Abstract. The method developed in this paper is motivated by Peskin's immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the two-dimensional incompressible Navier-Stokes equations using a high-resolution finite-volume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included. The IB method represents the flexible membrane using Lagrangian markers at which force-densities are computed, based on the position of the membrane, and spread to a Cartesian fluid grid by a set of discrete delta functions. The support of the delta function is comparable to the mesh width. The Navier-Stokes equations with this forcing are then advanced one time step on the Cartesian grid using a projection method. The velocities are first updated, ignoring the pressure gradient term in the Navier-Stokes equations in a transport step, and then a projection step is applied to compute the pressure and impose the incompressibility constraint on the velocity field. The resulting velocities are interpolated back to the Lagrangian markers using the same set of discrete delta functions. The markers are moved based on these velocities and the process is repeated in the next time step. This algorithm is written out in more detail in section 3.

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Air-driven core-annular fluid flows occur in many situations, from lung airways to engineering applications. Here we study, experimentally and theoretically, flows where a viscous liquid film lining the inside of a tube is forced upwards against gravity by turbulent airflow up the center of the tube. We present results on the thickness and mean speed of the film and properties of the interfacial waves that develop from an instability of the air-liquid interface. We derive a long-wave asymptotic model and compare properties of its solutions with those of the experiments. Traveling wave solutions of this long-wave model exhibit evidence of different mass transport regimes: Past a certain threshold, sufficiently large-amplitude waves begin to trap cores of fluid which propagate upward at wave speeds. This theoretical result is then confirmed by a second set of experiments that show evidence of ring waves of annular fluid propagating over the underlying creeping flow. By tuning the parameters of the experiments, the strength of this phenomenon can be adjusted in a way that is predicted qualitatively by the model.

An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N 2 ) to O(N ), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.

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