Numerical Methods and Modeling for Simulating the Motion of Self-propelled Swimmer in Incompressible Viscous Fluids

In the first chapter, we have studied, via direct numerical simulations, the equilibrium radial positions of neutrally buoyant balls moving in circular Poiseuille flows. For the one ball case, the Segre-Silberberg effect occurs at low Reynolds numbers (Re) as expected. However, at higher Re, the ball moves to one of two equilibrium positions. At even higher Re, the ball is pinched to a radial position closer to the central axis of circular cylinder. For the case of two neutrally buoyant balls placed on a line parallel with the central axis initially, this two-ball train is stable at low Re and its mass center moves to the outer Segre-Silberberg equilibrium position like the migration of a single neutrally buoyant ball. Moreover, for Re values greater than the critical value, $\mathrm{Re}c=435$, the two-ball train is unstable. The two balls interact periodically, suggesting a (kind of) Hopf bifurcation phenomenon. Nevertheless, the averaged mass center of the two balls is located at the inner equilibrium radial position. In the second chapter, we have studied, via direct numerical simulations, the moving behavior of a spherobot in Newtonian fluids. For the Newtonian cases, including two dimensions and three dimensions, those spherobots formed by two same size particles do not move due to the symmetry of the reciprocal motion. But for the case of two different size particles, the swimmer can move in the positive or negative x1-direction due to the asymmetry of reciprocal motion of the two particles. There is a critical Reynolds number so that the swimmer moves to the left (resp., right) for $($ resp.$,\mathrm{Re}>\mathrm{Re}c)$ if the larger particle is placed at the right side. In the third chapter, we have studied, via direct numerical simulations, the motion of a spherobot motion in a non-Newtonian shear-thinning fluid, we first tested the numerical method for the Poiseuille flows with no particles and swimmers. The accurate numerical solutions have been obtained for the Poiseuille flow of non-Newtonian shear-thinning fluids in a two-dimensional channel. Concerning a self-propelled swimmer formed by two disks, the effect of shear-thinning makes the swimmer moving faster and decreases the critical Reynolds number (for the moving direction changing to the opposite one) when decreasing the value of the power index in the Carreau-Bird model.