Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MM277	Special Topics in Fluid Mechanics	

Section	Semester	Hours: C+S+L	Category	Type	
Applied Mathematics	2	2+2+0		compulsory	
Mathematical Models in Mechanics and Astronomy - in English	1	2+2+0		compulsory	

Teaching Staff in Charge	Prof. KOHR Mirela, Ph.D., mkohr@math.ubbcluj.ro	
Aims	The aim of this course is to introduce and thoroughgoing study certain notions and fundamental results of fluid mechanics. The introduction in some special chapters of fluid mechanics with a special attention on the mathematical theory of viscous fluid flows at low Reynolds numbers. The presentation of some modern mathematical methods, such as the method of fundamental solutions and its applications in the theory of viscous incompressible fluid flows. The applications of the potential theory for Stokes, Stokes resolvent and Brinkman equations in the study of some viscous linearized flows in the presence of solid bodies, fluid interfaces, or in porous media. The presentation of some numerical methods that are useful in the tretment of the proposed problems, with a special attention on the boundary element method. The problems that will be treated have many applications in medicine, biology, chemical industry, geology, etc. The implication of the students in the research activity.  	
Content	1.	Fundamental notions and results of fluid mechanics.
-	Fluid, configuration, flow. 
-	Velocity and acceleration fields of fluid flow (continuous medium)
-	Principle of the mass conservation. The continuity equation.
-	The Cauchy stress principle. The Cauchy fundamental theorem.
-	The principle of the rate of change of momentum. The Cauchy equation.
2.	Constitutive equations of fluid mechanics.
-	The constitutive equation of ideal fluid. The Euler equations.
-	The constitutive equations of viscous Newtonian fluid. The Navier-Stokes 
        equations.
3.	Nondimensional analysis of the equations of viscous incompressible fluid flow 
-	Special forms of the Navier-Stokes equations.
-	Boundary and initial conditions for the problem of viscous incompressible fluid 
        flow.
4.	Uniqueness results for the Stokes system.
-	Uniqueness result of the classical solution to the Stokes system in a bounded 
        domain in R&#8319; (n=2,3). 
-       Uniqueness result of the classical solution to the Stokes system in an unbounded
        domain  in R&#8319; (n=2,3). 
5.      The method of fundamental solutions in fluid mechanics
-       The Green function, the pressure vector and the stress tensor for the Stokes 
        flow due to a point force. The Oseen-Burgers tensor in R&#8319; (n=2, 3).
-       The direct boundary integral representation of the velocity field of a Strokes 
        flow in a bounded or exterior domain in R&#8319; (n=2, 3).
-       Applications.
6.      The potential theory for the Stokes equations.
-       The theory of compact operators. The Fredholm alternatives. The double-layer 
        potential. The single-layer potential. Properties. 
-       Applications of the potential theory for the Stokes system: Existence and 
        uniqueness results to the boundary value problems for the Stokes system on 
        bounded or exterior domains in R&#8319; (n&#8805;2). Applications in the study of Stokes 
        flows.
-       The completed double layer boundary integral equation method in the study of 
        some Stokes flows. Existence and uniqueness results, as well as numerical 
        results.
7.	Boundary value problems for viscous incompressible flows at low Reynolds numbers 
        in porous media or past porous bodies.
-	Existence and uniqueness results in Hölder or Sobolev spaces.
-	Applications and numerical results. 
10.     Numerical methods in the study of some problems concerning linearized viscous 
        flows, with a special attention on the boundary element method.
	
Bibliography	1. Kohr, M., Pop, I., Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press 
(Wessex Institute of Technology Press)/Computational  Mechanics Publications, 
Southampton (UK) - Boston, 2004
2. Kohr, M., Probleme Moderne în Mecanica Fluidelor Vâscoase, Presa Universitară 
Clujeană, Cluj-Napoca, 2 vols., 2000
3. Kohr, M., Studiul unor Mişcări Fluide Vâscoase Incompresibile prin Metode Integrale pe Frontieră, Presa Universitară Clujeană, Cluj-Napoca, 1997
4. Wendland, W.L., Hsiao, G.C., Boundary Integral Equations, Springer, Heidelberg, 2008
5. Dragoş, L., Mecanica Fluidelor, vol. I, Editura Academiei, Bucureşti, 1999
6. Truesdell, C., Rajagopal, K.R., An Introduction to the Mechanics of Fluids, Birkhäuser, Basel, 2000
7. Truesdell, C., A First Course in Rational Continuum Mechanics, Vol. 1, Academic Press, New York, 1991
8. Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992
9. Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics, Oxford 
University Press, Oxford, 1997
10. Power, H., Wrobel, L.C., Boundary Integral Methods in Fluid Mechanics, WIT 
Press/Computational Mechanics Publications, Southampton (UK) &#8211; Boston, 1995
11. Varnhorn, W., The Stokes Equations, Akademie Verlag, Berlin, 1994.
	
Assessment	Exam (70%)+ student activity (30%).