Aspects of fluid dynamics and the fluid/gravity correspondence
This thesis considers various extensions to the fluid/gravity correspondence as well as problems fundamental to the study of fluid dynamics. The fluid/gravity correspondence is a map between the solutions of the Navier-Stokes equations of fluid dynamics and the solutions of the Einstein equations in one higher spatial dimension. This map arose within the context of string theory and holography and is a specific realisation of a much wider class of dualities known as the Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence.
The first chapter is an introduction; the second chapter reviews the fluid/gravity correspondence. The next two chapters extend existing work on the fluid/gravity map. Our first result concerns the fluid/gravity map for forced fluid dynamics in arbitrary spacetime dimensions. Forced fluid flows are of particular interest as they are known to demonstrate turbulent behaviour. For the case of a fluid with a dilaton-dependent forcing term, we present explicit expressions for the dual bulk metric, the fluid dynamical stress tensor and Lagrangian to second order in boundary spacetime derivatives. Our second result concerns fluid flows with multiple anomalous currents in the presence of external electromagnetic fields. It has recently been shown using thermodynamic arguments that the entropy current for such anomalous fluids contains additional first order terms proportional to the vorticity and magnetic field. Using the fluid/gravity map, we replicate this result using gravitational methods.
The final two chapters consider questions related to the equations of fluid dynamics themselves; these chapters do not involve the fluid/gravity correspondence. The first of these chapters is a review of the various constraints that must be satisfied by the transport coefficients. In the final chapter, we derive the constraints obtained by requiring that the equilibrium fluid configurations are linearly stable to small perturbations. The inequalities that we obtain here are slightly weaker than those found by demanding that the divergence of the entropy current is non-negative.