Physics of complex fluids and turbulence

course ID





14 Weeks

Semester DD


Course details

Fundamental equations
Conservation of mass and momentum. Stress tensor symmetry. Newtonian fluids constitutive relation. Navier Stokes equation for incompressible flows. Boundary conditions. Navier condition and slip length. Dimensionless form of the Navier-Stokes equations. Reynolds number. Stokes equation, linearity and symmetries. Notes on Purcell's theorem concerning the swimming of microorganisms. Poiseuille flow.

Brownian Motion
Diffusion of particles in a fluid. Conservation equation. Langevin equation for the motion of a single colloid. Fluctuation-dissipation theorem. Numerical methods for stochastic differential equations.

Complete system of equations for transporting charged species. Poisson-Boltzmann equation. Debye length. Ideal electroosmotic flow in a plane channel. Electroosmotic flows in nanopores. Applications for biosensors and blue energy.

Surface and dynamic tension of the interfaces
Definition of surface tension. Laplace equation. Young's equation and contact angle. Cassie and Wenzel states. Jurin's law. Capillary length. Taylor-Rayley instability. Overview on continuous models for two-phase flows (Continuum force model). Atomistic simulation techniques.

Description in Fourier space. Production, transfer and dissipation of turbulent kinetic energy. Kolmogorov theory for homogeneous and isotropic turbulence. Kolmogorov scale. Reynold-averages equations.


The course provides an introduction to advanced topics in fluid dynamics. The common thread of the course is the complexity and the methodologies to face it. The selected examples will be chosen from a multiscale perspective (different spatial and temporal scales relevant to the analysis of the phenomenon) and multi-physics (different effects contribute to the phenomenology). In particular, the following topics will be covered: turbulent motions for simple fluids, colloidal solutions of micrometric particles (Brownian motion), two-phase and electro-dynamic flows. The course provides conceptual and analytical tools to describe complex fluids and flows.


At the end of the course, the student will be able to understand the main phenomena related to the dynamics of complex fluids, in particular with regard to the description of the transport of particles in fluids and to electro-dynamics. Furthermore, the student will know the main phenomenologies associated with turbulent flows and their theoretical description.


The student will be able to recognize the range of validity of the various models proposed for the description of complex fluids and turbulence. He will also be able to apply the knowledge and understanding developed during the course to implement some simple numerical methods.

The transversal preparation provided by the course, together with a good knowledge of the technical scientific problems of the different aspects of the fluid dynamics of complex fluids implies: 1) the student's capabiltiy to integrate knowledge and manage complexity, 2) the student ability to identify and formulate the solution of problems in new and emerging areas in the study of complex fluids and turbulence and 3)
an understanding of the models suited for a given context and their limitations.

The student will be able to communicate the contents of the course to specialists in a clear and unambiguous way. It will also be able to communicate the main features of the models used and their limits to specialists in other related disciplines (example: other engineers, physicists, chemists).


The structure of the course contents, characterized by various topics apparently separated but connected by a multi-scale and multi-physics vision, will contribute to developing a systemic learning capacity that will allow the student to approach in a self-directed or autonomous way to other frontier problems concerning the fluiddynamics. More generally, the student will have conceptual tools to approach the study of mathematical models for complex problems. Furthermore, the student will be able to read and understand textbooks on advanced fluid dynamics and scientific publications.