Field theoretic viewpoints on certain fluid mechanical phenomena

TL;DR

This thesis explores the analogy between gauge theories and fluid mechanics, introduces a regularized model for gas dynamics that admits dispersive waves and solitons, and connects it to nonlinear Schrödinger equations, offering new insights into fluid phenomena.

Contribution

It establishes a novel analogy between Higgs mechanism and fluid added-mass effects, and develops a minimal conservative regularization for 3D gas flow that generalizes known nonlinear wave equations.

Findings

Physical analogy between gauge symmetry breaking and fluid added-mass tensor

Regularized 3D gas flow model admits dispersive waves and solitons

Numerical solutions show recurrence and phase-shift scattering in 1D for specific parameters

Abstract

In this thesis we study field theoretic viewpoints on certain fluid mechanical phenomena. In the Higgs mechanism, the weak gauge bosons acquire masses by interacting with a scalar field, leading to a vector boson mass matrix. On the other hand, a rigid body accelerated through an inviscid, incompressible and irrotational fluid feels an opposing force linear in its acceleration, via an added-mass tensor. We uncover a physical analogy between these effects and propose a dictionary relating them. The correspondence turns the gauge Lie algebra into the space of directions in which the body can move, encodes the pattern of symmetry breaking in the shape of an associated body and relates symmetries of the body to those of the vacuum manifold. The new viewpoint raises interesting questions, notably on the fluid analogs of the broken symmetry and Higgs particle. Ideal gas dynamics can develop shock-like singularities which are typically regularized through viscosity. In 1d, discontinuities can also be dispersively smoothed. In the 2nd part, we develop a minimal conservative regularization of 3d adiabatic flow of a gas with exponent $\gamma$, by adding a capillarity energy $\beta_* (\nabla \rho)^2/\rho$ to the Hamiltonian. This leads to a nonlinear force with 3 derivatives of $\rho$, while preserving the conservation laws of mass and entropy. Our model admits dispersive sound, solitary & periodic traveling waves, but no steady continuous shock-like solutions. Nevertheless, in 1d, for $\gamma = 2$, numerical solutions in periodic domains show recurrence & avoidance of gradient catastrophes via solitons with phase-shift scattering. This is explained via an equivalence of our model (for homentropic potential flow in any dimension) with a defocussing nonlinear Schr\"odinger equation (NLS, cubic for $\gamma = 2$). Thus, our model generalizes KdV & NLS to adiabatic gas flow in any dimension.