# MA3D1 Fluid Dynamics

Lecturer: Shreyas Mandre

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• Multivariate scalar and vector functions
• Differential identities
• Integral theorems
• Ability to perform line, surface and volumetric integrals

MA250 Introduction to Partial Differential Equations:

• Derivation and solution of various differential equations as applied to fluid dynamics, most notably Laplace equations
• Heat equation
• Understanding of appropriate boundary conditions that accompany the equations
• Methods of solution, including separation of variables and fundamental solution

Useful background:

• Cauchy-Riemann conditions
• Holonomic functions
• Complex integration
• Conformal maps

Synergies: Those who enjoy fluid dynamics may be interested in the following modules:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

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:

• Mathematical modelling of fluid flow: 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
• Additional conservation laws: Bernoulli's equations, Global conservation laws
• Vortex dynamics: Kelvin's circulation theorem, Helmholtz theorems, Cauchy-Lagrange theorem, 3D vorticity equation, vortex lines, vortex tubes and vortex stretching
• 2D flows: Flow in a pipe, Shear flows, jet flow by similarity solution, round vortices
• Irrotational 2D flows & classical aerofoil theory: Complex analysis methods in Fluid Dynamics, Blasius theorem, Zhukovskii lift theorem, force on a cylinder and force on an aerofoil
• Rotating flows: Navier-Stokes in a rotating frame, Rossby number, Taylor-Proudman theorem, geophysical flow
• Boundary layers: Prandtl's boundary layer theory, Ekman boundary layer in rotating fluids
• Waves: General theory of waves, sound waves, free-surface flows and surface tension, gravity-capillary water waves
• Instabilities: Rayleigh criterion, Orr-Sommerfeld equation, Kelvin-Helmholtz instability, stability of parallel flows

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)