Finite element formulations for nonequilibrium transport problems using fluctuating hydrodynamics

Nonequilibrium transport at micro- and mesoscopic scales plays a central role in many modern engineering systems, making accurate modeling of these processes essential. At such small scales, thermal fluctuations can significantly influence transport behavior and thus their effect must be taken into account by transport models. Fluctuating hydrodynamics provides a systematic framework for incorporating thermal noise into the conservation laws of mass, momentum, and energy, leading to stochastic transport equations.

In this seminar, we discuss engineering applications where this modeling framework is especially relevant, and we present finite element formulations for nonequilibrium transport based on fluctuating hydrodynamics. In particular, we focus on how to address the challenge of solving transport problems accurately when the solution has both long-ranged and short-ranged (delta-correlated) correlations, a common feature of nonequilibrium transport phenomena in engineering contexts. To illustrate this, we analyze the solution of a nonhomogeneous Fourier equation and demonstrate that decoupling local and non-local fluctuations is essential for achieving convergence. Finally, as a practical application, we present a finite element study of the rotational dynamics of an active colloid in a complex fluid. These, simulations are based on an advection-diffusion model with thermal fluctuations are conducted in a two-dimensional domain. The results indicate that this numerical approach successfully captures the key features observed in experimental studies of active colloids in polymer solutions.