Mathematical Analysis of Problems in Fluid Mechanics

The course addresses alternative mathematical formulation of flow problems based on integral representations of vector fields and variational principles.

Objectives

The course aims at familiarizing the students with, • The mathematical formulation of flow problems focusing on the proper definition of boundary conditions • The various types of formulations: velocity-pressure, velocity-vorticity • Integral representation of vector fields and its application to flow problem • Free-boundary problems • The use of variational principles in fluid mechanics

Prerequisites

Vector and tensor analysis – Differential Equations – Continuum Mechanics

Syllabus

The course includes the following topics: • Kinematics and dynamics of fluid flows, integral and differential formulations of the conservation laws • The Helmholtz decomposition theorem • Potential flow theory and formulation by means of integral equations • Application to classical aerodynamic theory the flow around a wing and the generation of its wake in constant density conditions, the development of the boundary layer and the strong viscous-inviscid interaction formulation • Free boundary problems: a) the case of aeroelastic coupling, b) the free-surface hydrodynamic problem • Weak formulations, the Lax Millgram theorem. Application to the flow in a rectangular cavity • Variational formulations: the Rietz theorem, application to the hydrodynamic free surface problem • Vortex based formulation of the flow equations: a) the constant density case for the flow around a wing profile, b) extension to compressible flows • Introduction to asymptotic methods and application to generic problems