Mathematical modelling of continuum media
Additional Info
- ECTS credits: 3
- University: University of L'Aquila
- Semester: 1
- Lecturer 1: Donatella Donatelli
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Objectives:
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. -
Topics:
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. -
Prerequisites:
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. -
Books:
- 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.
