Speaker: **Ondřej Kincl **and** Martin Šípka**

Title: **Novel multiscale tools in the modeling of superfluid helium**

Abstract: The second talk in the liquid helium miniseries will be composed of two parts which are new for the subject of superfluidity and are to be investigated in the scope of the START project. In the beginning, Ondřej Kincl will consider the application of the SPH (Smoothed particle hydrodynamics) theory for the modelling of superfluid phenomena. The implementation of the HVBK equations within the SPH framework will be shown and possible advantages and limitations shall be discussed.

The second part will be given by Martin Šípka, who will briefly introduce the use of a very recent tool in simulations: Machine Learning (ML). Machine learning is being applied to various parts of modelling, often with great success. In this talk, we shall illustrate its use in quantum chemistry, where the trained potentials speed up the subsequent calculations by orders of magnitude. We will show how we can use ML to capture the low-dimensional structure of a chemical reaction to produce so-called collective variables. We then outline the possible applications of ML on the Vortex Filament Method in superfluid helium. The method is often too slow when the number of vortices in the domain increase and would benefit from a similar speed-up as chemistry when using ML methods.

]]>The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place next Wednesday (May 26, 2021) from 9:00 till 10:00. The talk will be given by Šimon Midlík and Jakub Kvorka. Please see the details below.

Speakers: **Šimon Midlík** and **Jakub Kvorka**

Title: **Theoretical and experimental approach to superfluid 4He**

Abstract: In reaction to a newly started collaboration between members of the Mathematical Institute (division of Mathematical modelling), Department of geophysics, and Low-temperature physics department via the START project “Novel multiscale approach to the dynamics of superfluid helium”, we would like to introduce each member’s work to a wider audience.

In this talk we plan to introduce the basics of current theoretical models, vortex-filament method (VFM) and Hall-Vinen-Bekarevich-Khalatnikov model (HVBK), describing dynamics of superfluid 4-helium. We will further present some currently ongoing experiments in the laboratory of superfluidity. These experiments are to be understood by methods of mathematical modeling and numerics. In particular, we will present preliminary data on the second-sound waves in helium analyzed by means of the decomposition to spherical harmonics, which is a common numerical technique in geophysics.

]]>Speaker: **Christoph Allolio**

Title: **Multiscale Approach to Biomembrane Remodelling by Adsorbates**

Abstract: In this talk I give an introduction to the modelling of biomembranes and our current progress in the application of the Helfrich theory of curvature elasticity to important biological processes. I introduce the techniques we developed to extract specific, local continuum properties from molecular simulations. I show the local modification of membrane elastic properties inside nanodiscs of various compositions, demonstrating the possibilities of our method. In the following, I describe the modification of membrane curvature elastic properties by cationic adsorbates and their consequences to membrane structure: The change of the membrane elastic properties can have wide-ranging effects, including changes in topology. For example fusion is triggered by the formation of calcium clusters on the membrane surface. Another example is our recent collaboration with experimental groups with the goal of designing endosomal escape agents – we show how the efficiency of these agents is related to their modification of the membrane interface and draw consequences for the design of new agents. Furthermore, I introduce first results on our modeling of mitochondrial membranes and mitochondrial morphology under stress. For this purpose, we developed a new solver, whose technical aspects I will introduce. Finally, I will talk about ongoing projects to extract further parameters from membranes and a possible setup for measuring the polarity of biointerfaces in vivo.

]]>Speaker: **Martin Čížek**

Title: **Models of vibronic excitation of a molecule by collision with the electron and attempts to solve them with Krylov subspace methods**

Abstract: I will shortly review our current understanding of electron collisions with molecules with focus on vibrational excitation of molecules and induction of chemical reactions by electrons. Then I will explain the mathematical model of such collisions with reasonable compromise between generality and tractability of the resulting equations. Finally I will discuss our attempts to solve the equations with Krylov subspace methods. The talk will be presented from a physicist point of view, and I welcome any comments/suggestions from a mathematical perspective in the discussion.

]]>Speaker: **Tomáš Los**

Title: **On planar flows of viscoelastic fluids of the Burgers type**

Abstract: Rate-type fluid models involving the stress and its observer-invariant time derivatives of higher order are used to describe a large class of viscoelastic mixtures – geomaterials like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as SBR. A standard model that belongs to the category of viscoelastic rate-type fluid models of the second order is the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the first order. This viewpoint allows one to develop the whole hierarchy of generalized models of a Burgers type. We study one such generalization. Carrying on the study by Masmoudi (2011), who briefly proved the weak sequential stability of weak solutions to the Giesekus model, we prove long time and large data existence of weak solutions to a mixture of two Giesekus models in two spatial dimensions.

]]>Speaker: **Anna Abbatiello**

Title: **Generalized solutions to models of compressible viscous fluids**

Abstract: We consider general models of compressible viscous fluids with nonlinear viscosity tensor and inhomogeneous boundary conditions, for which the problem of existence of global in time weak/strong solutions is open. We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). This is a joint work with Eduard Feireisl and Antonin Novotny.

]]>Speaker: **Erika Maringová**

Title: **On dynamic boundary condition**

Abstract: In the talk, we discuss a response of the fluid on the boundary, which acts as a delayed slip due to material properties. In the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier-Stokes-like problems with dynamic slip boundary condition, which requires a proper generalisation of the Gelfand triplet and the corresponding function spaces setting. It is a joint work with Anna Abbatiello and Miroslav Bulíček.

]]>Speaker: **Josef Málek**

Title: **A thermodynamic framework for heat-conducting flows of a mixture of two fluids**

Abstract: In the lecture, we focus on the development of a model for the heat conducting mixture of two compressible fluids described in the terms of the densities and the velocities for each fluid and the temperature field for the whole mixture. We use a general thermodynamic framework that determines the response of the material from the knowledge of two pieces of information, namely how the material stores the energy and how the energy of material is dissipated. This information is expressed in the form of the constitutive equations for two scalars: the Helmholtz free energy and the entropy production. Additionally, we follow the goal to determine the response of a complex mixture from a small set of material parameters (bulk and shear viscosity, heat conductivity, the drag coefficient) that can be associated with the mixture as the whole. The same thermodynamic approach is used to obtain the model when the whole mixture responses as an incompressible material. Various variants based on different definitions of the velocity of the mixture are considered. The models are developed within the theory of interacting continua. The lecture is based on a joint work with Ondrej Soucek.

]]>Speaker: **Pablo Alexei Gazca Orozco**

Title: **A semismooth Newton method for implicitly constituted flow**

Abstract: We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; as a motivating example we consider the Bingham model for viscoplastic flow. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation introduced recently by Bulíček et al., for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the nonlinear iterations.

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