This dissertation introduces the new computational methods for two major topics. The first topic is computing the Stokes flow driven by an open immersed interface. The other topic is the simulation of the Stokes and Navier-Stokes fluid through an elastic tube driven by an internal source and sink.
For the first topic, we developed two second-order accurate method. One is for accurately evaluating boundary integral solutions at a point, and the other is for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N2 grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium.
For the second topic, we present numerical method for simulating both Stokes and Navier Stokes fluid flow through a compliant, closed tube, driven by an internal source and sink. The governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations.
In the Stokes fluid flow simulations, we solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface force and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface force yields an unsmooth solution, that part of the solution is computed by using the immersed interface method with the jump conditions for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. The source and sink are prescribed in the simulation. From the convergence test and oscillating frequency-amplitude study, we can demonstrate second-order accuracy and applicability of the method.
In the Navier-Stokes flow simulations, we adopt the velocity decomposition approach developed by Beale and Layton. The total velocity is decomposed into the Stokes part and the regular part. The Stokes part satisfies the Stokes equation and includes the boundary force. The regular part satisfies the modified Navier-Stokes equation that incorporate the source and sink terms, with the latter computed using the Hagen-Poiseuille equation. Convergence test, oscillating frequency-amplitude study and fluid viscosity-amplitude study are presented that demonstrate the accuracy of the method.
