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# Department of Mathematical Sciences

# MATH3101/4081 Fluid Mechanics III/IV

From our cytoplasm to climate dynamics to exploding stars, we are surrounded (indeed made up of) fluids. In this course we will study the equations modelling the dynamics of fluids as continuous media.

We start with a general kinematic description of fluid flow, followed by the Euler equations governing the motion of an ideal fluid (i.e. one with no internal friction). After deriving the equations, we will show how the behaviour of the Euler equations can be understood through the important concept of "vorticity". This is essentially the local rotation of a fluid, and we will show how it relates to phenomena such as whirlpools and tornadoes, as well as to the long-term existence of solutions to the Euler equations.

An important class of solutions to the fluid equations are waves: in this course we will consider both sound and water wave solutions. The mathematics of waves is similar to that of "stability" of fluid flows, which is also an important tool for understanding the behaviour of fluids in the real world.

Finally, we will derive the more realistic Navier-Stokes equations that take viscosity into account. Despite their fundamental importance for the natural world, and the presence of viscosity as a dissipative (smoothing) mechanism, the well-posedness of these equations remains a famously unsolved problem in mathematics.

## Outline of Course

Aim: To introduce a mathematical description of fluid flow and other continuous media to familiarise students with the successful applications of mathematics in this area of modelling. To prepare students for future study of advanced topics.

### Term 1

• Kinematics: continuum hypothesis, velocity field, material derivatives, particle paths and streamlines, compressibility.
• Dynamics: derivation of the Euler equations, vorticity and circulation, Kelvin's theorem, nonlinearity.
• Compressible flows: sound waves.

### Term 2

• Water waves: surface gravity waves, shallow water equations.
• Hydrodynamic stability: Kelvin-Helmholtz instability, astrophysical applications.
• Navier-Stokes equations: boundary layers, scaling and the Reynolds number, very viscous flows.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook: MATH3101, MATH4081.