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Identical coursesMathematical Introduction to Fluid Dynamics (B-KUL-G0N98A)
N. | Bacchini Fabio (substitute) Aims
1. The student is able to provide the mathematical formulation of basic concepts of fluid dynamics, he/she is able to compose the conservation laws for non-viscous ideal currents. He/she is able to connect these mathematical concepts and comparisons with the physical reality.
2. The student is familiar with (i) mathematics as a tool to model concrete problems of flows of gases and liquids (ii) he/she is aware of the limitations of the applicability of the relevant mathematical models, (iii) he/she is familiar with how physically relevant solutions can be found with simplified approximations (iv) he/she knows that eventually the mathematical result must be tested against the physical reality.
3. The student can come up with the mathematical solutions for non-compressible irrotational flows and flows with a free surface that he/she sees in everyday life around him and for example problems in sustainability.
4. The student knows that mathematical methods and models of fluid dynamics have applications outside of fluid dynamics. He/she can handle simple problems associated with traffic.
Previous knowledge
The required knowledge is vector calculus, calculus of real functions and differential equations.
Order of Enrolment
This course unit is a prerequisite for taking the following course units:
G0O05B : Eindproject
Is included in these courses of study
- Bachelor in de wiskunde (Leuven) 180 ects.
- Bachelor in de fysica (Leuven) (Minor sterrenkunde en informatica) 180 ects.
- Courses for Exchange Students Faculty of Science (Leuven)
Activities
4.4 ects. Mathematical Introduction to Fluid Dynamics (B-KUL-G0N98a)
Content
1. Introduction: Motivation, repetition of integral theorems of vector calculus, Continuum model.
2 Kinematics of fluid flow, visualization of a moving fluid; Eulerian and Lagrangian description; Reynolds transport theorem, Cauchy-Stokes decomposition theorem; Divergence and vorticity.
3. Equations for non-viscous ideal fluids: integral and differential versions of conservation equations for mass, momentum and energy; Euler equations.
4. Compressibility, vorticity and Bernoulli's theorem: Vorticity and irrotational flows, transport of vorticity in non-viscous liquids, tornadoes; Bernoulli's theorem.
5. Non-compressible irrotational flows: potential flows around spheres and cylinders, D'Alembert's paradox and lift.
6. Flows with a free surface: Linear surface gravity waves, Kelvin's ship and duck waves, tsunamis, surface gravity-capillaritity waves, wind over water: Kelvin-Helmholtz instability; Nonlinear flows with a free surface, hydraulic jumps.
7. Traffic flows: Continuum theory, Linear waves on a uniform flow, Initial value problem for a non-uniform traffic, method of characteristics, expansion wave in case of green light, cutting characteristics and shocks, traffic stopped by a red light; effect of diffusion on traffic shocks, advection-diffusion equation, Cole-Hopf transformation, Burgers equation
1.6 ects. Mathematical Introduction to Fluid Dynamics: Exercises (B-KUL-G0N99a)
Content
1. Introduction: Motivation, repetition of integral theorems of vector calculus, Continuum model.
2 Kinematics of fluid flow, visualisation of a moving fluid; Eulerian and Lagrangian description; Reynolds transport theorem, Cauchy-Stokes decomposition theorem; Divergence and vorticity.
3. Equations for inviscid ideal fluids: integral and differential versions of conservation equations for mass, momentum and energy; Euler equations.
4. Compressibility, vorticity and Bernoulli's theorem: Vorticity and irrotational flows, transport of vorticity in non-viscous liquids, tornadoes; Bernoulli's theorem.
5. Non-compressible irrotational flows: potential flows around spheres and cylinders, D'Alembert's paradox and lift.
6. Flows with a free surface: Linear surface gravity waves, Kelvin's ship and duck waves, tsunamis, surface gravity-capillaritity waves, wind over water: Kelvin-Helmholtz instability; Nonlinear flows with a free surface, hydraulic jumps.
7. Traffic flows: Continuum theory, Linear waves on a uniform flow, Initial value problem for a non-uniform traffic, method of characteristics, expansion wave in case of green light, cutting characteristics and shocks, traffic stopped by a red light; effect of diffusion on traffic shocks, advection-diffusion equation, Cole-Hopf transformation, Burgers' equation
Evaluation
Evaluation: Mathematical Introduction to Fluid Dynamics (B-KUL-G2N98a)
Explanation
The exam of this course is split up in two parts: theory and exercises. The theory will be assessed during the semester in two oral, open book exams. The exercise part will take place during the exam time by means of a written, open book exam.
Information about retaking exams
For the resit, theory and exercise exam will take place on the same day. During the exercise exam, the student will also have the opportunity to improve the score on the theory exam.