307 research outputs found
The micropolar Navier-Stokes equations: A priori error analysis
The unsteady Micropolar Navier-Stokes Equations (MNSE) are a system of
parabolic partial differential equations coupling linear velocity and pressure
with angular velocity: material particles have both translational and
rotational degrees of freedom. We propose and analyze a first order
semi-implicit fully-discrete scheme for the MNSE, which decouples the
computation of the linear and angular velocities, is unconditionally stable and
delivers optimal convergence rates under assumptions analogous to those used
for the Navier-Stokes equations. With the help of our scheme we explore some
qualitative properties of the MNSE related to ferrofluid manipulation and
pumping. Finally, we propose a second order scheme and show that it is almost
unconditionally stable
Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries
We develop stochastic mixed finite element methods for spatially adaptive
simulations of fluid-structure interactions when subject to thermal
fluctuations. To account for thermal fluctuations, we introduce a discrete
fluctuation-dissipation balance condition to develop compatible stochastic
driving fields for our discretization. We perform analysis that shows our
condition is sufficient to ensure results consistent with statistical
mechanics. We show the Gibbs-Boltzmann distribution is invariant under the
stochastic dynamics of the semi-discretization. To generate efficiently the
required stochastic driving fields, we develop a Gibbs sampler based on
iterative methods and multigrid to generate fields with computational
complexity. Our stochastic methods provide an alternative to uniform
discretizations on periodic domains that rely on Fast Fourier Transforms. To
demonstrate in practice our stochastic computational methods, we investigate
within channel geometries having internal obstacles and no-slip walls how the
mobility/diffusivity of particles depends on location. Our methods extend the
applicability of fluctuating hydrodynamic approaches by allowing for spatially
adaptive resolution of the mechanics and for domains that have complex
geometries relevant in many applications
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