Modeling and Simulation of Elastic Rods with Intrinsic Curvature and Twist Immersed in Fluid

Abstract: Understanding the dynamics of thin elastic rods that are immersed in fluid is fundamental in explaining many problems that arise in biology, physics and engineering. Solving the coupled system of rod-fluid in 3D is usually very costly, however in case of low Reynolds number, the three-dimensional problem can be reduced a one-dimensional problem on the centerline of the rod. In this thesis we examine the method of regularized Stokeslets which is a numerical algorithm for an elastic rod immersed in viscous, incompressible fluid at zero Reynolds number governed by Stokes equations. In this method, the elastic rod is represented by a space curve corresponding to the centerline of the rod. In addition, an orthonormal triad is varying along the curve, with one vector being tangent to the curve, and the others describing the material twist. The model that is used for the elastic forces based on this, allows for natural configurations for the rods that are far from straight, as described by curvature and torsion. In this way, the basic or equilibrium configuration for the rod can be e.g. a helix. The linearity of Stokes equations allows us to evaluate the linear and angular fluid velocity only at centerline of the rod. We also examine the dependency to the numerical parameters together with the accuracy and convergence properties of the method. As a bench mark, we compare the numerical result of this method to those produced by the non-local slender body method for the case of elastic rods with no intrinsic curvature and twist inside a planar shear flow. We also present the simulation of the extension of helical rods when they are placed within a constant background flow and we provide a fast converging formula for the periodic summation of the fundamental solutions to the Stokes equations.