Fluid Mechanics and Waves Seminar - Fall 2018

 
We describe an efficient algorithm for reconstruction of the electromagnetic parameters of an unbounded dielectric three-dimensional medium from noisy cross section data induced by a point source. The efficiency of our Bayesian inverse algorithm for the parameters is based on developing an offline high order forward stochastic model and also an associated deterministic dielectric media Maxwell solver. Underlying the inverse/offline approach is our high order fully discrete Galerkin algorithm for solving an equivalent surface integral equation reformulation that is stable for all frequencies.

The dynamics of particles and other objects play an important role in many flows of industrial interest, including the extraction of bitumen from oil sands, inkjet printing, micro- and nano-capsule drug delivery, soil remediation and many others. One of the most challenging aspects of modeling such flows is modelling the fluid-fluid interface and the effects of wettability at the contact line. While the Immersed Boundary Method (IBM) has been applied extensively to fluid-structure interaction problems, applications that include multiple fluid phases are relatively sparse. In this presentation, progress towards the development of an IBM coupled with a multliphase Volume-of-Fluid (VoF) solver complete with contact line modelling will be discussed, with special attention given to algorithm development and validation through select model problems.

In their 1967 seminal paper, Foias and Prodi captured precisely a notion of finitely many degrees of freedom in the context of the two-dimensional (2D) incompressible Navier-Stokes equations (NSE). In particular, they proved that if a sufficiently large low-pass filter of the difference of two solutions converge to 0 asymptotically in time, then the corresponding high-pass filter of their difference must also converge to 0 in the long-time limit. In other words, the high modes are "gradually enslaved” by the low modes. One could thus define the number of degrees of freedom to be the smallest number of modes needed to guarantee this convergence for a given flow, insofar as it is represented as a solution to the NSE. This property has since led to several developments in the long-time behavior of solutions to the NSE, particularly to the mathematics of turbulence, but more recently to data assimilation. In this talk, we will give a survey of rigorous studies made for a certain approach to data assimilation that exploits this asymptotic coupling property as a feedback control. We will discuss these issues in the specific context of the 2D NSE, as well as a geophysical equation called the 2D surface quasi-geostrophic equation.

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise interactions and (possibly degenerate) diffusion. In particular, models with hard height constraints (such as pedestrian crowd motion) attract significant interest. We prove that minimizers and gradient flows of constrained interaction energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We then apply this to develop numerical insight for open conjectures in a class of geometric variational problems.

This is a joint project with Katy Craig.

The magnetic equilibrium in axisymmetric fusion reactors can be described in terms of the solution to a semilinear elliptic Dirichlet boundary value problem. The equation is posed in a domain with a piecewise smooth curved boundary. This corresponds to a region of the reactor where the plasma remains confined.

The proposed solution method involves a high order Hybridizable Discontinuous Galerkin (HDG) solver on an unfitted polygonal mesh embedded within the confinement region. The curvature of the domain is handled by a high order transfer scheme that sidesteps the use of curved triangulations or isoparametric mappings. As long as the gap between the computational domain and the true boundary remains of the order of the mesh parameter, this method preserves the order of convergence of the solver.

The transferring algorithm however may fail to resolve sharp gradients close to the boundaries and local refinement becomes necessary. We propose an h-adaptive method that relies on a residual type estimator on the embedded computational mesh. The refinement is driven by a combination of Dorfler marking and the constraint that the gap between the mesh and the curved boundary must remain of the order of the local mesh diameter. This results on a nested sequence of unfitted grids that "grow" towards the physical boundary as refinement progresses.

This is an ongoing collaboration with Antoine Cerfon (NYU) and Manuel Solano (University of Concepcion).

We explore the impact of loss of symmetry in bimetallic Au-Pt rod-like microswimmers. These swimmers are known to exhibit complex individual and collective behaviors. As a proxy for change in swimmer type, e.g. pushers and pullers, we conduct experiments on swimmers with different relative lengths of their two metallic segments. We model the rods' reactive region as a region of fluid slip. Numerical simulations show that a non-centered position of the slip-region along the rod allow for a transition from an extensile to contractile force dipole in the disturbance fluid flow. The changes in the generated flow field, which affect interactions with other rods and boundaries, are here evidenced by the analysis of the swimmers rheotactic abilities and their motion near obstacles.