# Advanced Fluid Dynamics - Material for the year 2019-2020

Basic familiarity with fluid equations and stress tensors as provided, e.g., by Kinetic Theory

### Assessment type:

- Invigilated written examination in TT or no formal assessment; homework completion

Areas: CMT, Astro.

Prequels: Kinetic Theory (MT), an undergraduate course on Fluid Dynamics.

Please submit your homework for Advanced Fluid Dynamics via the 'assignments' tab.

Visiting and Short Option Students Only: Please e-mail your work to your TA/Tutor by the required deadline, copying in mmathphys@maths.ox.ac.uk

Part I. Magnetohydrodynamics (10 lectures)

MHD equations: conservation laws in a conducting fluid; Maxwell stress/magnetic forces; induction equation; Lundquist theorem, flux freezing, amplification of magnetic field. MHD in a strong guide field: MHD waves; high-beta and anisotropic limits and orderings; incompressible MHD, Elsasser MHD, Reduced MHD. Static MHD equilibria, force-free solutions, helicity, Taylor relaxation. Energy principle. Instabilities: interchange, Z-pinch.

Part II. Complex fluids (6 lectures) Fluid mechanics with general extra

stress. Dilute suspension of spheres: Einstein viscosity. Dilute suspension of bead-spring pairs leading to the Oldroyd-B model for polymeric liquids: Elastic waves and normal stress differences (anisotropic pressure). Jeffery's equation for orientable axisymmetric particles. Dilute suspensions of such particles: road map to liquid crystals, swimmers, active matter.