Contenidos teóricos y prácticos de la asignatura
PRELIMINARY NOTES:
(1). The program contains a number of optional complementary topics. They will be offered to the students depending on the general progress of the main topics of the course and, also, on the interest of the students.
(2) the optional practicals are of a simple numerical nature and will be carried out by programming using the Python language. The students will be offered, typically, one or two such practicals along the course.
Chapter 1 – THE DESCRIPTION OF A CONTINUOUS AND DEFORMABLE MEDIUM
1.1 – The Euler and Lagrange descriptions: multidimensional maps versus fluid element tracking. The Lagrange derivative.
1.2 – The relative motion of neighboring fluid elements. The expansion, rotation and strain tensors.
1.3 – Mass, momentum and energy of finite volumes and parcels of fluid. Volume forces and surface forces.
Optional numerical practical:
- fluid element tracking in a two-dimensional map.
Chapter 2 – THE CONSERVATION LAWS FOR AN IDEAL FLUID
2.1 – The variation in time of integrated quantities in finite fluid parcels: Reynolds Theorem
2.2 – The mass conservation law (i.e., the continuity equation). Precise definition of the concepts ‘volume density’ and ‘flux across surfaces’ of physical quantities.
2.3 – The momentum equation
2.4 – The total energy equation. Separation into equations for the kinetic and internal energy. The natural combination of mechanics and thermodynamics occurring in a fluid.
2.5 – The canonical conservation form for the equations of a continuous medium.
2.6 – Closure of the system of equations. Their intrinsically non-linear character.
Chapter 3 - IDEAL FLUIDS
3.1 - Euler equation
3.2 – Motion around obstacles. The classical potential flow problem. Ram pressure.
3.3 – Compressible motion: astrophysical example. The solar wind.
3.4 – Vorticity. Kelvin’s circulation theorem.
Optional complementary topics
– Aerodynamics. The lift force on aerodynamic profiles: Kutta-Zhukovski theorem. The Zhukovski transformation and the Kutta condition for the flow around wings.
– The static equilibrium of a gas sphere: stellar interiors.
Optional numerical practicals:
– Calculation of aerodynamic profiles. Lift force on actual aeroplanes. Force on windmill blades.
– The motion of the gas in solar coronal loops. Subsonic and supersonic regimes. Mach numbers.
– The numerical calculation of Parker’s solar wind solution including the sonic point and the sub- and supersonic regimes.
– The numerical solution of partial differential equations: heat conduction.
– The numerical solution of partial differential equations: the continuity equation.
Chapter 4 – THE MICROSCOPIC FOUNDATIONS OF THE FLUID EQUATIONS
4.1 – The continuum approximation: the criterion of scale separation in space and time. Local thermodynamic equilibrium.
4.2 – Statistical averages and macroscopic fluid quantities. The bulk kinetic energy of the flow. The internal energy due to the translational degrees of freedom.
4.3 – The calculation of pressure and viscosity in kinetic theory: elementary considerations.
4.4 – The entropy equation. Entropy flux. Sources and sinks of entropy. Irreversible processes in fluids. Irreversibility in thermodynamics and irreversibility in fluid dynamics.
Chapter 5 - VISCOSITY.
5.1 – Surface forces. Stress tensor. Cauchy theorem.
5.2 – The momentum equation for a generic stress tensor.
5.3 – The viscous stress tensor as a microscopic transport phenomenon. Newtonian fluids.
5.4 – The Navier-Stokes equation. Reynolds number.
5.5 – The energy equation for the viscous case. The irreversible heating through viscosity.
Optional complementary topics:
– Electromagnetism and fluid dynamics. The Maxwell stress tensor. Electromagnetic pressure and tension. Volume density and flux of the electromagnetic energy.
– Boundary layers
– Accretion discs around astrophysical objects.
– The flux density four-vector in Einstein´s relativity theory. The stress-energy tensor and the conservation laws for relativistic fluids.
Chapter 6 – Linear waves in gases
6.1 – Perturbation treatment of the non-linear equations. Linearization of the gas equations. Pressure waves (also known as sound waves).
6.2 – Fourier analysis. Eigenvalue equation. Dispersion relation. The eigenvectors as normal modes.
6.3 – Sound waves of finite amplitude. The nonlinearity and the spontaneous transition to shock waves.
Optional complementary topics:
– Inhomogeneous equilibrium. The WKB approximation. Phase speed and group speed. Ray tracing: geometrical acoustics.
– Gravity waves in stellar interiors.
Optional numerical practicals:
– normal mode decomposition of initial perturbations in a one-dimensional problem.
– Ray tracing of sound waves in an inhomogeneous gas.
Chapter 7 – SHOCK FRONTS
7.1 – Shock fronts: a ubiquitous and unavoidable phenomenon in the Universe.
7.2 – Conservation equations across a shock front. The Rankine-Hugoniot jump conditions. Mach numbers: supersonic and subsonic regimes.
7.3 – Weak shocks. Strong shocks: thermalization of the incoming kinetic energy flux.
7.4 – Examples: explosions in general. Supernova remnants. Accretion columns on white dwarf stars or neutron stars.
Optional complementary topic:
– The transmission of information in gases. Characteristic curves. The shock fronts as the natural, inescapable result of compressions in gases.
Optional numerical practical:
– gas element tracking across a shock.
Actividades a desarrollar en otro idioma
- All written material given by the lecturer to the students will be in English, including all course notes, the exercise and auxiliary sheets, practical scripts, computer programs and exam sheets.
- The theoretical lectures will be given in English. Support will be given to the students concerning specific technical terms pertaining to fluid dynamics, including pronunciation and correct spelling.
- The presentation of practicals or daily exercises by the students will be either in Spanish or English depending on the students’ preferences and their proficiency in those languages. The language to use will be chosen by the students on a case-by-case basis.
