Aspects of Acoustic Geometry in Inviscid and Slightly Viscous Fluids

Abstract

Analogies are one of the most powerful and fundamental tools often used in science to explain various difficult and often unrelated concepts. We explore the acoustic analogy by analysing the behaviour of barotropic and irrotational fluids under the effect of first- order acoustic perturbations. This thesis is divided into two chapters and an appendix. The first chapter entitled, ”Acoustic Geometry in Slightly Viscous Fluids” investigates the applicability of analogue gravity concepts in slightly viscous fluids by exploring the propagation of first order acoustic perturbation in this medium. We first introduce the principles of acoustic analogue gravity in inviscid fluid and explain its advantages in the context of exploring acoustic analogues of various fixed metric astrophysical phenomena. We then introduce viscosity in this inviscid system and examine the difficulties in deriving an acoustic metric in this case. Two approaches that were taken to address this problem namely, k-essence and Double Perturbation are introduced. We examine the approach based on k-essence and identify the lacunae of this approach towards achieving our final objective. The formalism of the double perturbation approach is then introduced and developed. A perturbation to the inviscid acoustic metric is calculated, using which a viscosity perturbed metric tensor is obtained. We further explore the properties of this metric tensor and discuss the issues associated with exploring acoustic analogues of various fixed metric astrophysical phenomena in viscous fluids. We then conclude by giving a summary of the obtained results and the future work that needs to be done to expand upon these results. The second chapter entitled, ”Irreducible Mass in Acoustic Analogue Gravity” explores the existence of an irreducible mass analogous to what we have in the case of Kerr black holes. The phenomena of Penrose process is introduced and the energy extraction from a rotating black hole due to this process is mathematically explained. Its wave analogue called ”Superradiant Scattering” is then introduced and the conditions under which a process like is possible are explained. We then explore this phenomena in a Draining Sink type flow and derive an expression for the irreducible mass which is shown to be non-decreasing. We also show that the perimeter of the acoustic horizon is also non-decreasing thus proving the acoustic analogue of Hawking’s area theorem. We then discuss these results in the context of General Relativity. We conclude by summarising the results and also discuss the work that is required to further explore these results. The appendix contains a research paper, entitled “Towards an Acoustic Geometry in Slightly Viscous Fluids”, published in the journal “Universe” based on the findings mentioned in Chapter 1.