# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 5 Optimal Strokes of \((m+1)\)-Linked-Sphere LRN Swimmers
### \((m+1)\)-Linked-Sphere PMPY Swimmers  
Next we consider \((m+1)\)-linked-sphere PMPY swimmers, which consist of \(m+1\) spheres with adjustable radius \(a_{i}(t)\)\((i=0,1,2,\cdots,m)\) and \(m\) linking-arms with adjustable length  
Figure 3: \((m+1)\)**-linked-sphere NG-type or PMPY-type of swimmers. (a)** Cartoon illustration of the swimmer, which consists of \(m+1\) spheres and \(m\) linking-arms. (b) Energy dissipation of optimal strokes \(\mathcal{E}(\overline{\tau})\) and (c) efficiency Eff of NG (blue dots) and PMPY (red dots) swimmers with \(m\) linking-arms. (d,e) The \(\gamma\)-path of the optimal stroke \(\overline{\tau}\) of a NG swimmer with 5 and 10 linked arms, respectively, where \(\gamma=(l_{1},l_{2},\cdots,l_{m})^{T}\). (f,g) The \(\gamma\)-path of the optimal stroke \(\overline{\tau}\) of a PMPY swimmer with 5 and 10 linked arms, respectively, where \(\gamma=(l_{1},l_{2},\cdots,l_{m},a_{1},a_{2},\cdots,a_{m})^{T}\), where the solid lines give the \(l_{i}\)-trajectories and dashed lines the \(a_{i}\)-trajectories. \(\varepsilon=0.2\) and \(X=5\times 10^{-4}\) in all simulations.  
\(1,2,\cdots,m\)) (Figure 3a). We assume initial shape \(a_{0}(0)=a_{1}(0)=\cdots=a_{m}(0)=A\) and \(l_{1}(0)=l_{2}(0)=\cdots=l_{m}(0)=L\). Under the volume conservation constraint of all spheres, we take \(a_{1},a_{2},\cdots,a_{m}\) to be controls while \(a_{0}(t)\) can be obtained from the volume conservation constraint. The equation system of the model follows that in Section 3.4. In our simulations, we test for \(m\in[1,20]\) and with small net translation \(X=5\times 10^{-4}\) and \(\varepsilon=A/L=0.2\) in all simulations.  
The energy dissipation of the optimal stroke reaches a minimum \(\min_{m}\mathcal{E}(\overline{\gamma})=1.70\times 10^{-2}\) at \(m=1\), i.e., a 2-sphere PMPY swimmer, which is the one we discussed in Section 3.4. \(\mathcal{E}(\overline{\gamma})\) increases as \(m\) increases (Figure 3b, red dots). Correspondingly, the most efficient swimmer is the 2-sphere NG swimmer with \(\mathrm{Eff}=29.5\times 10^{-3}\), and the efficiency decreases as \(m\) increases (Figure 3c, red dots). Comparing to NG type of swimmers with the same structure (i.e., same numbers of spheres and linking-arms), PMPY swimmers consumes less energy and is thus more efficient (Figure 3b,c).  
Similar to NG swimmers, for PMPY swimmers, the optimal strokes for different numbers of linking-arms have also made use of all degrees of freedom. Figure 3f,g show the \(\gamma\)-paths of the optimal strokes for 5 and 10 arms, respectively, which show that all linking-arms performed length deformations and all spheres performed radial deformations in the optimal stroke. In addition, we find that for large \(m\), the amplitude of sphere radial deformations \(a_{i}(t)\) are smaller than that of the linking-arms length changes \(l_{i}(t)\) (Figure 3f,g); on the other hand, in the most efficient PMPY type of swimmers, i.e., the 2-sphere PMPY swimmer, the amplitude of sphere radial deformation \(a_{1}(t)\) is larger than that of the linking-arm length change \(l(t)\) (Figure 2a,c).