Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.
Derivation of the governing equations: Euler and Navier-Stokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.