Modeling and Computation Seminar
Variational approach to fluid-structure interactions: tubes conveying fluid, friction, poromechanics and Darcy's laws
This work is motivated by the study of mechanics of elastic tubes conveying fluid, and the dynamics of moving porous media, which are both challenging and important problems involving fluid-structure interactions. We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we obtain
the fully three dimensional equations of motion. Next, we show how to incorporate similar variational approach to the dynamics of porous media by incorporating viscous forces in the variational principle. To elucidate the physics and mathematics of the problem, we study some simplified cases such as a pendulum with a moving viscous droplet. We show that the analogue of Darcy's law in these simplified models comes from the short-term convergence to a 'constraint manifold' in a singular perturbation problem, and the following long-term dynamics on that manifold. The resulting Darcy's law can reduce to either holonomic or non-holonomic constraint for the motion, depending the physical realization. We also demonstrate that care must be taken in formulating Darcy's law as the long-term dynamics can change drastically for small perturbations of the system. We discuss the relevance of these results for poromechanics and, time permitting, consider some simplified physical cases for the porous media motion.
This work was partially supported by NSERC and the University of Alberta.