ELEMNTS OF MATHEMATICAL PHYSICS I
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- MARIA CRISTINA PATRIA
- Credits
- 6
- Curriculum
- GENERALE
- Didactic period
- Primo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course is providing good knowledge of fluids thermomechanics and its applications.The study is treated from the spatial point of view. After obtaining mechanical and thermodynamic equations that govern the motion of a general continuum, one introduces and studies two fluids constitutive classes: inviscid fluids and classical viscous fluids.
At the end of the course the main knowledge acquired will be
• basic concepts of continuum mechanics and thermodynamics
• the most important properties of inviscid and linearly viscous fluids
• some particular flows of Newtonian fluids which are of relevant interest for the applications.
At the end of the course the student will be able:
• to formulate boundary-initial-value problems to study the flow of fluids belonging to different constitutive classes in different physical situations
• to point out the differences between different models of fluids
• to determine the exact solution in the cases of particularly simple motions and to study the influence of the material parameters on the flow
• to use dimensionless quantities in order to reduce the number of the parameters that must be taken into account
• to expose the topics of the course by using a correct scientific language. Prerequisites
- Good knowledge of differential and integral calculus. Basic concepts on tensor algebra and analysis.
Students not acquainted with tensor calculus are invited to look at the online lecture notes, that contain an Appendix with Recalls to Tensor Calculus. Course programme
- The course is scheduled in 48 hours.
The programme is the following.
KINEMATICS: definition of continuum body, motion of a continuum body in kinematic framework, material and spatial point of view, material and spatial time derivative, transport theorem, trajectories and streamlines, steady flows, circulation transport theorem, plane flows (9 hours);
KINETICS, DYNAMICS, THERMODYNAMICS: mass density and mass conservation equation, linear ed angular momentum balance equations, kinetic energy theorem, first and second thermodynamics axioms, themomechanics problem for a continuum (9 hours);
INVISCID FLUIDS: constitutive equations of compressible and incompressible inviscid fluids, problem of the flow for an inviscid fluid, barotropic fluids, ideal gases, properties of inviscid fluids in static conditions, first and second Bernoulli's theorems (7 hours);
CLASSICAL VISCOUS FLUIDS: Constitutive equations of compressible and incompressible classical viscous fluids, compatibility of the constitutive equations with second axiom of thermodynamics, formulation of the problem of the flow for a compressible and incompressible classical viscous fluids, differences between incompressible inviscid fluids and incompressible classical viscous fluids (5 hours);
CLASSICAL BOUNDARY-INITIAL-VALUE PROBLEM FOR AN INCOMPRESSIBLE HOMOGENEOUS NEWTONIAN FLUID: formulation of the problem, preliminary results, uniqueness and stability theorems (3 hours);
POISEUILLE AND POISEUILLE-COUETTE FLOW FOR AN INCOMPRESSIBLE NEWTONIAN FLUID: preliminaries, Poiseuille flow between two parallel planes and numerical simulations, Poiseuille-Couette flow between two parallel planes and numerical simulations (4 hours);
FLOWS OF AN INCOMPRESSIBLE NEWTONIAN FLUID PAST A ROTATING PLANE: preliminaries, non-symmetric solutions, numerical simulations (3 hours);
STAGNATION-POINT FLOWS OF A NEWTONIAN FLUID: preliminaries, plane orthogonal stagnation-point flow of an incompressible inviscid fluid, plane orthogonal stagnation-point flow of an incompressible Newtonian fluid (3 hours);
MAGNETOHYDRODYNAMICS: Maxwell equations and Lorentz forces, Hartmann flow and remarks (5 hours). Didactic methods
- The course is organized with lectures, exercises and examples on all topics of the program.
Learning assessment procedures
- The aim of the final exam consists in verifying the level of knowledge of the formative objectives previously stated. The final exam consists in an oral discussion on all subjects of the course.
Moreover the student will have to present an assigned topic and bring the resolution of some exercises assigned dopuring the course. Reference texts
- Lecture Notes available at the couse website.
Specific topics can be further developed on
M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981.
S. Forte, L. Preziosi, M. Vianello: Meccanica dei continui, Springer Italia, 2019.
