Applied Mathematics Seminars Autumn 2020

Seminars (unless otherwise stated) will take place online on Tuesdays at 3.00pm on Zoom via the link https://ucl.zoom.us/j/99614222402. If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

13 October 2020 at 2pm

Speaker: Marcus Waurick, University of Strathclyde

Title: Dynamic homogenisation problems of highly oscillatory type

Abstract:
In the talk we consider homogenisation problems for equations that rapidly oscillate between hyperbolic, elliptic and/or parabolic type. Mainly focussing on a simplified model for fluid-structure interaction (a coupling between hyperbolic and parabolic equations), we introduce an abstract homogenisation result, which will be applicable in the considered situation. The homogenised fluid-structure interaction model can be computed explicitly and (in dimensions greater than 1) describes a spatially constant integro-partial-differential equation, which is neither hyperbolic nor parabolic.

20 October 2020

Speaker: Karima Khusnutdinova,  Loughborough University

Title: Long surface and internal ring waves in stratified shear flows

Abstract:
In this talk I will first briefly overview some general results concerning the effects of the parallel shear flow on long weakly-nonlinear surface and internal ring waves in a stratified fluid (e.g., oceanic internal waves generated in narrow straits and river-sea interaction zones), generalising the results for surface ring waves in a homogeneous fluid [1]. We showed that there exists a linear modal decomposition (separation of variables) in the far-field set of Euler equations describing the waves in a stratified fluid, more complicated than the known decomposition for plane waves. We used it to describe the wavefronts of surface and internal waves, and to derive a 2D cylindrical Korteweg - de Vries (cKdV)-type model for the amplitudes of the waves [2,3].

Next, we consider a two-layer fluid with a rather general depth-dependent upper-layer current (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions (a 2D linear dispersion relation) in terms of the hypergeometric function [4]. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. We will also discuss a 2D generalisation of the long-wave instability criterion for plane interfacial waves on a piecewise-constant current [4], which on physical level manifests itself in the counter-intuitive squeezing of the wavefront of the interfacial ring wave.

REFERENCES
1. R.S. Johnson, Ring waves on the surface of shear flows: a linear and nonlinear theory, J. Fluid Mech., 215, 1638-1660 (1990).
2. K.R. Khusnutdinova, X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech., 794, 17-44 (2016).
3. K.R. Khusnutdinova, X. Zhang, Nonlinear ring waves in a two-layer fluid, Physica D, 333, 208-221 (2016).
4. K.R. Khusnutdinova, Long internal ring waves in a two-layer fluid with an upper-layer current, Russian J. Earth Sci. 20, ES4006 (2020).
5. L.V. Ovsyannikov, Two-layer shallow water model, J. Appl. Math. Tech. Phys. 20, 127-135 (1979).

27 October 2020

Speaker: Alex Doak, Bath University

Title: New exotic capillary free-surface flows

Abstract:
In this talk, we consider two-dimensional free-surface flows with the inclusion of capillary effects. All the flows considered are characterised by a point where the flow forcibly separates from a solid boundary and forms a free surface. We begin by reviewing previous works on the effects of capillarity on flow separation. In previously considered models, a one branch family of solutions is found. This branch starts from a free streamline (surface tension T=0) solution, and continues to a flat interface as T approaches infinity. We present geometry in which a second branch is found. As one continues along these branches of new solutions, one finds that there exist two new limiting cases for large values of T. These solutions are formed of arcs of circles, straight lines, and a point at which the flow turns near a boundary. This turning point allows for large values of the inertia to balance the large values of surface tension. 

03 November 2020

Speaker: Karima Khusnutdinova,  Loughborough University

Title: Long surface and internal ring waves in stratified shear flows

Abstract:
In this talk I will first briefly overview some general results concerning the effects of the parallel shear flow on long weakly-nonlinear surface and internal ring waves in a stratified fluid (e.g., oceanic internal waves

10 November 2020

NO SEMINAR - READING WEEK

17 November 2020

Speaker: Demetrios Papageorgiou, Imperial College London

Title: Flows in superhydrophobic channels and their stabilityFlows in superhydrophobic channels and their stability

Abstract:
Flows in micro channels structured with grooves that can provide superhydrophobic properties are of wide interest and particularly in thermal management applications for server data banks and other energy intensive cooling systems. Superhydrophobicity helps increase the flux and hence the convective part of cooling.

This presentation will begin with a general study of pressure-driven plows in channels structured with longitudinal grooves. Parallel basic states are considered first. The groove width is small enough to enable a stable meniscus to form that separates the sheared liquid in the main part of the channel with the trapped gas in the grooves (such states are known as Cassie states). If the meniscus is flat or slightly curved analytical progress is possible, but since arbitrary curvatures are required in applications (in fact operational regimes need maximization of the meniscus curvature for a given geometry) we take a computational approach. The methods are spectral and treat the singularities at triple contact points (gas, liquid, solid) in a semi-analytical way to increase efficiency and convergence.

The flow is driven by a constant pressure gradient that in turn induces a non-uniform pressure on the meniscus making the problem fully three-dimensional, in general. In microchannel applications the meniscus curvature varies slowly in the longitudinal direction and we will present an asymptotic solution in this case that captures the primary and secondary flows as well as inertial effects. The latter are shown to have significant impact on resulting flow rates.

Finally, the parallel basic states constructed will be studied for linear instabilities. In contrast to channel flows with smooth walls, new unstable modes are found that are a direct consequence of the two-dimensionality of the basic states and in particular the presence of inflection points due to the underlying spanwise periodicity. These modes coexist with the classical plane Poiseuille flow ones but remain unstable in the inviscid limit. Importantly, for small channel heights they become unstable at order one critical Reynolds numbers and are the main source of instability. This is compatible with experiments. A replacement of the exact basic flow with a hydrodynamic slip model would completely miss the instability, and so caution must be exercised.

24 November 2020

Speaker: Graham Benham (Cambridge University)

Title: Whatever floats your boat: some maths problems in sports

Abstract:

In this talk I will present several topics related to the maths of rowing and cycling sports:

Firstly, a century old paradox is revisited: how does one consistently describe the wave drag on an asymmetric boat? We demonstrate that introducing a boundary layer to the potential-flow formulation of J.H. Michell (1898) is sufficient to break the symmetry of motion, recovering close comparison with experimental observations.

Secondly, motivated by recent Olympic records in rowing race courses of different depths, we reveal a possible strategy for exploiting shallow-water wave resonance at certain moments during a race to boost overall performance. We trace out bifurcation diagrams, and compare with real parameter values, discussing the implications for boat races, as well as a possible connection to the famous dead-water problem.

Finally, on the topic of track-cycling, we re-interpret the classic Brachistochrone problem first proposed by Bernoulli in 1697, to calculate the shortest descent trajectory of a cyclist on the sloped surface of a velodrome. Some new analytical solutions are derived, and comparisons with real cyclist trajectories are made.

01 December 2020

Speaker: Artur L Gower (University of Sheffield)

Title: Ensemble average waves in random materials of any geometry

Abstract:
How do you take a reliable measurement of a material whose microstructure is random? When using wave scattering, the answer is often to take an ensemble average (average over time or space). By ensemble averaging we can calculate the average scattered wave and the effective wavenumber. Over the past 70 years, methods have been developed to calculate the average wave scattering, and effective wavenumber, for a plate filled with particles. One clear unanswered question was how to extend this approach to a material of any geometry and for any source? For example, does the effective wavenumber depend on only the microstructure, or also on the material geometry? In this talk, I'll demonstrate that the effective wavenumbers depend on only the microstructure and not the geometry, and further that beyond the long wavelength limit there are multiple effective wavenumbers. As an example, I will show how to calculate the average wave scattered from a sphere filled with particles.

08 December 2020

Speaker: Alexander Korobkin (University of East Anglia)

Title: Vertical impact onto floating ice 

Abstract:
The vertical impact of a rigid body onto a floating ice plate is investigated. The problem is unsteady, two-dimensional and coupled. The hydrodynamic loads and elastic ice response are determined simultaneously. The ice deflection is described by the Euler beam equation. The impact problem is regularized by introducing a thin viscoelastic layer between the impacting body and the elastic ice plate. The local reaction force in the viscoelastic layer is a given function of the local compression of the layer. The impact loads and the regions of contact between the impacting body and the soft layer are calculated together with the deflections and strains of the ice plate.  It is shown that the elastic response of the floating plate is weakly dependent on the thickness and rigidity of the soft layer on the top of the floating plate. 

15 December 2020

Speaker: Marianne Odlyha, (Birkbeck University)

Title: Advanced analytical techniques for the preservation of organic based cultural heritage

Abstract:
The talk will include examples of some of our past studies where we were involved in monitoring environmental conditions in museums and historical houses and assessing  the impact of inappropriate conditions on the state of preservation of organic based heritage objects and then examples of our current work. This has focused on evaluating effects of novel conservation treatment on objects, and in particular nanocellulose-based treatment for consolidation of  painting canvases. In addition to mechanical and dielectric studies neutron radiography was used to test the response of the treated canvases by subjecting them to programmed cycles of moisture sorption and desorption. Some nanomechanical studies of these samples will also be presented This work is part of a collaboration with paintings conservators at the Royal Danish Academy of Architecture, Design, and Conservation. Reference will also be made to a  preliminary study on of nanomechanics of varnished wood samples. This indicated differences due to the different materials and methods of varnish preparation used for violin varnishes. Some examples of our collaborative work with English Heritage will also be included e.g damage assessment of leather bookbindings and our current project dealing with evaluation of state of preservation of  their collection of archaeological bones.