PDE Theme Day

Our second theme day of this academic year will be the **PDE Theme Day** on the **9th March**. This event will take place in **Room 130 of the Huxley Building**, between **2 pm and 5.30 pm**.It will be a good opportunity to get together with our Imperial PDE community and to find out what other PhD students are working on. See below the title and abstracts of our speakers. The scheduled can be found here.

Marco Agnese:** Fitted Finite Element Discretization of Two–Phase (Navier-) Stokes Flow **

*We propose a novel fitted finite element method for two-phase (Navier-)Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm.*

Marina Ferreira: **Damped Arrow-Hurwicz Algorithm for Hard-Sphere Packing**

*Hard-particle packings are found in a wide range of situations, from planet formation to cell tissue and standing crowds. The search for a packing gives rise to nonconvex optimization problems. These type of problems become extremely hard as the number of particles increases. In this talk I will derive a novel multi-step scheme for the packing of N hard-spheres: the Damped-Arrow-Hurwicz algorithm. Numerical results on the efficiency, accuracy and robustness of the method to different initial configurations are going to be presented for the case of two spatial dimensions. Our method will be compared with two other methods based on the linearization of the constraints and on a formulation of the problem in terms of the velocities, respectively.*

Francesco Patacchini: **Existence of global minimisers for the interaction energy. **

*We show the existence of compactly supported global minimisers under almost optimal hypotheses for continuum models of particles interacting through a potential. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The class of potentials for which we prove existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. This is a joint work with J. A. Cañizo and J. A. Carrillo.*

Yong Sul Won: **Starvation Driven Diffusion**

*Many biological organisms start migrations towards foods to survive. Such dispersals increase as more individuals sense insufficient foods around them. So there are no reasons not to suppose that the diffusivity of population density depends on the level of starvation. This led to a more realistic model taking into account of interaction between species and their response to the environment, which was first suggested by E. Cho and Y-J. Kim in 2013. In this talk, I will present the existence theories of the above systems by providing suitable a priori bounds.*

Franca Hoffmann: **Asymptotic behaviour of diffusing and self-attracting particles**

*We study interacting particles behaving according to a reaction-diffusion equation with non-linear diffusion and non-local attractive interaction. This class of equations has a very nice gradient flow structure that allows us to make links to variations of well-known functional inequalities (Hardy-Littlewood-Sobolev inequality, logarithmic Sobolev inequality). Depending on the non-linearity of the diffusion, the choice of interaction potential and the dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behaviour of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. This is joint work with José A. Carrillo and Vincent Calvez.*

Thomas Johnson:** ****Linear Wave Equations on Black Hole Spacetimes**

*The celebrated Kerr family of solutions to the vacuum Einstein equations in General Relativity describe spacetimes that contain a rotating black hole. It is believed that that they should represent the end state of all dynamic gravitational processes, as evidenced by the recent black hole binary merger detected by LIGO. In particular, one would like to know whether the Kerr family is stable. This is one of the biggest open problems in mathematical relativity.*

*Since the Einstein equations can be viewed as a coupled system of quasilinear wave equations, a toy stability problem would be to prove a decay statement for solutions to the linear wave equation on Kerr. Such a result has been obtained only recently by Dafermos, Rodnianski and Shlapentokh-Rothman.*

*In the first part of the talk I will discuss one of the key decay mechanisms exploited in their proof, illustrated for the simpler case of the Schwarzschild family. In the second part, I present some of my own results establishing a similar decay mechanism for the full system of gravitational metric perturbations, linearised about Schwarzschild*

Thomas Holding: **Probabilistic coupling methods for hypocoercivity of the kinetic Fokker-Planck equation in the Wasserstein distance.**

*The kinetic Fokker-Planck equation describes the evolution of the probability density function of a particle interacting with a fluid background. A central object of study for this equation is the contraction property of the semigroup: Given a distance d(f,g), do two solutions of the kinetic Fokker-Planck equation converge together exponentially fast in this distance?*

*In L^2 and Sobolev type distances there are well-developed theories of *coercivity* in the spatially homogeneous setting, and* hypocoercivity* in the inhomogeneous kinetic setting.* *The more probabilistic Wasserstein distance is, however, `inaccessible’ from this analytic world, requiring more probabilistic techniques. In the spatially homogeneous setting these probabilistic coupling techniques are well understood. In contrast, in the inhomogeneous (kinetic) setting, no such probabilistic coupling methods exist, even in seemingly trivially easy cases.*

*In this talk I will attempt to shine some on the following questions:*

*Is the (simplest example of the) kinetic Fokker-Planck equation contractive in the Wasserstein distance?**To what extent can probabilistic coupling techniques be used to prove this?*

*This is joint work with Helge Dietert and Josephine Evans.*

Fluids Theme Day

Our first theme day this academic year will be the **Fluids Theme Day** on the **17th November**. This event will take place in **Room 139 of the Huxley Building**, between **12-3pm**. See below the title and abstracts of the currently confirmed speakers:

Elizaveta Dubrovina:**Two layer flow between corrugated electrodes**

*In recent years there has been growing interest in the miniaturisation of electronic tools and much research has gone into finding appropriate techniques for patterning at small scales. One such technique exploits the electrohydrodynamic instabilities of a system to induce ordered structures. In this talk we present a model of the evolution of the interface between two perfect dielectric fluids flowing between two electrodes one of which is corrugated. With the help of a Floquet stability analysis and of full time dependant numerical simulations, we will show how the amplitude and the shape of the topography influence interfacial patterns.*

Susana Gomes: **Controlling Spatiotemporal Chaos in Active Dissipative- Dispersive Nonlinear Systems **

*We present a novel generic methodology for the stabilization and control of infinite- dimensional dynamical systems exhibiting low-dimensional spatiotemporal chaos. The methodology is exemplified with the generalized Kuramoto-Sivashinsky equation, the simplest possible prototype that retains that fundamental elements of any nonlinear process involving wave evolution. The equation is applicable on a wide variety of systems including falling liquid films and plasma waves with dispersion due to finite banana width. We show that applying the appropriate choice of time- dependent feedback controls via blowing and suction, we are able to stabilize and/or control all stable or unstable solutions, including steady solutions, travelling waves and spatiotemporal chaos, but also use the controls obtained to stabilize the solutions to more general long wave models. *

(Dr.) Radu Cimpeanu: **The effect of thin liquid films on boundary-layer separation**

*In this study we develop the theory for understanding the process of boundary-layer separation in the presence of a thin liquid film. The investigation is physically motivated by the accumulation of water on aircraft surfaces as a result of flying during adverse weather conditions, with implications in aircraft safety, certification and performance. We present an extension of the asymptotic framework of viscous-inviscid interaction and formulate a modified triple-deck model accounting for the strong density and viscosity contrast between the fluids in the system. The primary goal of the study is to address the question of whether the thin liquid layer acts to suppress or promote boundary-layer separation. We find that an increase in liquid film height (within its asymptotic scaling) contributes to a delay in the onset of separation. Furthermore, the main flow features, represented by local extrema in the perturbed flow quantities, are shifted further downstream within the interaction region. The consequences of the presence of the liquid film are illustrated through two typical examples encountered in flows past aircraft wings, namely surface roughness elements and corners/flap junctions.*

Elena Luca: **Dragging cylinders in slow viscous flows**

*The so-called “dragging problem” in slow viscous fluids is an important basic flow with many applications. In two dimensions, the Stokes paradox means there is no solution to the dragging problem for a cylinder in free space. The presence of walls changes this; the solutions exist, but are not easy to find without purely numerical methods. This talk describes new “transform methods” that produce convenient, semi-analytical solutions to dragging problems for cylinders in various geometries. We apply the techniques to low-Reynolds-number swimming where dragging problem solutions can be combined with the reciprocal theorem to compute swimmer dynamics in confined domains.*

Sam Brzezicki: **Analysis of the triple contact point in Electrowetting**

*In electrowetting applications, it is often necessary to consider static conducting droplets sitting on substrates. In this talk we perform a detailed study of the triple contact point at which the droplet, the ambient medium and the substrate touch. A local analysis is performed to understand the nature of*

*the field singularities at the contact point. We also present some global solutions obtained using a novel transform formulation and gain insights into how those solutions depend on the system parameters.*

Simon Game: **Physical mechanisms of flow resistance in textured microchannels**

*Transport in microchannels can be enhanced by replacing flat, no-slip boundaries with boundaries etched with longitudinal grooves containing an inert gas, resulting in an effective slip flow. Various physical considerations which are often omitted from mathematical models play a significant role in the behaviour of this flow. Such considerations include: gas viscosity, meniscus curvature, finite channel cross-sections, molecular slip on the gas/liquid or gas/solid interfaces. Using a computationally efficient, multi-element, Chebyshev collocation method, we are able to quantify and combine each of these physical effects. We have shown that for physically realistic parameter values, including each of these effects significantly alters the volumetric flow rate, and hence these effects should not be ignored. Using this framework, we hope to manipulate these effects in order to minimise the flow resistance of the channel*

Adam Butler: **Non-Parallel Flow Effects of Stationary Crossflow Vortices at their Genesis**

*In this talk I will discuss the linear stability of stationary crossflow vortices whose spanwise wavenumber is sufficiently small that non-parallel flow effects play a leading order role in determining the growth rate of the vortices. The chordwise and spanwise variations of the baseflow and the perturbation are of equal importance, and must both be accounted for. Neutral modes can occur in this regime, which lies close to the leading edge. *

*If the effective pressure minimum falls within this regime, a new scaling for the lower deck must be determined, along with a new dispersion relation for the chordwise wavenumber. When the mode from the non-parallel regime is continued through the effective pressure minimum, it passes into a critical layer in the form of a Cowley, Hocking, & Tutty instability. Downstream of the effective pressure minimum, this critical layer will eventually move into the main body of the boundary layer. This CHT instability can occur in a more general setting, when the first three derivatives of the effective velocity profile are zero at the wall. *

Past Themed Events:

**Fluid Dynamics** (18/03/2015) [Poster] [Speaker&AbstractInfo]

**Biomathematics** (20/11/2014) [Poster] [Speaker&AbstractInfo]