# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 3 Results
### Comparison with Analytic Solutions
#### 3.2.3 Force Analysis  
For a Stokes swimmer, the swimming speed is determined by a balance of thrust and drag forces [26]. Here the thrust F\({}_{T}\) is defined as the anchoring force applied so that the undulating sheet is prevented from swimming. The drag is the force F\({}_{D}\) required to tow the sheet with a frozen shape at the swimming speed U. Due to linearity of Stokes flow, the drag force can be expressed as F\({}_{D}(\text{U})=\gamma\)U, where the magnitude of the drag coefficient \(\gamma\) is equal to F\({}_{D}(1)\). To compute the thrust force, we carry out simulations in which tether points are used to prevent IB points on a waving sheet from moving in the x direction. F\({}_{T}\) is computed as the total tethering force on the sheet, averaged over one wave period. Similarly, for simulations in which IB points on a frozen sheet are connected to tether points moving with velocity of 1 along the x direction, \(\gamma\) can be calculated from the time averaged towing force on the sheet. From the force balance equation F\({}_{T}+{}_{D}=0\), the swimming speed is given by U \(=-\frac{\text{F}_{T}}{\gamma}\). To confirm the accuracy of the thrust and drag calculation from the simulations, we list the simulated results in Tables 1 and 2 for non-deformable and deformable interfaces, respectively. The viscosity ratio is \(\beta=0.5\) for all calculations. As seen from the tables, the ratio between the thrust force and the drag coefficient from the simulations is very close to the swimming speed of the sheet. In Figure 8, we plot the thrust force and the drag coefficient for different values of kH from the simulations with non-deformable interfaces. Both F\({}_{T}\) and \(\gamma\) increase as the fluid interfaces get closer to the swimmer. The variation is approximately linear for \(\gamma\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**kH** & **Thrust Force (F\({}_{T}\))** & **Drag Coefficient (\(\gamma\))** & \(-\frac{\text{F}_{T}}{\gamma}\) & **Swimming Speed \(U\)** \\ \hline
0.16 & \(3.03\times 10^{-2}\) & \(2.43\times 10^{-1}\) & \(-1.247\times 10^{-1}\) & \(-1.262\times 10^{-2}\) \\
0.25 & \(1.41\times 10^{-2}\) & \(2.3\times 10^{-1}\) & \(-6.130\times 10^{-2}\) & \(-6.151\times 10^{-2}\) \\
0.4 & \(7.36\times 10^{-3}\) & \(2.12\times 10^{-1}\) & \(-3.472\times 10^{-2}\) & \(-3.479\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Non-deformable interface (\(\beta=0.5\)): thrust force, drag coefficient, their ratio, and the simulated swimming speed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**kH** & **Thrust Force (F\({}_{T}\))** & **Drag Coefficient (\(\gamma\))** & \(-\frac{\text{F}_{T}}{\gamma}\) & **Swimming Speed \(U\)** \\ \hline
0.16 & \(1.23\times 10^{-2}\) & \(2.43\times 10^{-1}\) & \(-5.062\times 10^{-2}\) & \(-5.073\times 10^{-2}\) \\
0.25 & \(9.54\times 10^{-3}\) & \(2.3\times 10^{-1}\) & \(-4.147\times 10^{-2}\) & \(-4.151\times 10^{-2}\) \\
0.4 & \(6.63\times 10^{-3}\) & \(2.12\times 10^{-1}\) & \(-3.127\times 10^{-2}\) & \(-3.124\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deformable interface (\(\beta=0.5\)): thrust force, drag coefficient, their ratio, and the simulated swimming speed. Figure 7: The scaled swimming speed vs. the nondimensional interface height kH for different \(\beta\). The fluid interfaces are deformable. The analytical solutions are from [11]. superlinear for F\({}_{T}\). As the result, swimmers near non-deformable fluid interfaces always move faster than those in a single fluid, as observed in Figure 3. The enhancement of the swimming speed is more dramatic at smaller values of kH, where the thrust force increases more significantly. For simulations in which the interfaces are deformable, the values of F\({}_{T}\) and \(\gamma\) are plotted in Figure 9 for \(\beta=0.5\) and \(0.1\). Comparing Figure 9a,b with Figure 8, we see that for a fixed viscosity ratio, the drag coefficient is almost identical for simulations with non-deformable and deformable interfaces.