Numerical methods for fluids Numerical methods for incompressible fluids EAFIT, Escuela de Verano, Medellin, June 17-25, 2015 Course instructors: • (LJK, Grenoble) • (UPMC, Paris) 1. Syllabus This class is an introduction to mathematical and computational aspects of incompressible fluid flow simulations. It is presented to the point of view that the students are (going to be) applied mathematicians, physicists or engineers.

Computational fluid dynamics is at the crossroad of many disciplines and, like many topics at the interfaces between disciplines, its access may seem a bit harsh for (under)graduate students in mathematics or engineering. Our goal is to cover the main aspects of finite element methods for incompressible flows. We have sought to achieve a right balance between theoretical concepts, numerical analysis, description of schemes and algorithms and engineering applications. Numerical experiments using FreeFem++ will help students to understand these concepts and see advanced numerical methods in action. The 7-days course is divided into 5 parts: • A fluid mechanics primer • notations, vectors, tensors • conservation laws • flow models and simplifications • The Stokes model • mathematical and numerical analysis • finite element approximation, resolution • unsteady Stokes problem • The Navier-Stokes model • analysis of the steady-state problem • discretization procedures • Two-fluid or two-phase flows • level set formalism • bifluid simulations • Shape optimization for fluids • Appendix • variational approximation • error estimates • mesh adaptation 2. Material Students can download • the of the course in PDF (Part I) • the of the course in PDF (Part II) • the numerical experiments in PDF • the documentation in PDF • and 3.

Bernard, Fluid dynamics, CUP, 2015. Paterson: A first course in fluid dynamics, CUP, 1983. Kundu & I.M Cohen: Fluid mechanics, AP, 2002. (Electronic resource). Best Of Platinum Edition Dj Otzi Music. Acheson: Elementary fluids dynamics, OUP, 1990. Batchelor: An introduction to fluid dynamics,. The study of the dynamics of fluids is a central theme of modern applied mathematics. It is used to model a vast range of physical phenomena and plays a vital role in science and engineering. This textbook provides a clear introduction to both the theory and application of fluid dynamics, and will be suitable for all.

References • Functional and numerical analysis • Allaire G., Numerical analysis and optimizaton, Oxford Science Publishing, (2007) • Brezis H., Analisis funcional, Teoria y applicaciones, Allianza Editorial, (1983). • Ciarlet P.G., The finite element methods for elliptic problems, SIAM classics, 40, (2002) • Ern A., Guermond J.L., Theory and practice of finite elements Applied Mathematical Series, 159, Springer, (2004) • Evans L.C., Partial differential equations, AMS, (2002).

The study of the dynamics of fluids is a central theme of modern applied mathematics. It is used to model a vast range of physical phenomena and plays a vital role in science and engineering. This textbook provides a clear introduction to both the theory and application of fluid dynamics, and will be suitable for all undergraduates coming to the subject for the first time. Prerequisites are few: a basic knowledge of vector calculus, complex analysis, and simple methods for solving differential equations are all that is needed. Throughout, numerous exercises (with hints and answers) illustrate the main ideas and serve to consolidate the reader's understanding of the subject. The book's wide scope (including inviscid and viscous flows, waves in fluids, boundary layer flow, and instability in flow) and frequent references to experiments and the history of the subject, ensures that this book provides a comprehensive and absorbing introduction to the mathematical study of fluid behaviour.