## Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

## Academic calendar

The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technical-administrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.

## Course calendar

The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..

Period | From | To |
---|---|---|

I semestre | Oct 1, 2015 | Jan 29, 2016 |

II semestre | Mar 1, 2016 | Jun 10, 2016 |

Session | From | To |
---|---|---|

Sessione straordinaria Appelli d'esame | Feb 1, 2016 | Feb 29, 2016 |

Sessione estiva Appelli d'esame | Jun 13, 2016 | Jul 29, 2016 |

Sessione autunnale Appelli d'esame | Sep 1, 2016 | Sep 30, 2016 |

Session | From | To |
---|---|---|

Sess. autun. App. di Laurea | Oct 12, 2015 | Oct 12, 2015 |

Sess. autun. App. di Laurea | Nov 26, 2015 | Nov 26, 2015 |

Sess. invern. App. di Laurea | Mar 15, 2016 | Mar 15, 2016 |

Sess. estiva App. di Laurea | Jul 19, 2016 | Jul 19, 2016 |

Sess. autun. 2016 App. di Laurea | Oct 11, 2016 | Oct 11, 2016 |

Sess. autun 2016 App. di Laurea | Nov 30, 2016 | Nov 30, 2016 |

Sess. invern. 2017 App. di Laurea | Mar 16, 2017 | Mar 16, 2017 |

Period | From | To |
---|---|---|

Festività dell'Immacolata Concezione | Dec 8, 2015 | Dec 8, 2015 |

Vacanze di Natale | Dec 23, 2015 | Jan 6, 2016 |

Vacanze Pasquali | Mar 24, 2016 | Mar 29, 2016 |

Anniversario della Liberazione | Apr 25, 2016 | Apr 25, 2016 |

Festa del S. Patrono S. Zeno | May 21, 2016 | May 21, 2016 |

Festa della Repubblica | Jun 2, 2016 | Jun 2, 2016 |

Vacanze estive | Aug 8, 2016 | Aug 15, 2016 |

## Exam calendar

Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.

To view all the exam sessions available, please use the Exam dashboard on ESSE3.

If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.

Should you have any doubts or questions, please check the Enrolment FAQs

## Academic staff

Dos Santos Vitoria Jorge Nuno

jorge.vitoria@univr.itSquassina Marco

marco.squassina@univr.it +39 045 802 7913## Study Plan

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University. Please select your Study Plan based on your enrolment year.

Modules | Credits | TAF | SSD |
---|

Modules | Credits | TAF | SSD |
---|

Modules | Credits | TAF | SSD |
---|

1° Year

Modules | Credits | TAF | SSD |
---|

2° Year

Modules | Credits | TAF | SSD |
---|

3° Year

Modules | Credits | TAF | SSD |
---|

Modules | Credits | TAF | SSD |
---|

#### Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

### Fluid dynamics (2015/2016)

Teaching code

4S00258

Teacher

Coordinatore

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/07 - MATHEMATICAL PHYSICS

Period

II semestre dal Mar 1, 2016 al Jun 10, 2016.

## Learning outcomes

Derivation of the fluid-dynamic equations from conservation laws in Physics; discussion on the rheological structure of fluids and the model for Newtonian fluids; different flows and simplifications of the governing equations; Bernoulli theorem in all forms and for all cases; some exact solutions; vorticity dynamics; laminar boundary layer; stability and transition; turbulence; hyperbolic equations in fluid dynamics.

## Program

1. Introduction to fluids: definitions, continuous hypothesis and properties of fluids; differences between fluid, flux, flow; some kinematics (stream-lines, trajectories, streak-lines), forces and stresses (Cauchy Theorem and symmetry of the stress tensor), the constitutive relation for Newtonian fluids (viscous stress tensor).

2. Governing equations: Eulerian vs Lagrangian approach; control volume and material volume, conservation of mass in a fixed volume, time derivative of the integral over a variable domain, Reynolds Theorem (scalar and vectorial forms), conservation of mass in a material volume, from conservation laws to the Navier-Stokes equations, the complete Navier-Stokes equations (in conservative, tensorial form), substantial derivative, conservative vs convective form of the equations, alternative forms of the energy equation, dimensionless equations, initial and boundary conditions.

3. Particular cases of the governing equations: time dependence, effect of viscosity, thermal conduction, entropy, compressibility, barotropic flows, incompressible flows, ideal flows, Euler equations irrotational flows, barotropic and non-viscous flows: Crocco's form, Bernoulli theorem in all cases and forms.

4. Some exact solutions: incompressible and parallel flows, infinite channel flow, Couette and Poiseuille flows, flow in a circular pipe, Hagen-Poiseuille solution.

5. Vorticity dynamics: preliminary definitions, vorticity equation in the general case, special cases (constant density, non-viscous flow with conservative external field), Kelvin's theorem, Helmholtz's theorems and their geometrical meaning.

6. Laminar boundary layer: Prandtl theory, boundary layer past a flat plate, derivation of Blasius' equation (similar solutions), boundary-layer thickness, drag due to skin-friction, characteristics of a boundary layer (displacement thickness, momentum thickness, shape factor), integral von Kàrmàn equation, numerical solution of the 2D steady equations for the boundary layer past a flat plate:

(a) parabolic PDE + BC (Prandtl's equations): marching in space

(b) ODE + BC (Blasius' equation): nonlinear boundary value problem

(c) comparison between the two methods.

7. Stability and transition: flow in a pipe - Reynolds' experiment, transition in a laminar boundary layer, linear stability for parallel flows (Orr-Sommerfeld equation),

Squire's theorem, non-viscous stability (Rayleigh's criteria), viscous stability, linear stability curves.

8. Turbulence: phenomenological characteristics, turbulent scales, energy cascade, Kolmogorov's theory, DNS (Direct numerical simulation), RANS (Reynolds-Averaged-Navier-Stokes equations), the problem of closure for the RANS, closure models, Boussinesq hypothesis for the tutbulent viscosity (models of order 0, 1 and 2), LES (Large Eddy Simulation).

9. Hyperbolic differential equations in fluid dynamics: main characteristics and comparison with parabolic and elliptic equations, conservation laws, transport equation, characteristic lines, Riemann problem, Burgers' equation, weak solutions, shock waves, rarefaction waves, comparison between conservative and non-conservative numerical methods, method of characteristics, usage of an applet for the visualization of shock and rarefaction waves, hyperbolic linear and non-linear systems, genuine nonlinearity, linear degeneration, contact discontinuity, solution of the Riemann for the Euler equations.

## Examination Methods

The exams is an oral interview. During the oral part the students have to provide the solution to the exercises assigned during the course and to be able to discuss about them, because they contribute to the final grade together with the oral part.

## Type D and Type F activities

**Modules not yet included**

## Career prospects

## Module/Programme news

##### News for students

There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details.

## Further services

I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.

## Graduation

## Attachments

Title | Info File |
---|---|

1. Come scrivere una tesi | 31 KB, 29/07/21 |

2. How to write a thesis | 31 KB, 29/07/21 |

4. Regolamento tesi (valido da luglio 2020) | 259 KB, 29/07/21 |

5. Regolamento tesi (valido da luglio 2022) | 171 KB, 17/02/22 |

## List of theses and work experience proposals

theses proposals | Research area |
---|---|

Formule di rappresentazione per gradienti generalizzati | Mathematics - Analysis |

Formule di rappresentazione per gradienti generalizzati | Mathematics - Mathematics |

Mathematics Bachelor and Master thesis titles | Various topics |

Stage | Research area |
---|---|

Internship proposals for students in mathematics | Various topics |

## Attendance

As stated in point 25 of the Teaching Regulations for the A.Y. 2021/2022, except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended on-site.Please refer to the Crisis Unit's latest updates for the mode of teaching.