# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 4 Optimal Strokes of NG, PMPY and VE Swimmers
### Numerical Results

Table 1: Energy dissipation \(\mathcal{E}(\overline{\gamma})\) and efficiency Eff. Figure 2: **Optimal strokes of NG 3-sphere, PMPY 2-sphere and VE 3-sphere swimmers. ** (**a**,**c**) give the \(\gamma\)-paths of the optimal strokes of the NG (blue), PMPY (red) and VE (green) swimmers, when \(X(\gamma)=0.001,\ 0.01\), respectively. (**b**,**d**) give the optimal stroke trajectories \(X(\overline{\gamma}(t))\), \(t\in[0,1]\) (dashed lines) corresponding to the \(\gamma\)-paths (solid lines). \((\gamma_{1},\gamma_{2})=(l_{1},l_{2})\) for NG swimmer; \((\gamma_{1},\gamma_{2})=(l,a_{1})\) for PMPY; \((\gamma_{1},\gamma_{2})=(a_{1},a_{3})\) for VE. All strokes start from \((\gamma_{1}(0),\gamma_{2}(0))=(1,1)\). **Remark 1**: _At LRN, different microorganisms adopt various propulsion mechanisms and directed locomotion strategies for searching for food and running from predators. While many microorganisms including bacteria use a flagellated or ciliated mode of swimming [2; 3; 46; 47; 48; 5; 49; 50; 51; 52; 53], some cells use an amoeboid swimming strategy, which relies on the generation, protrusion and even travel of pseudopodia or blebs [6; 7; 54]. Such shape deformations propagating over the whole cell body can usually be generalized as a combination of two modes: stretching of the cell body, and mass transportation along the cell body. Comparing with the linked-sphere types of models, cell body stretching can be considered as mode \(\dot{\mathbf{i}}\) while mass transportation as mode \(\dot{\mathbf{a}}\). According to the scallop theorem [1], a minimal LRN swimmer has at least two degrees of freedom in its shape deformations, therefor we have three possible combinations: two single-mode of shape changes: both in body stretching (both are \(\dot{\mathbf{i}}\)-type of shape deformations), both in mass transportation (both are \(\dot{\mathbf{a}}\)-type of shape deformations), and a mixed-modes of shape change: body stretching combined with mass transportation \((\dot{\mathbf{i}},\dot{a})\). It is clearly seen that the three combinations correspond to the three linked-sphere swimmers (NG, VE and PMPY) we discussed here, and our numerical results indicate that:_  
_Comparing to single-mode shape deformations, mixed-modes of shape deformations, i.e., body stretching combined with mass transportation is more efficient when swimming at LRN._  
_An asymptotic analysis study on the three swimmers that derives the same conclusion can be found in [55], which illustrates that a PMPY swimmer has a net translation \(X(\gamma)\sim O(1)\), comparing to \(X(\gamma)\sim O(\varepsilon^{2})\) for a NG or a VE swimmer. Therefore the mixed control strategy as is adopted by a PMPY swimmer is superior to uni-mode of control strategies as are adopted by a NG or a VE swimmer._  
**Remark 2**: _A more popular definition of efficiency of NG swimmer in its dimensional form is given by [30]:_  
\[\text{Eff}^{-1}=\frac{\frac{1}{T}\int_{0}^{T}P(\gamma)dt}{6\pi\mu AX^{2}}\]  
_which measures the ratio of the average energy dissipation in the stroke \(\gamma\) to that applied to a rigid sphere of radius \(A\) traveling the same distance \(X\). However, this definition does not apply well to our discussion in that (1) we are considering non-rigid spheres where the spheres are allowed to deform radially, and (2) we are comparing swimmers of different numbers of spheres. Therefore we adopt the definition given by Equation (14) instead._