# Course Details

#### Name:

**Mathematical fluid dynamics / Mathematical fluid dynamics**

### Basic information

##### Credits:

*Master Degree in Mathematics:* 6 Ects (c)

##### Degree(s):

Elective 2^{nd} year Master Degree in Mathematics curriculum Generale

##### Language:

English

### Course Objectives

This course is designed 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 analysis of other partial differential equations.

### Course Content

- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Vorticiy equation in 2D and 3D
- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Fluid dynamic modeling in various fields: magnetohydrodynamics, combustion, astrophysics, biofluids.
- Existence of solutions for viscid and inviscid fluids

### Learning Outcomes (Dublin Descriptors)

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.

### Prerequisites and Learning Activities

Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

### Teaching Methods

**Language**: English

5 hours per week during the whole semester.

### Assessment Methods and Criteria

Written exam.

### Textbooks

- 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. * *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

*Course information last updated on: 18 ottobre 2017, 09:58*