# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 3 Results
### Swimming Near Deformable Interface in Fluid Mixtures  
For all simulations presented so far, the continuum medium on each side of the interface is modeled as a single phase viscous fluid. Such models may not be suitable for some applications. For example, many biological fluids such as gastric mucus are mixtures composed of a polymer network immersed in a fluid solvent. The mixture can not be adequately described as a single phase homogeneous medium if the composition of the mixture has large spatial variations, or when there is significant relative motions between different components in the mixture. In this section, we investigate problems for which analytical solutions are not available. Specifically, we study the dynamics of the swimming sheet in mixtures of fluids separated by free interfaces. The main focus is on the effect of fluid composition and frictional drag on the swimming motion. A relatively large wave amplitude \(\mathrm{b}=0.048\) is chosen for the swimming sheet. At the initial time, the network volume fraction in the bulk fluid away from the swimmer is set to \(\theta_{\mathrm{n}}^{out}=0.8\). The value of \(\theta_{\mathrm{n}}^{\mathrm{in}}\) for the mixture around the swimmer is varied between simulations. We also vary the frictional coefficient \(\xi\) between the network and the solvent to investigate the influence of phase separation on locomotion. For all simulations in this section, the viscosity ratio between the solvent and the network is \(\beta=0.25\). In Figure 10, the distributions of \(\theta_{\mathrm{n}}\) and \(\mathbf{u}_{\mathrm{n}}\) are plotted at different time for the simulation in which \(\mathrm{H}=0.16\), \(\xi=100\), and \(\theta_{\mathrm{n}}^{\mathrm{in}}\) is set to 0.2 at the initial time. To give a better contrast, white color is used to fill regions with \(\theta_{\mathrm{n}}>\theta_{\mathrm{n}}^{in,max}\), where \(\theta_{\mathrm{n}}^{in,max}=0.26\) is the largest value of \(\theta_{\mathrm{n}}\) observed near the sheet during the simulation. In Figure 10b-e, the boundaries of the colored region (contour lines of \(\theta_{\mathrm{n}}=\theta_{\mathrm{n}}^{in,max}\)) are referred to as the \"fluid interfaces\". As time progresses, spatial inhomogeneities in \(\theta_{\mathrm{n}}\) and deformations of fluid interfaces appear with the waving motion of the sheet. Compared with the plots at \(\mathrm{t}=0.25\) and \(\mathrm{t}=0.75\), there are more significant inhomogeneities in the spatial distribution of \(\theta_{\mathrm{n}}\) at the half wave period \(\mathrm{t}=0.5\). Interestingly, it is also at \(\mathrm{t}=0.5\) that the fluid interfaces and the swimmer have approximately the same phase. At one wave period \(\mathrm{t}=1.0\), the distribution of network volume fraction roughly returns to its initial value of 0.2 over the entire inner layer. At \(\mathrm{t}=1.0\), although the fluid interfaces do not fully return to their horizontal positions, their vertical deformations are the smallest within the wave period. Relative to the results at earlier time, the network velocity \(\mathbf{u}_{\mathrm{n}}\) tends to have smaller x-components at \(\mathrm{t}=1.0\). The distributions of \(\theta_{\mathrm{n}}\) over the outer layers of fluid and the solvent velocity \(\mathbf{u}_{\mathrm{s}}\) are plotted in Figure 11. Here white color is used to fill the region where \(\theta_{\mathrm{n}}>\theta_{\mathrm{n}}^{in,out}\), with \(\theta_{\mathrm{n}}^{in,out}=0.72\) being approximately the smallest value of \(\theta_{\mathrm{n}}\) in the outer layer. The fluid interfaces and \(\theta_{\mathrm{n}}\) in the plot exhibit similar periodic patterns in time as observed in Figure 10. Notice that due to the numerical smearing, there is a small vertical displacement (about 2 grid blocks) between the fluid interfaces visualized in Figures 10 and 11. At one wave period \(\mathrm{t}=1.0\), x-components of the solvent velocity have smaller magnitude than those at earlier time. Due to its low viscosity, the solvent has greater motion than the network in the vicinity of the swimmer.