right)$ vs the reduced mean swimming velocity $\overline{U}/\left({\varepsilon }^2\omega a\right)$ for $d=5a,{\eta }_{*}=0.01$, and $0<\varepsilon \le 3$.

Figure 6

As in Fig. 4 for the efficiency $E_T=\eta \omega a^2\overline{U}/\overline{\mathcal{D}}$.

Figure 7

Plot of the impetus $\mathcal{I}\left(t\right)$ (solid curve), the resistive contribution to the impetus (short dashes), and of the center velocity $U\left(t\right)$ (long dashes) as functions of time during a period for the three-sphere swimmer for $d=5a,{\eta }_{*}=0.01$, and amplitude factor $\varepsilon =3$.

Figure 8

Plot of the absolute values of the Fourier coefficients $f_n$ of harmonics of frequency $n\omega$, normalized to $f_0=1$, of the center velocity $U\left(t\right)$ of the three-sphere swimmer for $d=5a$ and ${\eta }_{*}=0.01$ with amplitude factor $\varepsilon =3$.

Figure 9

Plot of the positions of the three spheres found by numerical integration of the equations of motion Eq. (2.7) for $d=5a,{\eta }_{*}=0.01,\sigma =10,\varepsilon =1.5$ for 50 periods of time.

Figure 10

Plot of the mean value of the kinetic energy for successive periods $k=1,...,50$ corresponding to Fig. 9.

Figure 11

Plot of the orbit during the last period of Fig. 9 (solid curve) compared with the elliptical orbit given by Eq. (8.2) (dashed curve).