# Low Reynolds Number Swimming Near Interfaces in Multi-Fluid Media
## 1 Introduction  
Recently, there has been a surge of research interest in the locomotion of microorganisms through their fluidic environment. Examples include the swimming of the ulcer-causing Helicobacter pylori through the gastric mucus layer [1], and run-and-tumble dynamics of _Escherichia coli_ in polymeric solutions [2]. Experimental studies have revealed important features of the movement of microorganisms in biological fluids [3]. In addition to experiments, mathematical modeling, analysis, and computational simulations are playing important roles in giving new insights on the mechanics behind small scale swimming in complex fluids [4, 5].  
Many biological fluids such as mucus and cytoplasm are mixtures of multiple components, which exhibit complicated non-Newtonian characteristics. The composition and rheology of the these mixtures have significant impact on the speed and efficiency of biological locomotion. On the other hand, the swimming motion is also greatly affected by physical boundaries and material interfaces near the swimmer. For example, _E. Coli_ tends to exhibit circular trajectories close to a solid boundary [6]. Sperm cells near a liquid-solid interface swim slower than those near a liquid-gas interface [7]. Furthermore, some micro-swimmers may drastically enhance their motility by creating heterogeneous layers of fluids around them. For instance, self-propelling helical swimmers in shear-thining fluids may move 50% faster than in a Newtonian fluid. The speedup is believed to come from the viscosity stratification around the swimmer [8]. Gastric mucin forms a layer of protective gel for the stomach lining against harmful bacteria. To swim through the mucus layer, _H. pylori_ hydrolyzes ambient urea to produce ammonia. This causes an elevation of pH, and the transition of the local gastric mucus from a viscoelastic gel to a viscous fluid. Therefore, instead of boring its way across a mesh of polymer network, the bacterium essentially swims through a pocket of fluid surrounded by mucus gel [1]. Motivated by these experimental discoveries, there have been a number of theoreticalanalysis on the dynamics of swimmers near a plane surfactant-laden interface [9], in a viscous fluid enclosed by a Brinkman medium [10], within two layers of fluids separated by elastic deformable membranes [11] or by stationary interfaces [12]. By contrast, there are only few computational studies on locomotion in multi-component fluids, especially for problems involving dynamical fluid interfaces. In [13,14], different variations of the Immersed Boundary Method are proposed to simulate the interactions between elastic boundaries and a two-phase fluid mixture. However, no free interfaces are included in those studies.  
In this paper, we investigate the classical [PERSON]'s swimming sheet problem near interfaces in a two-fluid environment, using the computational method developed in [13]. The mixture model provides a simple and unified framework for the study of swimming dynamics near interfaces in a multi-fluid mixture. Our simulation results reveal important characteristics of locomotion in heterogeneous fluid media, which are very different from those of swimming in a homogeneous environment without interfaces. In Section 2, the model equations and numerical methods are presented. The simulation results are presented in Section 3. The discussions about the results and potential extensions of the computational method are given in Section 4.