Chaos is the erratic behavior of simple nonlinear dynamical systems. This course covers the basic fundamentals of chaos theory, including the concept of scaling and universal route to chaos, and its connection to these non-linear, deterministic but unpredictable dynamical systems. From simple chaotic maps, to the physics of fractals and multi-fractals via the concept of renormalization group. Applications in fluid dynamics, plasma and fusion physics will be discussed.

# Category Archives: master

# 3MT120 - Advanced Computational Fluid and Plasma Dynamics

The course guides students to the hand-on discovery of several classical numerical methods commonly used to study the dynamics of fluids and plasmas: from continuum computational fluid dynamics (CFD) approaches (Spectral and finite volume), to the Lattice Boltzmann (LBM), particles based methods (Molecular Dynamics, Brownian and Stokesian dynamics) and Particle-in-cell Monte Carlo (PIC-MC).

# 3CS03 - Caput Theoretical Physics: Theory of liquids

Liquids lack the long-range order typical for solids. Collisional processes and short-range correlations distinguish liquids from dilute gases. Therefore, no idealized models comparable with the perfect gas or the harmonic solid are available for even simple liquids. During the last half of the 20th century a rapid progress has been made in our understanding of the microscopic structure and the dynamics of simple liquids. With advances in experiments (light and neutron scattering), theoretical analysis (statistical mechanics, kinetic theory of strongly correlated systems) and numerical tools (Molecular Dynamics and Monte Carlo simulations) a rather clear and complete picture emerged on the properties of simple atomic liquids. Since the last few decades a variety of more complicated systems are being studied: ionic, molecular and polar liquids, liquid metals, liquid-vapor interfaces, liquid crystals, and colloidal suspensions. In this lecture we will address the basic theory of the liquid state based on a statistical mechanical description of liquids. Topics that will be discussed include static properties of liquids, distribution function theories, perturbation theory and inhomogeneous fluids. We will conclude with an outlook to more complex fluids.

Introduction (week 1-2)

Liquid state, intermolecular forces

Liouville equation, BBGKY hierarchy

Statistical mechanics and ensemble averages

Static properties of liquids (week 3-4)

Particle densities and distribution functions

Computer simulations (MD and MC)

Diagrammatic expansions, virial expansion of the equation of state

Equation of state of a hard sphere fluid

Distribution function theories (week 5-6)

Static structure factor

Ornstein-Zernike direct correlation function

Percus-Yevick solution for hard spheres, mean spherical approximation

Outlook (week 7)

Perturbation theories

Complex liquids

# 3T350 - Chaos

This course is the continuation of 3T220 (Chaos), but now with a greater emphasis on the application of Chaos concepts to fluids. Chaos is the seemingly erratic behavior of simple deterministic, but nonlinear dynamical systems. We will first discuss the route to chaos, where we will already encounter the scaling concepts that will return in the description of chaotic fluid flow and turbulence. After a discussion of basic concepts, such as the sensitivity to variation in initial conditions, and the multifractal organization of phase space, we will introduce chaotic behavior and synchronization in coupled systems. These scaling ideas will then be carried over to the description of turbulence, the erratic flow of a fluid. We will do this in both the Eulerian and Lagrangian frame, where we move with the flow. While turbulence is wild chaos, also stirred viscous fluids may be chaotic, which may help to efficiently stir tracers. Also this case will be analyzed with the tools introduced in this course, such as local sensitivity to perturbations, and scaling of the concentration field of the stirred material. Central to the course is the exposure to the modern literature, in particular papers which appeared in Physical Review Letters (the most famous Physics journal). These papers can serve as inspiration for student presentations. Of course, adequate coaching is offered here.

# 3T380 - Advanced Computational Fluid Dynamics

Quick links: [Link to course schedule] [Link to lecture notes] [Link to additional books and material] [Link to exercises] [Link to possible individual assignments]

The 3T380 course provides an introduction to advanced computational methods useful to investigate fluid flows from the micro to the macro-scales under laminar and turbulent flow regime. The course will present an overview of several complementary computational methods. Reduced dimensionality methods can be helpful in understanding the qualitative feature of the phenomena and particularly for chaotic flows do allow high statistics with small computational resources. Lattice kinetic methods can provide relative computational efficiency with the ease of modeling complex geometries and complex physics. Spectral methods are the method of choice when accuracy is the requirement in spite of a much smaller flexibility on the flow geometry.

Students are expected to be able to describe, select, adapt and apply the following numerical methodologies (commonly used for the study of large- and small-scale flow problems:

- Advanced methods for numerical integration
- Pseudo-spectral methods for fluid flows and turbulence
- Finite volume methods
- Lattice Boltzmann methods for large and small scale flows
- Particle based methods.

During the course students will implement / modify simple numerical codes based on the above methods, perform collaborative work in small groups, exercise in accessing the modern literature on the subject, produce written reports and oral presentation.

Link to additional books and material

Link to possible individual assignments

Contact information:

prof.dr. F. Toschi - CC2.22 Phone 3911

prof.dr. H.J.H. Clercx - CC2.19 Phone 2680

dr. J.D.R. Harting - CC3.17 Phone 3766

dr.ir. J.H.M. ten Thije Boonkkamp - MF 7.95 Phone 4123

F. Tesser - CC2.11

M.P.J. Wouters