# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 3 Results
### Swimming Near Deformable Interface in Fluid Mixtures
In Figure 12, we plot the distributions of \(\theta_{\mathrm{n}}\) and the velocity difference \(\mathbf{u}_{\mathrm{n}}-\mathbf{u}_{\mathrm{s}}\) for two simulations with \(\xi=100\) and \(\xi=1000\), at \(\mathrm{t}=0.5\). With a 10-fold increase of the frictional coefficient, there is a drastic reduction in both the extent of phase separation and the relative motion between the two fluids. The region with the largest velocity difference is near the fluid interfaces. A close look at the plot indicates that the interface deformation is larger in the simulation with a larger frictional coefficient (not shown). In Figure 13, we show the distributions of \(\theta_{\mathrm{n}}\) and \(\mathbf{u}_{\mathrm{s}}\) from two simulations with \(\mathrm{H}=0.32\) and \(\mathrm{H}=0.5\) at \(\mathrm{t}=0.5\) and \(\mathrm{t}=1\). Regions with \(\theta_{\mathrm{n}}>0.25\) are filled with white color. Comparing the plots in Figure 13a,b, we see that with the locations of the fluid interfaces further away from the swimmer, both the interface deformation and the extent of phase separation are reduced. The vortices of \(\mathbf{u}_{\mathrm{s}}\) for the simulation with \(\mathrm{H}=0.32\) are stronger than than those in the simulation with \(\mathrm{H}=0.5\). As indicated in Figure 13c,d, at one wave period \(\mathrm{t}=1\), the distributions of \(\theta_{\mathrm{n}}\) return approximately to their initial profile for both simulations. Furthermore, at this moment, all fluid interfaces are horizontal approximately, especially for the simulation with the larger value of \(\mathrm{H}\). Figure 10: \(\mathrm{H}=0.16\), \(\beta=0.25\), \(\xi=100\), \(\phi_{\mathrm{n}}^{\mathrm{in}}=0.2\), and \(\theta_{\mathrm{n}}^{\mathrm{out}}=0.8\). Distributions of \(\theta_{\mathrm{n}}\) and \(\mathbf{u}_{\mathrm{n}}\) at different time. All vectors have the same scale. Figure 11: \(\mathrm{H}=0.16\), \(\beta=0.25\), \(\xi=100\), \(\theta_{\mathrm{n}}^{\mathrm{in}}=0.2\), and \(\theta_{\mathrm{n}}^{\mathrm{out}}=0.8\). Distributions of \(\theta_{\mathrm{n}}\) and \(\mathbf{u}_{\mathrm{s}}\) at different time. All vectors have the same scale. In Figure 14, the scaled swimming speed \(\frac{U}{U_{0}}\) is plotted as a function of the network volume fraction \(\theta_{\text{n}}^{\text{in}}\) for H = 0.5, H = 0.32, and H = 0.16. The value of \(\theta_{\text{n}}^{out}\) is 0.8 for all simulations. For all values of H, the swimming speed always increases with the increase of the frictional coefficient for a given \(\theta_{\text{n}}^{\text{in}}\). This is consistent with the conclusion from [27], in which the swimming sheet problem is investigated in a two-fluid mixture without interfaces. In the simulations with the largest drag of \(\xi=10^{5}\), the network and the solvent move approximately with the same velocity. Therefore, the inner fluid layer behaves like a single phase fluid with the effective viscosity of \(\mu_{\text{eff}}^{\text{in}}=\theta_{\text{n}}^{\text{in}}\mu_{\text{n}}+(1- \theta_{\text{n}}^{\text{in}})\mu_{\text{s}}\). Similarly, the effective viscosity of the outer layer is given by \(\mu_{\text{eff}}^{out}=\theta_{\text{n}}^{out}\mu_{\text{n}}+(1-\theta_{ \text{n}}^{out})\mu_{\text{s}}\). Based on the viscosity ratio \(\beta=\frac{\mu_{\text{eff}}^{in}}{\mu_{\text{eff}}^{out}}\), we can use the analytical formula from [11] to compute the swimming speed. The results are plotted as the solid black curves in Figure 14. Despite of the relatively large amplitude of the waving sheet and large interface deformations, the simulation results agree very well with the analytical solutions. Notice that for the simulations with the largest drag, the swimming speed in a single phase fluid is recovered(\(\frac{II}{U_{0}}\approx 1\)) when \(\theta_{\rm n}^{\rm in}=\theta_{\rm n}^{out}=0.8\). For all other values of \(\theta_{\rm n}^{\rm in}<0.8\), the sheet always swims faster than that in a single fluid, due to the confinement effect from the interfaces.