# 2018-2019 Intensive courses

**Theory of the Hydraulic Jump**

by Dr Bernhard Scheichl, Institute of Fluid Mechanics and Heat Transfer, Technische Universität Wien, Vienna, Austria

Starts **14 January** (9am to 1pm), ends **15 January** (9am to 1pm) / Venue: 20 Bedford Way, UCL

The hydraulic jump represents a ubiquitous phenomenon in shallow free-surface flows of viscous (here Newtonian) fluids over a rigid wall, orientated nominally perpendicular to the direction of gravity, its essential source.

Most easily produced along a plate held under the open tap of a kitchen sink, it is widely unwelcome in more complex flow situations in technical applications as associated with flow reversal, thus dissipative losses and e.g. hampering the desired removal of particles from an engineering surface.

It is introduced in its classical sense as a discontinuity of the flow quantities or a weak solution of the governing equations in the hydraulic (inviscid) limit. As intrinsically tied in with streamline curvature, however, this often stressed analogy with gasdynamical shocks, viz the Froude number playing the role of the Mach number, forms a conceptual shortcoming. A local description of the jump phenomenon shows that this is surmounted rigorously by the identification of viscous–inviscid flow interaction that facilitates the required bifurcation process.

Despite the substantial progress made in this direction, the upstream influence, equally necessary for a conclusive understanding, has only been coped with satisfactorily for a weak jump in accordingly weakly viscous flow over an accordingly reduced streamwise length. These findings of the last decades are reappraised, and an account of very recent ones is given. Specifically, the upstream influence provoking a jump on larger scales in practical situations, such as in developed flow over a rotating disc with finite radius, and the impact of capillarity are discussed.

- The course addresses PhD students confronted with mathematical modelling of real world problems, but not necessarily in fluid mechanics.
- Requirements for students attending: typical mathematical skills; specifically, solution methods for ODEs/PDEs; basic knowledge in fluid mechanics expected.
- Learning objectives: training in mathematical modelling by starting with inspection and dimensional analysis; identifying key effects to properly set up a realistic (least-degenerate) problem formulation; application and peculiarities of asymptotic concepts and methods, specifically matched asymptotic expansions; deepening understanding of and getting in touch with frontiers of research in theoretical fluid mechanics.

# Leverhulme Lectures: Non-local Equations and Conformal Geometry

by Professor Mariel Saez Trumper, Pontificia Universidad Católica de Chile.

Starts **15 January** (1pm to 5pm), ends **16 January** (9am to 1pm) / Venue: Chadwick Building, UCL

This intensive course will include the following topics:

- Asymptotically hyperbolic manifolds and conformally covariant operators. Conformal invariants.
- The fractional laplacian in R^n.
- The conformal fractional laplacian on asymptotically hyperbolic manifolds.
- Fractional curvatures and the fractional Yamabe problem

Conformal geometry can be understood as the study of angle preserving transformations in a space and appears naturally in several branches of physics. In particular, on a Riemannian manifold the conformal property may be described by an equivalence relation among possible metrics.

Curvature properties related to conformal classes of metrics are often described through partial differential equations and they have been proven to be deeply related to the underlaying topology.

Independently, in recent years, the PDE community has devoted attention to the study of the so-called non-local pseudo-differential operators.

In this lecture series we aim to describe connections between these two fields and some interesting open problems.