A universal framework for hydrodynamics

In this thesis, we present a universal framework for hydrodynamics starting from the fundamental considerations of symmetries and the second law of thermodynamics, while allowing for additional gapless modes in the low-energy spectrum. Examples of such fluids include superfluids and fluids with surfaces. Typically, additional dynamical modes in hydrodynamics also need to be supplied with their own equations of motion by hand, like the Josephson equation for superfluids and the Young-Laplace equation for fluid surfaces. However, we argue that these equations can be derived within the hydrodynamic framework by a careful off-shell generalisation of the second law. This potentially provides a universal framework for a large class of hydrodynamic theories, based on their underlying symmetries and gapless modes. Motivated by this newly found universality, we present an all-order analysis of the second law of thermodynamics and propose a classification scheme for the allowed hydrodynamic transport, including arbitrary gapless modes, independent spin current, and background torsion. In the second half of this thesis, we look at the construction of null fluids which are a new viewpoint of Galilean fluids. These are essentially fluids coupled to spacetime backgrounds carrying a covariantly constant null isometry, but with additional constraints imposed on the background gauge field and affine connection to reproduce the correct Galilean degrees of freedom. We discuss the Galilean version of quantum anomalies and their effect on hydrodynamics. Finally, we follow our relativistic discussion to allow for arbitrary gapless modes in Galilean hydrodynamics and present a classification scheme for the second law abiding hydrodynamic transport at all orders in the derivative expansion. We apply these abstract ideas to review the theory of ordinary relativistic/Galilean hydrodynamics and provide novel constructions for relativistic/Galilean (non-Abelian) superfluid dynamics and surface transport. We also comment on the possible application to the theory of magnetohydrodynamics.