## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Boundary integral methods in fluid mechanics
 Code Semes-ter Hours: C+S+L Credits Type Section MM021 8 2+2+0 7.5 optional Matematică
 Teaching Staff in Charge
 Assoc.Prof. KOHR Mirela, Ph.D., mkohr@math.ubbcluj.ro
 Aims The aim of this course is to realize an introduction in some special chapters of fluid mechanics, with a special accent on the theory of viscous linearized flows. The first part of the course is devoted to the basic results concerning the flows of viscous fluids. It would be presented the linearized method of Stokes, the singularity method and also, the theory of hydrodynamical potential. A special chapter of this course is concerned with certain boundary integral methods for Stokes flows in the presence of solid bodies and walls. The boundary integral methods for Stokes flows due to interfaces are also obtained.
 Content 1. Preliminaries. -Description of the fluid flows -General principles of fluid flows -Constitutive equation of a Newtonian fluid. The Navier-Stokes equations -Linearization of the equations of fluid flow. The equations of Stokes and Oseen flows 2. The linearization method due to Stokes. -Properties of steady and unsteady Stokes flows -The generalized Lorentz reciprocal identity for Stokes flows -Uniqueness of solution of steady Stokes flow -Uniqueness of solution of unsteady Stokes flow -The flow of an incompressible Newtonian fluid past a circular cylinder. The Stokes paradox -The Stokes flow of an incompressible Newtonian fluid past a solid sphere. The Stokes law 3. The singularity method. -Green's functions associated to steady and unsteady Stokes flows. Properties of Green's functions -Examples of Green's functions -The fundamental solutions of Stokes flow. Applications 4. The theory of hydrodynamical potential. -Properties of the single-layer potential -Properties of the double-layer potential 5. Direct boundary integral representations of Stokes flows -The boundary integral representation of the velocity field of a Stokes flow -The boundary integral representation of the pressure filed of a Stokes flow 6. Indirect boundary integral representations of Stokes flows. -Representation of a Stokes flow in terms of a single-layer potential -Representation of a Stokes flow in terms of a double-layer potential -The completed double-layer boundary integral equations method. Applications: the study of Stokes flows past or due to the motion of solid bodies -Existence and uniqueness results -Numerical methods applied to certain Stokes flow problems 7. Boundary integral methods for Stokes flows due to interfaces -Direct boundary integral representation for a Stokes flow due to an interface -Indirect boundary integral representations -Existence and uniqueness results -The discontinuity in the interfacial surface force -Examples of interfaces
 References 1. C.A. Brebbia, J.C. Telles, L.C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering, Berlin:Springer-Verlag, 1984. 2. L. Dragos, Principiile Mecanicii Mediilor Continue, Ed. Tehnica, Bucuresti, 1983. 3. L. Dragos, Mecanica Fluidelor, Editura Academiei, 1999. 4. S. Kim, S.J. Karrila, Microhydrodynamics: Principles and Selected Applications, London: Butherworth-Heinemann, 1991. 5. M. Kohr, Studiul unor Miscari Fluide Vascoase prin Metode Integrale pe Frontirea, Presa Universitara Clujeana, Cluj-Napoca, 1997. 6. M. Kohr, Probleme Moderne in Mecanica Fluidelor Vascoase, Presa Universitara Clujeana, Cluj-Napoca, 2000. 7. A. Ladyzshenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. 8. C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, 1992. 9. C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, 1997. 10. H.Power, L.C. Wrobel, Boundary Integral Methods in Fluid Mechanics, Sauthampton: Computational Mechanics Publications, 1995
 Assessment Exam.