# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 3 Results
### Comparison with Analytic Solutions
#### 3.2.2 Swimming Near Deformable Fluid Interface  
The assumption that the fluid interfaces remain flat near a waving sheet is only valid for large \(\mathrm{H}\) or small \(\beta\). In [11], the locomotion of an infinite swimmer close to a deformable interface between two viscous fluids is investigated analytically. The interface condition is given by the continuity of stress and velocity. To illustrate the accuracy of our interface  
Figure 3: The scaled swimming speed vs. the nondimensional interface height \(\mathrm{kH}\) for different \(\beta\). The fluid interfaces are non-deformable. The analytical solutions are from [12].  
Figure 2: Distribution of network volume fraction \(\theta_{\mathrm{n}}\) and fluid velocity \(\mathbf{u}_{\mathrm{s}}\) at \(\mathrm{t}=0.5\) for different \(\mathrm{H}\). All vectors have the same scale. The viscosity ratio between the two fluids is \(\beta=0.5\).  
capturing scheme, we remove the IB walls at \(y=\pm\mathrm{H}\) and redo the simulations as described in the previous section. In Figures 4 and 5, the distributions of \(\theta_{\mathrm{n}}\) and \(\mathbf{u}_{\mathrm{s}}\) are plotted for simulations with \(\beta=0.5\) and \(\beta=0.1\), respectively. As an indication of the interface locations, the jump in \(\theta_{\mathrm{n}}\) is slightly smeared over \(1-2\) mesh blocks at \(t=0.5\) relative to its initial sharp profile. The interfaces deform from their horizontal configurations due to the waving motion of the swimmer. For interfaces closer to the swimmer, the deformations are more significant. Comparing Figure 5 with Figure 4, there are less interface deformations in the simulation with a smaller viscosity ratio \(\beta\). Because of the smearing effect, we choose to visualize the fluid interfaces using the contour lines of \(\theta_{\mathrm{n}}=0.5\), a value in the middle of the network volume fractions within the inner and outer layers of fluids. The swimming sheet, the \"fluid interfaces\", and the fluid velocity are plotted in Figure 6 for two simulations with \(\beta=0.5\) and \(\beta=0.1\) at different time. The initial swimmer-interface distance is \(\mathrm{H}=0.16\). It can be noticed that for the simulation with a smaller viscosity ratio (right column of Figure 6), the less deformed fluid interfaces provide more effective confinements to the swimmer. Thus, the fluid velocity exhibits stronger counter-rotating vortices than those in the simulation with a larger \(\beta\) (left column of Figure 6). By the time at \(t=1.0\) (which is exactly one wave period), all fluid interfaces return approximately back to their initial horizontal positions. In Figure 7, the simulated swimming speed is compared with the 2 nd order analytical result in [11]. The agreement is very good.  
Figure 6: \(\mathrm{H}=0.16\). Distributions of \(\mathbf{u}_{s}\), the profiles of swimming sheet (black curves) and fluid interfaces (red curves) for different \(\beta\). (**a**,**c**,**e**,**g**) are for \(\beta=0.5\). (**b**,**d**,**f**,**h**) are for \(\beta=0.1\). All vectors have the same scale.