# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 4 Optimal Strokes of NG, PMPY and VE Swimmers
### Numerical Results  
The numerical results of optimal strokes of a NG 3-sphere, PMPY 2-sphere and VE 3-sphere swimmers are given in Table 1 and Figure 2, where we take \(\varepsilon=A/L=0.2\) in all simulations. For different values of the net translation \(X\), the optimal strokes \(\overline{\gamma}\) of the three swimmers are computed by the SM method described in Section 4.1 and Appendix A, the energy dissipation of the optimal strokes \(\mathcal{E}(\overline{\gamma})\) is recorded in Table 1, and the \(\gamma\)-paths (solid lines) and the \(X(\overline{\gamma}(t))\), \(t\in[0,1]\) trajectories (dashed lines) are given in Figure 2. The simulation results show that [PERSON] is the most efficient swimmer among the three, next is [PERSON], the last is [PERSON]:  
\[\text{Eff}_{\text{PMPY}}>\text{Eff}_{\text{NG}}\sim\text{Eff}_{\text{VE}}\]  
where we define the _efficiency_ of an LRN swimmer as the net translation to energy dissipation ratio of the optimal stroke of the swimmer:  
\[\text{Eff}=\frac{X(\overline{\gamma})}{\mathcal{E}(\overline{\gamma})},\qquad \text{where}\ \overline{\gamma}\ \text{is the optimal stroke}. \tag{14}\]  
The dimensional form of Eff has the unit of force\({}^{-1}\). In addition, from Table 1 we find that for single loop optimal strokes, the efficiency for each swimmer is about the same for different values of \(X(\overline{\gamma})\), with a slight increase in the efficiency for NG and PMPY swimmers as \(X(\overline{\gamma})\) increases:  
\[\text{Eff}_{\text{PMPY}}\sim 29.5-33.0\times 10^{-3},\quad\text{Eff}_{\text{NG} }\sim 2.6-3.1\times 10^{-3},\quad\text{Eff}_{\text{VE}}\sim 1\times 10^{-3}\]  
As \(X\) increases, geodesic strokes of multiple loops show up for NG and PMPY swimmers. However, we do not catch multi-loop solutions in our numerical simulations for VE swimmers, as large shape deformations of VE models easily lead to negative volume of the spheres. Comparing to single loop geodesic strokes, multi-loop geodesic strokes are less efficient (see Appendix B). We summarize our main conclusions from the above numerical results as follows:  
1. A PMPY model that adopts a mixed-mode of shape deformations is the most efficient among the three; next is NG and the last is VE, both of which adopt a single-mode of shape deformations (see Remark below). 2. Single-loop geodesic strokes are more efficient than multi-loop geodesic strokes. 3. The efficiency of a given LRN swimmer is almost the same for different \(X\), as long as the optimal stroke is single-looped. Finally, we numerically compare the optimal strokes of the three model swimmers to stepwise square strokes. The results are presented in Appendix C, which clearly shows the improvement of swimming efficiency in the optimal strokes comparing to the stepwise square strokes. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(X(\overline{\gamma})\) & \multicolumn{2}{c}{**NG 3-Sphere**} & \multicolumn{2}{c}{**PMPY 2-Sphere**} & \multicolumn{2}{c}{**VE 3-Sphere**} \\ \hline  & \(\mathcal{E}(\overline{\gamma})\) & **Eff** & \(\mathcal{E}(\overline{\gamma})\) & **Eff** & \(\mathcal{E}(\overline{\gamma})\) & **Eff** \\ \hline
0.001 & \(37.61\times 10^{-2}\) & \(2.66\times 10^{-3}\) & \(3.37\times 10^{-2}\) & \(29.67\times 10^{-3}\) & \(95.87\) & \(1.04\times 10^{-3}\) \\ (Figure 2a,b) & & & & & \\
0.005 & \(182.96\times 10^{-2}\) & \(2.73\times 10^{-3}\) & \(16.82\times 10^{-2}\) & \(29.73\times 10^{-3}\) & \(539.33\times 10^{-2}\) & \(0.93\times 10^{-3}\) \\
0.01 & \(358.28\times 10^{-2}\) & \(2.79\times 10^{-3}\) & \(33.55\times 10^{-2}\) & \(29.81\times 10^{-3}\) & - & - \\ (Figure 2c,d) & & & & & \\
0.02 & \(695.46\times 10^{-2}\) & \(2.88\times 10^{-3}\) & \(66.77\times 10^{-2}\) & \(29.95\times 10^{-3}\) & - & - \\
0.05 & \(1636.25\times 10^{-2}\) & \(3.06\times 10^{-3}\) & \(164.91\times 10^{-2}\) & \(30.33\times 10^{-3}\) & - & - \\
0.1 & - & - & \(320.94\times 10^{-2}\) & \(31.16\times 10^{-3}\) & - & - \\
0.2 & - & - & \(606.30\times 10^{-2}\) & \(32.99\times 10^{-3}\) & - & - \\ \hline \hline \end{tabular}
\end{table}