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A single particle representation of a self-propelled microorganism in a viscous incompressible fluid is derived based on regularised Stokeslets in three dimensions. The formulation is developed from a limiting process in which two regularised Stokeslets of equal and opposite strength but with different size regularisation parameters approach each other. A parameter that captures the size difference in regularisation provides the asymmetry needed for propulsion. We show that the resulting limit is the superposition of a regularised stresslet and a potential dipole. The model framework is then explored relative to the model parameters to provide insight into their selection. The particular case of two identical particles swimming next to each other is presented and their stability is investigated. Additional flow characteristics are incorporated into the modelling framework with in the addition of a rotlet double to characterise rotational flows present during swimming. Lastly, we show the versatility of deriving the model in the method of regularised Stokeslets framework to model wall effects of an infinite plane wall using the method of images. 

The general system of images for regularized Stokeslets (GSIRS) developed by Cortez and Varela [] is used extensively to model Stokes flow phenomena such as microorganisms swimming near a boundary. Our collaborative team uses dynamically similar scaled macroscopic experiments to test theories for forces and torques on spheres moving near a boundary and uses these data and the method of regularized Stokeslets (MRS) created by Cortez [] to calibrate the GSIRS. We find excellent agreement between theory and experiments, which provides experimental validation of exact series solutions for spheres moving near an infinite plane boundary. We test two surface discretization methods commonly used in the literature: the 6-patch method and the spherical centroidal Voronoi tessellation (SCVT) method. Our data show that a discretization method, such as SCVT, that uniformly distributes points provides the most accurate results when the motional symmetry is broken by the presence of a boundary. We use theory and the MRS to find optimal values for the regularization parameter in free space for a given surface discretization and show that the optimal regularization parameter values can be fit with simple formulas when using the SCVT method. We also present a regularization function with higher-order accuracy when compared with the regularization function previously introduced by Cortez []. The simulated force and torque values compare very well with experiments and theory for a wide range of boundary distances. However, we find that for a fixed discretization of the sphere, the simulations lose accuracy when the gap between the edge of the sphere and the wall is smaller than the average distance between discretization points in the SCVT method. We also show an alternative method to calibrate the GSIRS to simulate sphere motion arbitrarily close to the boundary. Our computational parameters and methods along with our matlab and python implementations of the series solution of Lee and Leal [], MRS, and GSIRS provide researchers with important resources to optimize the GSIRS and other numerical methods, so that they can efficiently and accurately simulate spheres moving near a boundary. 

Microorganisms often navigate a complex environment composed of a viscous fluid with suspended microstructures such as elastic polymers and filamentous networks. These microstructures can have similar length scales to the microorganisms, leading to complex swimming dynamics. Some microorganisms secrete enzymes that dynamically change the elastic properties of the viscoelastic networks through which they move. In addition to biological organisms, microrobots have been engineered with the goals of mucin gel penetration or dissolving blood clots. In order to gain insight into the coupling between swimming performance and network remodeling, we used a regularized Stokeslet boundary element method to compute the motion of a microswimmer consisting of a rotating spherical body and counter-rotating helical flagellum. The viscoelastic network is represented by a network of points connected by virtual elastic linkages immersed in a viscous fluid. Here, we model the enzymatic dissolution of the network by bacteria or microrobots by dynamically breaking elastic linkages when the cell body of the swimmer falls within a given distance from the link. We investigate the swimming performance of the microbes as they penetrate and move through networks of different material properties, and also examine the effect of network remodeling. 

This article presents a mathematical modeling activity for students related to the process of memorization in which students collect their own data to drive their model development, parameterization, and validation. Engaging in the data collection gives them insight to critique and evaluate various models. This task is a low-floor high ceiling problem that offers both a relatable context and a window to quantitative approaches in cognitive science. Experimental results of students’ participation in this activity are discussed. This article also includes pedagogical recommendations with a focus on fostering equitable teaching practices and a detailed analysis of the situation comprised of several mathematical approaches to model the memorization process that highlight the richness of the problem. Instructors can adapt and implement this modeling exploration for use in various undergraduate courses, from introductory to advanced, depending on the emphasis of the lesson.