Tues AM, HT weeks 1-8.
Lectures 9:00-10:00 and 11:15-12:15 Peter Howell
Supervisions 1:30-2:30 Rob Style
Assessment method: Presentations at the end of the course
This course introduces students to advanced theoretical methods for building, simplifying and analysing mathematical models based on differential equations. Practical industrial problems will be use to motivate each of the models and techniques. Many industrial processes involve free boundary problems, where the shape of the region in which the equations are posed must be determined as part of the solution. These will be motivated and studied using real- world examples including welding and electropainting. Perturbation methods are key to analysing and simplifying such problems. In examples such as engine bearings, coating flows and optical fibre drawing, the geometry is thin and geometrical reduction is possible using lubrication theory. On the other hand, problems such as filtration and bubbles in liquid glass involve thin obstacles or inclusions which can be analysed using slender body theory. The study of thin elastic objects such as beams, rods and plates will be motivated by industrial processes involving rope, hair, cables and disk drives. Finally, models involving nonlinear waves and friction will be motivated by problems including river flow and high-speed trains in tunnels.
Stefan and Hele-Shaw free boundary problems and generalisations. Stability, regularisation and mushy regions. Co-dimension 2 free boundary problems. Examples including welding and electropainting.
Lubrication theory: bearings, coating flows, contact line movement. Extensional flows: fibre drawing and extrusion.
Slender body theory, both inviscid and viscous. Examples including flow past porous cylinders, deformation of slender bubbles.
Mechanics of elastic beams, rods, plates. Examples including rope, hair, cables, disk drives.
One-dimensional gas dynamics and shallow water theory. Friction laws, Fanno flow. Examples including river flow, roll waves, high-speed trains.