# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 2 Model Equations and Numerical Method  
### Two-Fluid Mixture Model  
In this work, the fluid medium surrounding a swimmer is described as a mixture of two immiscible fluids. We call the less viscous fluid in the mixture as the solvent phase (denote by s), and the more viscous fluid component as the network phase (denoted by n). Two-fluid models of this kind have been successfully used for the investigation of many biofluids such as blood clot, biofilm and cytoplasm [15,16]. At any spatial location \(\mathbf{x}\), the relative amounts of the two fluids are given by their volume fraction, \(\theta_{\mathrm{s}}(\mathbf{x},\mathbf{t})\) and \(\theta_{\mathrm{n}}(\mathbf{x},\mathbf{t})\) for the solvent and network, respectively. The solvent and network fluids move with their own velocity fields, \(\mathbf{u}_{\mathrm{s}}(\mathbf{x},\mathbf{t})\) and \(\mathbf{u}_{\mathrm{n}}(\mathbf{x},\mathbf{t})\):  
\[\frac{\partial\theta_{\mathrm{s}}}{\partial\mathbf{t}}+\
abla\cdot(\theta_{ \mathrm{s}}\mathbf{u}_{\mathrm{s}})=0, \tag{1}\]  
\[\frac{\partial\theta_{\mathrm{n}}}{\partial\mathbf{t}}+\
abla\cdot(\theta_{ \mathrm{n}}\mathbf{u}_{\mathrm{n}})=0. \tag{2}\]  
Since \(\theta_{\mathrm{n}}+\theta_{\mathrm{s}}=1\), the sum of (1) and (2) gives the volume averaged incompressibility condition:  
\[\
abla\cdot(\theta_{\mathrm{s}}\mathbf{u}_{\mathrm{s}}+\theta_{\mathrm{n}} \mathbf{u}_{\mathrm{n}})=0. \tag{3}\]  
For a small Reynolds number, the force balance equations for the two fluids are given by:  
\[\
abla\cdot(\theta_{\mathrm{s}}\mathbf{\sigma}_{\mathrm{s}})-\theta_{\mathrm{s}} \
abla\mathrm{P}+\xi\theta_{\mathrm{n}}\theta_{\mathrm{s}}(\mathbf{u}_{ \mathrm{n}}-\mathbf{u}_{\mathrm{s}})+\mathbf{f}_{\mathrm{s}}=0. \tag{4}\]  
\[\
abla\cdot(\theta_{\mathrm{n}}\mathbf{\sigma}_{\mathrm{n}})-\theta_{\mathrm{n}} \
abla\mathrm{P}+\xi\theta_{\mathrm{n}}\theta_{\mathrm{s}}(\mathbf{u}_{ \mathrm{s}}-\mathbf{u}_{\mathrm{n}})+\mathbf{f}_{\mathrm{n}}=0, \tag{5}\]  
Here, \(\sigma_{\mathrm{s}}\) and \(\mathbf{\sigma}_{\mathrm{n}}\) are the viscous stress tensors for the solvent and network, respectively. \(\xi\theta_{\mathrm{n}}\theta_{\mathrm{s}}(\mathbf{u}_{\mathrm{n}}-\mathbf{u}_{ \mathrm{s}})\) represents the frictional drag force between the two fluids due to relative motions, with \(\xi\) being the frictional coefficient. \(\mathbf{f}_{\mathrm{s}}\) and \(\mathbf{f}_{\mathrm{n}}\) are the force densities generated by immersed elastic structures (swimmer) on the two fluids. Their formulations will be discussed in the next section. \(\mathbf{\sigma}_{\mathrm{s}}\) and \(\mathbf{\sigma}_{\mathrm{n}}\) are taken to be those of Newtonian fluids:  
\[\mathbf{\sigma}_{\mathrm{s}} = \mu_{\mathrm{s}}(\
abla\mathbf{u}_{\mathrm{s}}+\
abla\mathbf{u} _{\mathrm{s}}{}^{\mathrm{T}})+(\lambda_{\mathrm{s}}\
abla\cdot\mathbf{u}_{ \mathrm{s}})\mathbf{I}, \tag{6}\] \[\mathbf{\sigma}_{\mathrm{n}} = \mu_{\mathrm{n}}(\
abla\mathbf{u}_{\mathrm{n}}+\
abla\mathbf{u} _{\mathrm{n}}{}^{\mathrm{T}})+(\lambda_{\mathrm{n}}\
abla\cdot\mathbf{u}_{ \mathrm{n}})\mathbf{I}. \tag{7}\]  
Here I is the identity tensor, \(\mu_{\mathrm{s}}\) and \(\mu_{\mathrm{n}}\) are the shear viscosities and \(\lambda_{\mathrm{s,n}}+2\mu_{\mathrm{s,n}}/d\) are the bulk viscosities of the solvent and network (\(d\) is the space dimension). We choose \(\lambda_{\mathrm{s,n}}=-\mu_{\mathrm{s,n}}\) so that the bulk viscosities in both fluids are zero. We nondimensionalizeall model equations using the characteristic length scale of \(L=1\)\(\mu\)m, time scale of \(T=1\) second, and stress scale of \(\mu_{s}/T\). The viscosity of the solvent \(\mu_{s}\) is taken as the viscosity of water.