We develop a multiscale Eulerian–Lagrangian localized adjoint method for transient linear advection– diffusion equations with oscillatory coefficients, which arise in mathematical models for describing flow and transport through heterogeneous porous media, composite material design, and other applications.

We present a Lagrangian-Eulerian strategy for proving uniqueness and local existence of solutions of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.

In this work, we develop numerical schemes for mechano-chemical fluid-structure interactions with long-term effects. We investigate a model of a growing solid interacting with an incompressible fluid. A typical example for such a situation is the formation and growth of plaque in blood vessels. This application includes two particular difficulties: First, growth may lead to very large deformati...

We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing modified adjoint problem for the test function and performing integration of PDE over space-time region partitioned time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation can then be taken into account flux term, integrated method-of-...

Our work has a number of objectives: • To determine the predictability time scales of the meandering of oceanic fronts • To determine the ways in which predictability in such a flow is lost, the physical processes responsible, and the sites of most rapid error growth • To develop simple, efficient data assimilation schemes that can be used to forecast the behavior of such fronts • To develop a ...

In this thesis, we study stability of inviscid fluids in the Lagrangian sense. That is, we look at how paths of particles diverge from each other under a small perturbation of the initial velocity field. The more usual question of stability is in the Eulerian sense, when one looks at how velocities at each point diverge from a steady state under a small perturbation. We demonstrate, for certain...

We present the dynamic formulations for sliding beams that are deployed or retrieved through prismatic joints. The beams can undergo large deformation, large overall motion, with shear deformation accounted for. Until recently, the sliding beam problem has been tackled mostly under small deformation assumptions, or under quasi-static motion. Here we employ geometrically-exact beam theory. Two t...

The free surface flow of an Oldroyd-B viscoelastic fluid is considered, following [1]. When removing a term in the extra-stress constitutive relation, the description of an elastic incompressible solid is obtained, in Eulerian coordinates, with the velocity field as unknown, rather than the usual deformation field. Two simulations are proposed, a bouncing ball and an oscillating beam.

Contents 1. Introduction 1 2. Reminders on the Cauchy-Lipschitz theory 4 3. Nonsmooth vector fields in Euclidean spaces 4 4. Abstract setup 7 5. Derivations and their regularity 10 6. Eulerian side 11 6.1. Existence of solutions to the continuity equation 11 6.2. Uniqueness of solutions to the continuity equation 14 7. Lagrangian side 16 7.1. ODE's associated to derivations and Regular Lagrangi...

In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to...