L3

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane. 

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier

Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

Most sensing systems exhibit so-called ‘sidelobe’ responses, which can be interpreted as an inevitable effect in one domain of truncation of the signal in the Fourier-complement domain.  Perhaps the best-known example is in antenna theory where sidelobes are an inevitable consequence of the fact that the antenna aperture must be finite.  The effect also appears in many other places, for example in time-frequency conversions and in the range domain of a pulse-compressed radar which radiates a signal only over a finite frequency band.  In the range domain these sidelobes extend over twice the length of the transmitted pulse.  For a conventional radar with relatively short pulses the effect of these unwanted returns is thus confined to a relatively short part of the range swathe.

Some of the most modern radar techniques, however, use continuous, noise-like transmissions.  ‘Primary’ noise-modulated radars are in their infancy but so-called ‘Passive’ radars using broadcast transmissions as their power source receive similar signals.  The sidelobes of even a small target at very short range can be larger than the main return from a target at much greater range.  This limits the dynamic range of the radar.

Since, however, the sidelobe pattern is predictable if the illuminating signal is known sufficiently accurately, the expected sidelobes due to a large target can be estimated and removed to tidy up the image.  This approach was first described formally in:

Hoegbom, J. A., ‘Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines,’ Astrom. Astrophys. Suppl. 15, pp417-26, 1974.

And is generally known by the name of the ‘CLEAN’ algorithm.

The seminar will outline the problem, outline the basic form of the algorithm and ask questions about what is possible with non-iterative versions of the algorithms, how to process the data coherently and how to understand any stability issues associated with the algorithm.

Abstract: Motivated loosely by the problem of carbon sequestration in underground aquifers, I will describe computations and analysis of one-sided two-dimensional convection of a solute in a fluid-saturated porous medium, focusing on the case in which the solute decays via a chemical reaction.   Scaling properties of the flow at high Rayleigh number are established and rationalized through an asymptotic model, that addresses the transient stability of a near-surface boundary layer and the structure of slender plumes that form beneath.  The boundary layer is shown to restrict the rate of solute transport to deep domains.  Knowledge of the plume structure enables slow erosion of the substrate of the reaction to be described in terms of a simplified free boundary problem.

Co-authors: KA Cliffe, H Power, DS Riley, TJ Ward

1. A Hybrid Monte-Carlo Partial Differential Solver for Stochastic  Volatility Models (Cozma)

In finance, Monte-Carlo and Finite Difference methods are the most popular approaches for pricing options. If the underlying asset is modeled by a multidimensional system of stochastic differential equations, an analytic solution is rarely available and working under a given computational budget comes at the cost of accuracy. The mixed Monte-Carlo partial differential solver introduced by Loeper and Pironneau (2009) is one way to overcome this issue and we investigate it thoroughly for a number of stochastic volatility models. Our main concern is to provide a rigorous mathematical proof of the convergence of the hybrid method under different frameworks, which in turn justifies the use of Monte-Carlo simulations to compute the expected discounted payoff of the financial derivative. Then, we carry out a quantitative assessment based on a European call option by comparison with alternative numerical methods.

2. tbc (Brackmann)