Minimal models for finite particles in fluctuating hydrodynamics

Zusammenfassung

This thesis is devoted to the development of efficient numerical solvers for fluctuating hydrodynamics, in particular, for flows with immersed particles. In the first part of the thesis we develop numerical solvers able to work in a broad number of flow regimes with a high computational performance. To derive thermodynamically consistent set of equations in the continuum and discrete settings we have made use of the augmented Langevin formalism. This formalism predicts the form of the stochastic and thermal drift terms which should be included to satisfy the fluctuation dissipation balance. In this work the dynamic of the fluid is described by the fluctuating Navier-Stokes equations which we discretize with a finite volume method in a staggered grid. We found the staggered grid far superior to collocated grids in ensuring numerical stability for the solvers. Several numerical solvers have been developed to simulate both compressible and incompressible flows, in some cases allowing to use very large time steps thanks to the use of semi-implicit discretizations. The fluid solvers have been extended to solve the dynamics of solute particles. We have developed minimal resolution particle models based on the immersed boundary method. The fluid-particle coupling is based on a no-slip constraint which allows to capture the hydrodynamic forces from the Brownian limit to turbulence and even fast oscillating acoustic flows. All the solvers have been implemented in CUDA to run efficiently on graphical processing units (GPUs) and made available under a free software licence. In the second part of the thesis we have applied our solvers to some interesting non-trivial problems. A common characteristic of all the problems considered is the high disparity of time scales involved; only by using efficient solvers and minimal models like the ones we derived in the first part of the thesis it is possible to tackle these problems. We have studied giant concentration fluctuations whose characteristic times spans over more than 6 orders of magnitude, proving that approximate Galerkin theories though very useful present discrepancies with experimental results of up to a 50%. Acoustic forces have also been studied, showing a good agreement between numerical results and the Gor'kov predictions. More interesting, we have found that the spatial distribution for a particle immersed in an acoustic potential is given by the Gibbs-Boltzmann distributions, a result that can be explained by the time scale separation between the sound modes and the particle diffusion. As a last application we studied the polymer tumbling under shear flow. This problem is better approached by specialized solvers for the infinite Schmidt number limit, which have also been derived. We found that the dynamic of the polymer in shear flow can be described by three characteristic times related with the flow strength and the polymer interactions.