MATHEMATICAL PHYSICS I
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- MARIA CRISTINA PATRIA
- Credits
- 8
- Curriculum
- APPLICATIVO
- Didactic period
- Primo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course is providing good knowledge of continuum thermomechanics and of its applications. In the first part of the course the study is made 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. In the second part general equations governing continuum thermomechanics are rewritten from the material point of view and some constitutive classes of solids are studied: thermoelastic solids, elastic solids, linearly elastic solids.
At the end of the course the main knowledge acquired will be
• basic concepts of continuum mechanics and thermodynamics
• the most relevant properties of ideal and linearly viscous fluids
• some particular flows of Newtonian fluids which are of relevant interest for the applications
• basic concepts of hydrodynamic stability
• the most important properties of thermoelastic, elastic and linearly elastic solids
• wave propagation in linear elastic solids.
At the end of the course the student will be able:
• to study the motion of a continuum both from the spatial point of view and from the material point of view
• to formulate boundary-initial-value problems to study the flow of fluids or solids belonging to different constitutive classes in different physical situations
• to point out the differences between different models of continua
• to determine the exact solution in the case 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 on Tensor Calculus. Course programme
- The course is scheduled in 64 hours.
The program consists of three parts: in the first part the study is made from the spatial point of view and from the material point of view; in the second part some aspects of fluid mechanics are studied and in the third part the elastic solids are studied
PART I
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 and incompressible flows, plane flows and stream function (6 hours);
CONTINUUM THEMOMECHANICS FROM MATERIAL POINT OF VIEW: analysis of deformation of a continuum, incompressibility condition and mass conservation equation from material point of view, linear and angular momentum balance equations, local equation and inequality consequences of two axioms of thermomecanics from the material point of view (6 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 (6 hours).
PART II
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 houres);
CLASSICAL BOUNDARY-INITIAL-VALUE PROBLEM FOR AN INCOMPRESSIBLE HOMOGENEOUS NEWTONIAN FLUID: formulation of the problem, preliminary results, uniqueness and continuous dependence 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);
PART III
THERMOELASTIC AND ELASTIC MATERIALS: constitutive class of the thermoelastic materials, some properties of the thermoelastic materials, elastic and hyperelastic materials, elasticity tensor and its properties (6 hours);
LINEARLY ELASTIC MATERIALS: definition of linearly elastic materials, particular subclasses of linearly elastic materials, linear elastostatics, work and energy theorem, mixed boundary-value problem of linear elastostatics and uniqueness theorems, linear elastodynamics, theorem of power and energy, mixed boundary-initial-value problem of linear elastodynamics and uniqueness theorem (7 hours);
WAVE PROPAGATION IN LINEARLY ELASTCIC MATERIALS: definition of wave, acoustic tensor of a linearly elastic body, eigenvalues and eigenvectors of a symmetric tensor of second order, plane progressive waves, elastic plane progressive waves, Fresnel-Hadamard propagation condition, Fedorov-Stippes theorem (3 hours). Didactic methods
- The course is organized with lectures and exercises 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 student will have to present an assigned topic and bring the resolution of some exercises assigned.
The final exam consists in an oral discussion on all subjects of the course.
Reference texts
- Lecture notes available at the course 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.
M. E. Gurtin, E. Fried, L. Anand: The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2010.
