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Course, academic year 2023/2024
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Thermodynamics and Mechanics of Non-Newtonian Fluids - NMMO402
Title: Termodynamika a mechanika nenewtonovských tekutin
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Note: enabled for web enrollment
Guarantor: RNDr. Karel Tůma, Ph.D.
Class: M Mgr. MOD
M Mgr. MOD > Povinné
Classification: Mathematics > Differential Equations, Potential Theory
Incompatibility : NDIR057
Interchangeability : NDIR057
Is interchangeable with: NDIR057
Annotation -
Last update: T_MUUK (14.05.2013)
Description of non-Newtonian phenomena and explanation how to model these phenomena within the complete thermomechanical framework using the concept as natural configuration, maximization of rate of entropy production, implicit constitutive theory. Basic mathematical insights on equations describing steady and unsteady flows of Newtonian and non-Newtonian incompressible fluids will be also given.
Aim of the course -
Last update: T_MUUK (14.05.2013)

The course aims to describe basic phenomena that cannot be captured by Newtonian (Navier-Stokes) fluids and then to provide a derivation of models that have the ability of capturing these phenomena.

Course completion requirements -
Last update: RNDr. Karel Tůma, Ph.D. (11.06.2019)

Credit must be received before the exam. Credit is obtained for successfully solved homeworks. The exam consists of a written test and an oral part.

Literature - Czech
Last update: T_MUUK (14.05.2013)

[1] W. R. Schowalter: Mechanics of Non-Newtonian Fluids, Pergamon Press (Oxford), 1978.

[2] R. R. Huilgol: Continuum mechnaics of viscoelastic liquids, Hindusthan Publishing Co. (Delhi), 1975.

[3] J. Malek, K. R. Rajagopal: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Handbook of Differential Equations, Evolutionary Equations, Vol. 2 (eds. C. Dafermos and E. Feireisl),

Elsevier, 2005, 371-459.

Teaching methods -
Last update: T_MUUK (14.05.2013)

Lecture course

Requirements to the exam -
Last update: RNDr. Karel Tůma, Ph.D. (30.04.2020)

You can enroll for the exam only if you already got credits for the tutorials.

There will be two variants of the exam (depending on whether the Covid-19 measures will be relaxed):

(1 - classical full-time variant) The exam consists of the test and the oral part. The test will contain four examples corresponding to the syllabus of the lecture and examples trained at the tutorial. The requirements to the oral part of the exam correspond to the syllabus of the lecture in the range presented in the lecture.

(2 - remote variant) The test will be sent by email, computed and sent back. Then we will discuss the calculation and an additional question will follow.

Syllabus -
Last update: RNDr. Karel Tůma, Ph.D. (22.02.2022)

1. Introduction: What is a fluid? Balance equations: Balance of mass, balance of linear momentum, balance of angular momentum, balance of energy. Constitutive equations: differential type, rate type, integral type. Navier-Stokes equations. Incompressibility. Examples of boundary conditions.

2. Viscosity. Non-Newtonian phenomena part 1: Shear thinning, shear thickening: Singular and degenerated generalized viscosity, power-law model. Pressure thickening: Barus model. Presence of activation/deactivation criteria: Bingham model, Herschel-Bulkley model, locking. Presence of non-zero normal stress differences in a simple shear flow: Rod climbing (Weissenberg effect), die swelling, delayed die swelling, pressure hole error, flow over inclined plane, inverted secondary flow.

3. Non-Newtonian phenomena part 2: Stress relaxation: Stress relaxation function, relaxation time. Non-linear creep: creep function, retardation time. Typical response of viscoelastic fluid and viscoelastic solid. Mechanical analogs: Linear spring and linear dashpot, Kelvin element, Maxwell element, Oldroyd element, Burgers element. Stress-strain relation, initial conditions.

4. Application of the Laplace transform. Responses of the Kelvin-Voigt, Maxwell and Oldroyd elements in the stress-relaxation test and creep test. General stress-strain relation.

5. Constitutive theory: Basic considerations. Principle of frame indifference (PFI) and Principle of material frame indifference (PMFI). Euclidean and Galilean change of observers. Objectivity. Consequences of PFI and PMFI on constitutive theory.

6. Constitutive theory: Incompressibility. Objective derivatives of tensor quantities: Gordon-Schowalter, Jaumann-Zaremba (corrotational), upper-convected Oldroyd, lower-convected Oldroyd. Full three-dimensional Oldroyd-A and Oldroyd-B models. Relation to Oldroyd mechanical analog: How to generalize?

7. Rod climbing experiment. Calculation of Oldroyd-A and Oldroyd-B. Oldroyd-B enables climbing, Oldroyd-A does not. Since 90\% of fluids climb (Oldroyd 1950), upper-convected derivative in Oldroyd-B is prefered.

8. Continuum thermodynamics: Specific energy, temperature, entropy, second law of thermodynamics, Clausis-Duhem inequality, balance of entropy, entropy flux, entropy production. Entropy production of Navier-Stokes-Fourier fluid (compressible and incompressible).

9. Derivation of the constitutive relation from the knowledge how the body stores and dissipates energy: Linear closures, assumption of the maximization of the rate of entropy production. Derivation of Navier-Stokes-Fourier system (using both affinities and fluxes). General scheme to obtain the constitutive relations.

10. Derivation of compressible Eulerian elastic and Kelvin-Voigt solids. Compressible Kelvin-Voigt with temperature-dependent material parameters: consistent evolution equation for temperature.

11. Thermodynamically consistent incompressible rate-type fluid models. Natural configuration. Derivation of model by Rajagopal and Srinivasa (2000). Its linearization to Oldroyd-B model.

12. Compressible rate-type fluid models. Thermodynamic derivation of Maxwell and Oldroyd-B model. Helmholtz free energy, reduced thermodynamic identity. Apriori estimates using thermodynamical approach. Multiple natural configurations and termodynamic derivation of Burgers model.

13. Thermodynamic derivation of compressible and incompressible Navier-Stokes-Fourier-Korteweg model and its properties.

(14. Gibbs potential based thermodynamical approach. Legendre transform. Derivation of Maxwell model with corrotational objective derivative.)

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