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Course, academic year 2023/2024
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Theory of Mixtures - NMMO541
Title: Teorie směsí
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: RNDr. Ondřej Souček, Ph.D.
Class: M Mgr. MOD
M Mgr. MOD > Povinně volitelné
Classification: Mathematics > Mathematical Modeling in Physics
Incompatibility : NMOD043
Interchangeability : NMOD043
Is interchangeable with: NMOD043
Annotation -
Last update: T_MUUK (05.05.2015)
The goal of the course is to present several approaches to modelling of multi-component materials in the framework of mechanics and thermodynamics of continua. The general theory will be presented together with derivation of simplified models.
Course completion requirements -
Last update: prof. RNDr. Josef Málek, CSc., DSc. (12.10.2017)

Credit for exercises based on the active participation at the seminars and on the successful solution of homeworks has to be assigned before the beginning of the exam.

Literature - Czech
Last update: Mgr. Dalibor Šmíd, Ph.D. (05.05.2015)

K. Hutter, K. Johnk, Continuum methods of physical modelling, Springer-Verlag Berlin- Heidelberg, 2004.

R.M. Bowen, Theory of mixtures in continuum physics III, ed. A.C. Eringen. Academic Press, New York, 1976.

C.Truesdell, Rational thermodynamics, Springer-Verlag, New York, 1984.

L.Schneider, K. Hutter, Solid-fluid mixtures of frictional materials in geophysical and geo-technical context, Springer-Verlag Berlin-Heidelberg, 2009.

K.R. Rajagopal, L. Tao, Mechanics of mixtures, World scientific publishing, Co. Singapore, 1995.

D.A. Drew, S.L. Passman, Theory of multicomponent fluids, Springer, 1998.

I. Samohýl, Racionální termodynamika chemicky reagujících směsí, Academia, Praha 1982.

I. Muller, A thermodynamic theory of mixtures of fluids, Arch. Rat. Mech. Anal., 28, 1-39, 1968.

Requirements to the exam -
Last update: prof. RNDr. Josef Málek, CSc., DSc. (12.10.2017)

The exam is oral and consists of answering two questions from selected group of topics. More details is available at , item Syllabus and general remarks.

Syllabus -
Last update: Mgr. Dalibor Šmíd, Ph.D. (05.05.2015)

1. Continuum theory of mixtures: Motivation, basic applications, assumption of co-occupancy, kinematics

2. Measures and relations between them, balance of mass, momentum, angular momentum, energy and entropy for the mixture components and for the mixture as a whole, definition of mixture velocity, mixture classes I,II,III,IV

3. Class I mixtures, derivation of Fick’s law and Fick-Navier-Stokes-Fourier (Fick-NSF) model

4. Fick-NSF model with boundary conditions

5. Quasi-incompressible approximations, quasi-incompressible variants of the Fick-NSF model

6. Derivation of the Cahn-Hilliard-NSF model

7. Chemical reactions - introduction, stoichiometry

8. Chemical reactions - mixture of ideal gasses, chemical potential, chemical equilibrium, mass action law, chemical kinetics

9. Derivation of the Allen-Cahn-NSF model

10. Mechanical interaction between the phases, basic mechanical analogues of the interaction mechanisms in mixtures, drag, lift, Magnus force, virtual mass effect

11. Class II mixtures, derivation of the Darcy law and other simplified models for flows through porous media (Forchheimer, Brinkman)

12. A thermodynamic framework for the mixture of two liquids

13. Balance equations at interfaces, generalized interface conditions

14. Multi-phase mixture theory: formalism, averaging, balance equations, structure of the interaction terms

Entry requirements -
Last update: Mgr. Dalibor Šmíd, Ph.D. (05.05.2015)

Recommended prerequisities: Single-component continuum mechanics and thermodynamics.

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