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.

This information is intended exclusively for students already enrolled in this course.
If you are a new student interested in enrolling, you can find information about the course of study on the course page:

Laurea in Matematica applicata - Enrollment from 2025/2026

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 enrollment year.

2° Year  activated in the A.Y. 2013/2014

ModulesCreditsTAFSSD
6
A
MAT/02
6
B
MAT/03
6
B
MAT/06
One course chosen from the following two
6
C
SECS-P/01
6
C
FIS/01
One course chosen from the following two
6
C
SECS-P/01

3° Year  activated in the A.Y. 2014/2015

ModulesCreditsTAFSSD
One course of 12 ECTS or two courses of 6 ECTS chosen from the following three
6
C
MAT/06 ,SECS-P/05
Prova finale
6
E
-
activated in the A.Y. 2013/2014
ModulesCreditsTAFSSD
6
A
MAT/02
6
B
MAT/03
6
B
MAT/06
One course chosen from the following two
6
C
SECS-P/01
6
C
FIS/01
One course chosen from the following two
6
C
SECS-P/01
activated in the A.Y. 2014/2015
ModulesCreditsTAFSSD
One course of 12 ECTS or two courses of 6 ECTS chosen from the following three
6
C
MAT/06 ,SECS-P/05
Prova finale
6
E
-
Modules Credits TAF SSD
Between the years: 1°- 2°- 3°
Other activities
6
F
-
Between the years: 1°- 2°- 3°

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.




S Placements in companies, public or private institutions and professional associations

Teaching code

4S00258

Coordinator

Simone Zuccher

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/07 - MATHEMATICAL PHYSICS

Period

II sem. dal Mar 2, 2015 al Jun 12, 2015.

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.

Students with disabilities or specific learning disorders (SLD), who intend to request the adaptation of the exam, must follow the instructions given HERE