# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 3 Linked-Sphere LRN Swimmers
### Fundamental Solutions for Translating and Radially Deforming Spheres  
In either of the three models, shape deformations either come from linking-arm length change (the NG and the PMPY models) or radial deformations of the spheres (the PMPY and the VE models). When a sphere of radius \(a(t)\) is pulled through a quiescent fluid with a steady force \(\mathbf{f}\) under no-slip conditions at the surfaces, the resulting flow field is given by [37]:  
\[\mathbf{u}(\mathbf{r})=\mathbf{f}\cdot\Big{(}1+\frac{a^{2}}{6}\
abla^{2} \Big{)}\frac{\mathbf{G}(\mathbf{x},\mathbf{x}_{0})}{8\pi\mu}=\frac{\mathbf{f }}{8\pi\mu r}\cdot\Big{[}\mathbf{I}+\frac{\mathbf{r}\mathbf{r}}{r^{2}}+\frac {a^{2}}{3r^{2}}\Big{(}\mathbf{I}-3\frac{\mathbf{r}\mathbf{r}}{r^{2}}\Big{)} \Big{]} \tag{3}\]  
where \(\mathbf{x}_{0}\) is the position of the center of the sphere and \(\mathbf{r}=\mathbf{x}-\mathbf{x}_{0}\), and the resulting velocity of the sphere is given by [PERSON] law  
\[\mathbf{f}=6\pi\mu a\mathbf{U} \tag{4}\]  
A second fundamental solution is the velocity field \(\mathbf{u}\) produced by a radially-expanding sphere, which can be generated by a point source at the center \(\mathbf{x}_{0}\) of the sphere. The corresponding velocity is [36]:  
\[\mathbf{u}=\dot{a}\big{(}\frac{a}{r}\big{)}^{2}\frac{\mathbf{r}}{r}=\frac{ \dot{v}}{4\pi r^{2}}\hat{\mathbf{r}} \tag{5}\]  
where \(v=4\pi a^{3}/3\) is the volume of the sphere and \(\hat{\mathbf{r}}=\mathbf{r}/r\). By combining the two fundamental solutions (3) and (5), we obtain the solution of the flow velocity generated by a sphere of radius \(a\), centered at \(\mathbf{x}_{0}\), subject to a pulling/pushing force \(\mathbf{f}\), and radially deforming at the rate \(\dot{a}\):  
\[\mathbf{u}(\mathbf{r};a,\mathbf{f},\dot{a})=\frac{1}{24\pi\mu r}\Big{[}\Big{(} 3+\frac{a^{2}}{r^{2}}\Big{)}\mathbf{f}+3\Big{(}1-\frac{a^{2}}{r^{2}}\Big{)} \big{(}\mathbf{f}\cdot\hat{\mathbf{r}}\big{)}\hat{\mathbf{r}}\Big{]}+\dot{a} \big{(}\frac{a}{r}\big{)}^{2}\hat{\mathbf{r}}\]  
Next, the fundamental solution involves the interaction between two spheres. Suppose that at time \(t\) the \(i\)th (\(i=1,2\)) sphere has radius \(a_{i}(t)\), centered at \(\mathbf{x}_{i}(t)\), subjected to force \(\mathbf{f}_{i}(t)\) and with radially deforming rate \(\dot{a}_{i}(t)\). In the scenario that the two spheres are far apart, i.e., \(a_{i}/l\ll 1\)where \(l=|\mathbf{x}_{1}-\mathbf{x}_{2}|\), the translational velocity of the \(i\)th sphere can be obtained by the _Reflection Method_[37; 38; 39; 40; 41; 42]:  
\[\mathbf{U}_{i}\sim\frac{\mathbf{f}_{i}}{6\pi\mu a_{i}}+\delta\mathbf{U}_{i}^{t} \{j\}+\delta\mathbf{U}_{i}^{e}\{j\}+O\Big{(}\frac{1}{l^{4}}\Big{)} \tag{6}\]  
where \(\delta\mathbf{U}_{i}^{t}\) is due to the translation of the other sphere, which arises from the flow given by Equation (3) and is  
\[\delta\mathbf{U}_{i}^{t}\{j\}=\Big{(}1+\frac{a_{i}^{2}}{6}\
abla^{2}\Big{)} \mathbf{u}(\mathbf{r})\big{|}_{\mathbf{r}=\mathbf{x}_{i}-\mathbf{x}_{j}}= \frac{1}{8\pi\mu l}\Big{[}\Big{(}1+\frac{a_{i}^{2}+a_{j}^{2}}{3l^{2}}\Big{)} \mathbf{I}+\Big{(}1-\frac{a_{i}^{2}+a_{j}^{2}}{l^{2}}\Big{)}\mathbf{\hat{I}} \Big{]}\mathbf{f}_{j} \tag{7}\]  
where \(\mathbf{I}=\mathbf{x}_{i}-\mathbf{x}_{j}\). The velocity \(\delta\mathbf{U}_{i}^{e}\) is resulted from a flow generated by the radial deformation of the other sphere, defined by Equation (5):  
\[\delta\mathbf{U}_{i}^{e}\{j\}=\Big{(}1+\frac{a_{i}^{2}}{6}\
abla^{2}\Big{)} \mathbf{u}(\mathbf{r})\big{|}_{\mathbf{r}=\mathbf{x}_{i}-\mathbf{x}_{j}}= \dot{a}_{j}\Big{(}\frac{a_{j}}{l}\Big{)}^{2}\mathbf{\hat{I}} \tag{8}\]  
Finally, the power consumption \(P\) of a sphere with radius \(a\), dragged by a force \(\mathbf{f}\), translating at velocity \(\mathbf{U}\) and radially expanding/contacting at a rate of \(\dot{a}\) comprises two parts: \(P^{t}=\mathbf{f}\cdot\mathbf{U}\) that results from the drag force on the sphere, and \(P^{e}=16\pi\mu a\dot{a}^{2}\) that results from the radial deforming [15; 40].