- Unit Coordinator: Donatella Donatelli
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Aims:
Learning Objectives:
The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.Learning Outcomes:
On successful completion of this course, the student should:- understand the basic principles governing the dynamics of non-viscous fluids;
- be able to derive and deduce the consequences of the equation of conservation of mass;
- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;
- understand the concept of vorticity and the conditions in which it may be assumed to be zero;
- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;
- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.
- demonstrate capacity for reading and understand other texts on related topics. - Content:
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)
- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Fluidi di tipo Poiseulle e Couette
- Vorticity equation in 2D and 3D
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.
- Pre-requisites:
PREREQUISITES for Mathematical Modelling of Continuum Media:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces. - Reading list:
- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.
- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.
- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.
- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.
- Unit Coordinator: Donatella Donatelli
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
This course is designed to give an overview of luid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of luid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular luid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.
- Content:
- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Poiseulle and Couette fluid types
- Vorticity equation in 2D and 3D
- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms. - Pre-requisites:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
- Unit Coordinator: Marko Lindner
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: Hamburg University of Technology
- Language: English
- Aims:
Students are able to characterize and compare diffusion equations, explain elementary methods of image processing, explain methods of image segmentation and registration, sketch and interrelate basic concepts of functional analysis.
- Content:
- Basic methods of image processing
- Smoothing filters
- The diffusion / heat equation
- Variational formulations in image processing
- Edge detection
- De-convolution
- Inpainting
- Image segmentation
- Image registration
- Pre-requisites:
- Analysis: partial derivatives, gradient, directional derivative
- Linear Algebra: eigenvalues, least squares solution of a linear system
- Reading list:
Will be announced in the lecture
- ECTS Credits: 5
- Year: 2
- Campus: Brno University of Technology
- Language: English
