Mathematical Methods in Fluid Dynamics 1

Garant předmětu:

Pavel Strachota

Lecturer:

Pavel Strachota

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

First, the differential equations representing the conservation laws of fluid flow are briefly derived and reviewed. Next,

the problems for the resulting equations are formulated, focusing on boundary conditions specification. The reference

problem undergoes numerical analysis with emphasis on explaining the weak solution and its role in describing real

phenomena. In the second part, important problems are introduced, involving fluid flow and other effects (heat transfer,

chemical reactions, multiphase nature) and an adequate mathematical description is chosen.

Requirements:

Basic courses of calculus and differential equations (in the extent of the courses 01DIFR, 01MA1, 01MAA2-4, 01RMF held at the FNPE

CTU in Prague).

Syllabus of lectures:

1. Formulation and brief derivation of the conservation laws in a fluid (continuity equation, Navier-Stokes equations,

energy equation).

2. Euler equations, boundary conditions for problems of viscous and inviscid flow.

3. Irrotational flow, potential equation.

4. Fundamental qualitative aspects of Navier-Stokes equations – classical and weak solutions, questions of existence

and uniqueness of solutions in the stationary and non-stationary case.

5. Fluid flow problems, formulation of conservation laws in less dimensions, boundary conditions in less dimensions.

6. Turbulent flow and turbulence modeling, Reynolds averaging of NS equations and filtering.

7. Fluid thermodynamics, heat transfer, radiation.

8. Reacting multicomponent flow, combustion modeling.

9. Multiphase flow, phase transitions.

10. Dimensionless numbers characterizing the flow.

11. Moving boundary flow problems.

Syllabus of tutorials:

Study Objective:

To learn basic principles of mathematical modelling in fluid dynamics, to learn and understand mathematical models of various type sof flows (compressible or incompressible, viscous or non-viscous, laminar or turbulent, etc.), to learn about basic methods and results in the field of qualitative properties of the Navier-Stokes equations.

Study materials:

Key references:

[1] J. N. Reddy, Principles of Continuum Mechanics: Conservation and Balance Laws with Applications. Cambridge

University Press, 2017.

[2] Y. A. Cengel, J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications (4th ed.), McGraw-Hill Education,

2017.

[3] C. Pozridikis, Computational Fluid Dynamics: Theory, Computation, and Numerical Simulation (2nd ed.), Springer

Science + Business Media LLC, 2017.

[4] E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids (2nd ed.), Birkhäuser, Springer, 2017.

Recommended references:

[5] Y. A. Cengel, A. J. Ghajar, Heat and Mass Transfer: Fundamentals & Applications. McGraw-Hill Education, 2015.

[6] M. Gurtin, An Introduction to Continuum Mechanics. Vol. 158. Academic Press, 1981.

[7] C. R. Doering, J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations. Cambridge University Press, 1995.

[8] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis (2nd ed.), SIAM Philadelphia, 1995.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Matematické inženýrství (compulsory course in the program)
• Fyzikální elektronika - Počítačová fyzika (elective course)