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Vakhtang Putkaradze (University of Alberta), Applied Mathematics Colloquium
November 11, 2016 @ 4:00 pm - 5:00 pm
Tea at 3:30 in 330 Phillips Hall
Title: Exact geometric approach to the discretization of fluid-structure interactions and the dynamics of tubes conveying fluid
Abstract: Variational integrators for numerical simulations of Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there are obstacles in direct applications of these integrators to systems involving fluid-structure interactions. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). We start by deriving 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. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. Time permitting, we shall also discuss some fully nonlinear solutions and the results of experiments.
Joint work with F. Gay-Balmaz (ENS and LMD, Paris). The work was partially supported by NSERC and the University of Alberta Centennial Fund.