# MA3D1 Fluid Dynamics

**Lecturer: **Shreyas Mandre

**Term(s):** Term 2

**Status for Mathematics students:** List A

**Commitment:** 30 lectures

**Assessment:** 3 hour exam

**Prerequisites:** MA231 Vector Analysis and MA250 PDEs.

MA3B8 Complex Analysis is desirable.

**Leads To: **

**Content**: The lectures will provide a solid background in the mathematical description of fluid dynamics. They will cover the derivation of the conservation laws (mass, momentum,energy) that describe the dynamics of fluids and their application to a remarkable range of phenomena including water waves, sound propagation, atmospheric dynamics and aerodynamics. The focus will be on deriving approximate expressions using (usually) known mathematical techniques that yield analytic (as opposed to computational) solutions.

The module will cover the following topics:

Specification of the flow by field variables; vorticity; stream function; strain tensor; stress tensor. Euler's equation. Navier-Stokes equation. Introduction of non-dimensional parameters*Mathematical modelling of fluid flow.*Bernoulli's equations. Global conservation laws.**Additional conservation laws**.Kelvin's circulation theorem. Helmholtz theorems. Cauchy-Lagrange theorem. 3D vorticity equation, vortex lines, vortex tubes and vortex stretching.**Vortex dynamics**.. Flow in a pipe. Shear flows. Jet flow by similarity solution. Round vortices.**2D flows**Complex analysis methods in Fluid Dynamics. Blasius theorem. Zhukovskii lift theorem. Force on a cylinder and force on an aerofoil.**Irrotational 2D flows & classical aerofoil theory**.Navier-Stokes in a rotating frame. Rossby number. Taylor-Proudman theorem. Geophysical flow.**Rotating flows**.Prandtl's boundary layer theory. Ekman boundary layer in rotating fluids.*Boundary layers.*General theory of waves. Sound waves. Free-surface flows & surface tension. Gravity-capillary water waves.**Waves**.

Rayleigh criterion. Orr-Sommerfeld equation. Kelvin-Helmholtz instability. Stability of parallel flows.*Instabilities.*

**Aims**:

An important aim of the module is to provide an appreciation of the complexities and beauty of fluid motion. This will be highlighted in class using videos of the phenomena under consideration (usually available on YouTube).

**Objectives**: It is expected that by the end of this module students will be able to:

- be able to understand the derivation of the equations of fluid dynamics

- master a range of mathematical techniques that enable the approximate solution to the aforementioned equations

- be able to interpret the meanings of these solutions in 'real life' problems

**Strongly recommended texts:
**D.J. Acheson, Elementary Fluid Dynamics, OUP. (Excellent text with derivations, examples and solutions)

S. Nazarenko,

*Fluid Dynamics via Examples and Solutions*, Taylor and Francis. (Great source of questions and detailed solutions.)

**Further Reading:**

A.R. Paterson, *A First Course in Fluid Dynamics*, CUP. (Easier than Acheson.)

L.D. Landau and E.M. Livshitz, *Fluid Mechanics*, OUP. (A classic for those with a deep interest in fluid dynamics in modern physics.)

D.J. Tritton, Physical Fluid Dynamics , Oxford Science Publs. (The emphasis is on the physical phenomena and less on the mathematics.)