## Applied Seminars 2010 - 2011

### Programme

#### Meshfree Modeling of Dynamic Fracture

26 October 2010

**Speaker:** T. Rabczuk (Bauhaus-University Weimar).

**Abstract:** To be confirmed

#### Existence and equilibration of global weak solutions to kinetic models for dilute polymers

16 November 2010

**Speaker:** Endre Suli (University of Oxford).

**Abstract:**We prove the existence of global-in-time weak solutions to a general
class of coupled microscopic-macroscopic bead-spring chain models (including finitely extensible nonlinear elastic (FENE) models), which arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational.

With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian, we prove the existence of global-in-time weak solutions to the coupled Navier-Stokes-Fokker-Planck system, satisfying the initial condition, such that the velocity belongs to the classical Leray space and the probability density function has bounded relative entropy and square-integrable Fisher information over any time interval. The key analytical tools in our proof are a weak-compactness argument based on a relative entropy method, and Dubinskii's compactness theorem in seminormed sets. It is also shown using a logarithmic Sobolev inequality and the Csiszar-Kullback inequality that, in the absence of a body force, global weak solutions decay exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett (Department of Mathematics, Imperial College London).

**References:**

John W. Barrett and Endre Süli:

Existence and equilibration of global weak solutions to finitely

extensible nonlinear bead-spring chain models for dilute polymers

arXiv:1004.1432v1, 8 April 2010. http://arxiv.org/abs/1004.1432

Accepted for publication in M3AS, 17th September 2010.

John W. Barrett and Endre Süli:

Existence and equilibration of global weak solutions to Hookean-type

bead-spring chain models for dilute polymers.

arXiv:1008.3052v1, 18 August 2010. http://arxiv.org/abs/1008.3052

Submitted for publication, 2010.

#### Multilevel Simulation under Uncertainty

18 January 2011

**Speaker: **Robert Scheichl (University of Bath).

**Abstract:** The quantification of uncertainty in groundwater flow plays a central role in the safety assessment of radioactive waste disposal and of CO2 capture and storage underground. Stochastic modelling of data
uncertainties in the rock permeabilities lead to elliptic PDEs with
random coefficients. A typical computational goal is the estimation of
the expected value or higher order moments of some relevant quantities
of interest, such as the effective permeability or the breakthrough
time of a plume of radionuclides. Because of the typically large
variances and short correlation lengths in groundwater flow
applications, methods based on truncated Karhunen-Loeve expansions are
only of limited use and Monte Carlo type methods are still most
commonly used in practice. To overcome the notoriously slow
convergence of conventional Monte Carlo, we formulate and implement
novel methods based on (i) deterministic rules to cover probability
space (Quasi-Monte Carlo) and (ii) hierarchies of spatial grids
(multilevel Monte Carlo). It has been proven theoretically for certain
classes of problems that both of these approaches have the potential
to significantly outperform conventional Monte Carlo. A full
theoretical justification that the groundwater flow applications
discussed here belong to those problem classes are under current
investigation. However, experimentally our numerical results show that
both methods do indeed always clearly outperform conventional Monte
Carlo even within this more complicated setting, to the extent that
asymptotically the computational cost is proportional to the cost of
solving one deterministic PDE to the same accuracy.

#### An Arbitrary Lagrangian Eulerian Method for Moving-Boundary Problems

8 February 2011

**Speaker: **Jie Li (Cambridge).

**Abstract:** Phenomena involving moving boundaries are ubiquitous.
They are a rich source of research topics and of importance in industrial
applications. In this talk, we present an Arbitrary Lagrangian Eulerian method,
which employs a body-fitted grid system. The moving boundaries are lines of the
grid system, and complicated dynamic boundary conditions are incorporated
naturally and accurately in a Finite-Element formulation. Our method is
validated on the uniform flow passing a cylinder and several problems of bubble
dynamics for both steady and unsteady flows. Good agreement with other
theoretical, numerical and experimental results is obtained. Our method is
applied to the investigation of the impulsive motion of a small cylinder at
an interface, the bubble dynamics in a surfactant polymeric solution, and
three-phase flows with triple junction points.

#### Exchange flow of two immiscible fluids and the principle of maximum flux

15 February 2011

**Speaker: **Rich Kerswell (Bristol).

**Abstract:** TBC

#### Simulating the Bubble Scale Rheology of Foams

22 March 2011

**Speaker: **Simon Cox (Aberystwyth).

**Abstract:** TBC