Dissertation Defence: A lattice-Fokker-Planck model of thermal noise in non-ideal liquids

Kasper Juel Petersen, supervised by Dr. Joshua Brinkerhoff, will defend their dissertation titled “A lattice-Fokker-Planck model of thermal noise in non-ideal liquids” in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering.

An abstract for Kasper Juel Petersen’s dissertation is included below.

Examinations are open to all members of the campus community as well as the general public. Registration is not required for in person defences.

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ABSTRACT

Boiling and cavitation–the phenomena which describe thermally- and pressure-driven, first-order phase transitions of liquids to their gaseous states–are central to a myriad of industrial, natural, and medical flows. As an example of boiling, “rapid phase transitions” can cause detrimental vapour explosions in paper and metal processing, nuclear reactors, and cryogenics technologies when superheated volatile liquids become perturbed out of their equilibrium condensed state. In particular, phase-transitions are (i) metastable, instigated by (ii) thermal fluctuations, and are (iii) multiscale, (iv) heterogeneous processes; sub-micron vapour bubbles form preferentially at microscopic nucleation sites, and are fully grown on the macroscale (cm). From two comprehensive literature reviews of phase-change modelling and simulation, it was found that (i-iv) are rarely considered in engineering numerical simulations except for (i,ii) in fluctuating hydrodynamics theories computationally solved with the finite-volume method. Rather, the physics are most frequently modelled by empirical models, leaving little opportunity for peering into the realistic dynamics of phase-changing flows. Recently, the lattice-Boltzmann method (LBM) has become a popular choice for studying thermal fluctuations, specifically in the fluctuating LBM (FLBM). However, similar to fluctuating hydrodynamics, the FLBM relies on advanced sampling techniques of probability distributions for imposing thermal noise in to the simulation variables. In an alternative effort, the current research revolves around modelling thermal fluctuations by the stochastic Fokker-Planck equation (FPE) where lattice-projection techniques from the LBM are repurposed to solve the FPE on the lattice. To prospectively introduce metastability and accommodate a high level of thermo-hydrodynamic non-ideality into the model, its kinetics are recast with Particles-on-Demand—a recently proposed semi-Lagrangian realization of the LBM. The primary contribution in this dissertation is the mathematical derivation and interpretation of the resulting lattice-Fokker-Planck-Boltzmann model. Asymptotic analysis shows that the limit of the lattice model is the stochastic Navier-Stokes-Fourier equations with a stochastic stress tensor analogous to the deterministic equivalent derived on the basis of the Boltzmann equation. In addition, an initial fluctuation-dissipation theorem was formulated indicating equivalent behaviour to the FLBM. Initial numerical tests of the model are reported and show promise in computing thermal fluctuations evolved from the initial conditions in simulations, without the need for ad-hoc stochastic sampling. The insights are preliminary and more development is required. Nevertheless, the research opens the door to intriguing numerical, theoretical, and computational results via the application of the Fokker-Planck equation, which has not yet been employed to investigate phase-transitioning liquids.