MPhys Physics with Theoretical Physics / Course details

A. Introduction to fluid mechanics (13 lectures)

1. Basic concepts and governing equations of fluids (4 lectures) Fluids as continua; streamlines and pathlines; conservation of mass and the equation of continuity; rate of change following the fluid; conservation of momentum and the stress tensor; the constitutive equations and the Navier-Stokes equations.

2. Dynamical similarity and the Reynolds number (1 lecture) Dynamical similarity and the Reynolds number; scaling of the Navier-Stokes equations.

3. Unidirectional flows (2.5 lectures) Boundary conditions for viscous flow; unidirectional flows in two dimensions; Poiseuille and Couette flow; Poiseuille flow in a tube; unsteady flow.

4.  Viscous flows (2 lectures) Stokes flow past a sphere; flow reversibility; swimming at low Reynolds number.

5. Inviscid flows (2.5 lectures) Euler and Bernoulli’s equations; vorticity and its physical meaning; Kelvin’s theorem; potential flow; the stream function; 2D irrotational flows; lift force; flow around aerofoils.

6. Boundary layers and instability (1 lecture)

B. Phase transitions from classical to quantum (10 lectures)

1. Introduction: recap on forces and bonding, order of a phase transition, critical exponents, concept of universality, critical point. (2 lectures)

2. van der Waals theory: London, Hamaker and modern dispersion theory, equation of state, phase transition. (1 lecture)

3. Electric double layer: Gibbs adsorption equation, Debye-Huckel and modern theories (1 lectures)

4. Landau theory: First and second order phase transition, phase coexistence, Gibbs dividing surface. (2 lectures)

5. Superfluids: Liquid 4He, properties of superfluid 4He, Landau theory of the superfluid phase transition, role of fluctuations, BKT phase transition for vortices. (3 lectures)

6. Other phase transitions: water and ice. (1 lectures)

This unit introduces fundamental concepts in two key areas of physics: fluid mechanics and phase transitions. These topics recur in many areas of modern physics research beyond Condensed Matter Physics, and the unit equips students with the knowledge and analytical tools to understand their significance. The fluid mechanics component of this unit focuses on the fundamental principles governing flow behaviour of Newtonian fluids across scales, with applications from astrophysics and environmental science to quantum fluids. The course will introduce the continuum mechanics framework, which models fluids at a macroscopic level, and delve into key concepts such as viscous and inviscid flows. The unit emphasizes both the mathematical complexity of fluid mechanics and its wide-ranging applications across disciplines. The unit also explores the emergence of order in simple models of interacting systems, offering insights into how complex behaviours arise from basic principles. Through this course, students will develop a deeper appreciation of the underlying physical mechanisms that govern phase transitions and their applications across diverse scientific disciplines.

On the successful completion of the course, students will be able to:  

ILO 1

Describe and use key concepts in fluid dynamics to solve the Navier-Stokes equations in specific scenarios.

ILO 2

Apply key concepts to the viscous limit, such as Stokes settling and inertialess swimming, and to the inviscid limit, such as Bernoulli's equations, vorticity, irrotational flow and lift force.

ILO 3

Apply mean-field theories to describe phase transitions in simple interacting models.

ILO 4

Use Landau theory to describe phase transitions in condensed matter.

Two one hour, live in-person lectures per week where the core material will be delivered with examples. The recordings of these lectures will be available on Podcast and linked to the course online page. The lectures are accompanied by lecture notes and for some of the material explanatory videos. This is augmented by a set of weekly problems with solutions. Problems will also be released for example classes, where students will be able to work on them together for three hours over the course of the semester. A Piazza discussion forum is also provided where students can ask questions with answers provided by other students and the unit lead. 

Guyon E, Hulin J-P, Petit L. and Mitescu C.D., Physical hydrodynamics, (OUP)

Steven H. Simon, The Oxford Solid State Basics (OUP)

Acheson, D.J. Elementary Fluid Dynamics, (OUP)

Davidson, P. Incompressible Fluid Dynamics (OUP)

J. M. Yeomans, Statistical Mechanics of Phase Transitions (Clarendon Press)

H. Nishimori, G. Ortiz, Elements of Phase Transitions and Critical Phenomena (Oxford Graduate Texts)