Applied Mathematics Seminars Autumn 2013

All seminars (unless otherwise stated) will take place on Tuesdays at 3.00pm in Room 505 which is located on the 5th floor of the Mathematics Department. See How to Find Us for further details. There will be tea afterwards in Maths Room 606. If you require any more information on the Applied seminars please contact Prof Slava Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

8 October 2013

Dr. Yurij. Semenov - UCL

Integral Hodograph Method for Solving Nonlinear Free Boundary Problems

Abstract:
Historically, a progress in solving problems of two-dimensional free boundary potential flows is based on a development of the theory of complex variable. Since any analytical function meets the requirements of a fluid incompressibility and zero vorticity, the problem is to find such analytical function which satisfies to given boundary conditions. The present talk will be focused on determination of a complex function from its modulus and argument or its real part and argument, which are given on the boundary of a simply connected domain. In combination with Chaplygins singular point method it makes possible to determine expressions for a complex velocity and for a derivative of the complex potential of an arbitrary unsteady free boundary flow. These expressions contain in an explicit form the modulus and argument of the velocity defined as functions of a parameter variable and time. The dynamic and kinematic boundary conditions lead to a system of integral and integro-differential equations for determination of these unknown functions. The proposed method recently been applied for solving steady gravity flows, self-similar water entry problems and time-dependent problems of Hele-Shaw flows with and without surface tension.

15 October 2013

Dr. Ch. Geuzaine - University of Liege, Belgium

Title: A Domain Decomposition Method for the Helmholtz Equation

Abstract:
We present a domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal.  These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions. A preconditioner based on the analytic inversion of the iteration operator is combined with this algorithm, which, for topologically 1-D decompositions, leads to convergence in a fixed number of iterations, independently of the frequency and the number of subdomains.

22 October 2013 - CANCELLED

29 October 2013

Dr. Rebecca Shipley - UCL

Title: Mathematical Models of Transport in Vascular Tissues: From Discrete to Continuum

Abstract:
The vasculature is a 3D multiscale network comprised of a hierarchy of vessels that is frequently categorized according to vessel size. Although the geometry and topology of the vasculature is organ-specific, blood flows into an organ from a feeding artery, through the arterioles into the microcirculation, and exits through the venules then veins. Gas exchange occurs primarily in the microcirculation and, indeed, the function of the vasculature is to bring oxygenated blood within a small distance of every tissue point in the body in order to meet metabolic demands. Modelling the flow of blood and solutes through these networks could play a crucial role in, for example, controlling drug dosage and predicting efficacy, as well as understanding pathological scenarios such as myocardial ischaemia or local tumour microenvironments.

Traditional modelling approaches have employed a discrete approach by solving equations for blood flow in each vessel of a network. However, recent advances in imaging methods have led to a wealth of data that describe vascular structure in a highly detailed way. As the resolution of this data increases, it is becoming too computationally intensive to simulate flow and mass transport in the complete vascular tree using a discrete approach. As such, continuum models must be developed that capture the key functional properties of blood flow.

5 November 2013

READING WEEK - NO SEMINAR

12 November 2013

Dr. Andrew Nachbin - IMPA, Rio de Janeiro, Brasil

Title: Solitary waves in branching channels

Abstract:
The dynamics of solitary water waves is studied in two-dimensional open channels with a branching point. We rationalize the wave characteristics at a branching point by using the Jacobian |J| of  the Schwarz-Christoffel transformation. It is observed that |J| acts in a similar fashion as a topography. Computational  results illustrate the two-dimensional reflection-transmission wave-dynamics at the branching point, which are then compared to a reduced one-dimensional model for solitary waves on a graph/network. An approximate compatibility condition is used at the node of the 1D network. Numerical experiments show that the one-dimensional graph-like model captures well the effective  reflection-transmission properties of the solitary wave. Different regimes are considered though various values of branching angles and channel widths.

19 November 2013

Dr. Zhan Wang - UCL

Title: Focusing phenomenon in three-dimensional capillary-gravity waves

Abstract:
The capillary-gravity wave problem exhibits a large variety of phenomena, one of which is the combination of geometric and nonlinear self-intersection focussing. The underlying mathematical language is the focussing 2+1 nonlinear Schrodinger equation in the small amplitude limit. In this talk, we perform simulation of the water wave equations, using the cubic truncation of the Dirichlet to Neumann operator. Three examples are shown: transverse instability of plane solitary waves which finally evolve into fully localised structures or an intermediate state between plane solitary waves and lumps, super-Gaussian initial data and high-energy solitary waves, both of which break up into a complex set of localised structures.
 

26 November 2013 - CANCELLED

03 December 2013

Prof. Carolina Lithgow-Bertelloni -  UCL

Title: Experimental perspectives on Earth's internal dynamics

Abstract:
The concept of mantle plumes has been one of the most powerful for understanding Earth's geochemical heterogeneity and possible scales of mantle flow. To this day many questions remain about their basic fluid dynamical structure and capacity for stirring and entrainment. I will provide a different perspective, viewing plumes from the laboratory using a unique system for 3-D visualisation and measurement of temperature and velocity and re-examining long-held views about the structure of plumes and our ability to view them seismically.
 

10 December 2013

Dr Philippe H. Trinh - OCIAM, Mathematical Institute, University of Oxford

Title: New gravity-capillary waves at low speeds

Abstract:
When water flows past an obstruction such as a ship or a step in a channel, waves are often produced behind or ahead of the disturbance. Recently, special techniques in asymptotic analysis (exponential asymptotics) have allowed us to explore the regime of low-speed flows, and to predict the theoretical existence of new classes of gravity-capillary waves. These waves have never been seen before -- in nature or in the digital world. Do they truly exist? Come and decide for yourselves! This talk will also introduce the audience to the idea of exponential asymptotics, and the application of such techniques to the study of free-surface flows.