Shock and Rarefaction Waves in a Hyperbolic Model of Incompressible Fluids
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1 Shock and Rarefaction Waves in a Hyperbolic Model of Incompressible Fluids Andrea Mentrelli Department of Mathematics & Research Center of Applied Mathematics (CIRAM) University of Bologna, Italy
2 Summary Compressible vs. incompressible fluids Thermodynamic restrictions Models of incompressibility (perfect incompressibility, QTI model, EQTI model) An example of EQTI equation of Shock and rarefaction waves in EQTI materials The Riemann problem in EQTI materials
3 Introduction Incompressibility is a useful idealization when modeling materials characterized by extreme resistance to volume changes Incompressible fluids are treated as the limit case of compressible fluids Purely mechanical problems The limit is fine :) Solutions of incompressible model equations are obtained as limit of compressible ones (P.-L Lions, N. Masmoudi, J. Math. Pures Appl. 77, , 1998; S. Klainerman, S. Majda, Comm. Pure Appl. Math. 35, , 1982; etc. etc.) Thermo-mechanical problems The limit is ambiguous :( There are several definitions of incompressibility (perfect incompressibility; quasithermal-incompressibility; extended-quasi-thermal-incompressibility)
4 Introduction From a mathematical point of view, compressible and incompressible materials have different treatments: compressible materials the pressure is a constitutive function incompressible materials the pressure is a Lagrange multiplier (associated to the constraint of incompressibility) In order to compare the solutions of compressible/incompressible materials, it is convenient to choose the pressure (instead of the density) as a thermodynamic variable along with temperature
5 Introduction Euler equations (perfect fluid in the absence of external body forces) (conservation of mass) (conservation of momentum) (conservation of energy) Independent variables: pressure temperature velocity components Equations of : density (or specific volume) specific internal energy
6 Thermodynamic restrictions The constitutive equations of a material must satisfy the entropy principle and the conditions for the thermodynamic stability 1. Entropy principle Gibbs equation (entropy) Chemical potential Once the thermal equation of is known: with or
7 Thermodynamic restrictions 2. Thermodynamic stability Enthalpy Specific heat The chemical potential must be a concave function of pressure and temperature: Thus, thermodinamic stability requires:
8 Thermodynamic restrictions After introducing - as usual when studying the compressibility of a material - the two parameters thermal expansion coefficient compressibility factor The concavity of the chemical potential requires Thus, the thermodinamic stability conditions are Adiabatic sound velocity:
9 Models of incompressibility Perfectly incompressible material According to this model, a material is incompressible when its constitutive functions do not depend on the pressure It is easily seen that: the specific volume must be constant! (Müller's paradox) (I. Müller, Thermodynamics, Pitman, London, 1985)
10 Models of incompressibility Quasi-Thermal-Incompressible (QTI) Material According to this model, only the density is assumed to be indipendent on the pressure In this case: If then The Müller's paradox is removed, and perfectly incompressible materials are obtained as a limit of QTI materials The chemical potential is not concave: QTI materials are thermodynamically unstable (K. R. Rajagopal et al., Math. Model. Meth. Appl. Sci. 6, , 1996; H. Gouin, A. Muracchini, T. Ruggeri, Continuum Mech. Thermodyn. 24, , 2012)
11 Models of incompressibility Extended-Quasi-Thermal-Incompressible (EQTI) Material A compressible material satisfying the thermodynamic conditions is called an extendedquasi-thermal-incompressible (EQTI) material if: with An EQTI material is a stable compressible material that approximates a Müller incompressible material to the second order Assuming reference It is easily obtained:
12 Models of incompressibility Extended-Quasi-Thermal-Incompressible (EQTI) Material Thermodynamic stability is guaranteed if If And EQTI materials approximate a perfectly incompressible material (if the pressure is smaller than a critical value) Thermodynamically stability is guaranteed (H. Gouin, T. Ruggeri, Internat. J. Non-Linear Mech. 47, , 2012; A. M., T. Ruggeri, Internat. J. Non-Linear Mech. 51, 87-96, 2013)
13 An example of EQTI fluid We perform a linear expansion of the specific volume near the reference Modified Bousinnesq equation is an equation of of a EQTI material with:
14 An example of EQTI fluid Modified Bousinnesq equation Thermal expansion coefficient and compressibility factor: Condition for thermodynamic stability
15 Shock waves in EQTI fluids Shock wave, perturbed shock velocity, s unperturbed Euler equations (1D) Eigenvalues and eigenvectors
16 Shock waves in EQTI fluids Shock wave, perturbed shock velocity, s unperturbed Euler equations (1D) Rankine-Hugoniot conditions Hugoniot locus One-parameter family of solutions (parameter: ) Selection rule?
17 Shock waves in EQTI fluids Shock wave, perturbed shock velocity, s unperturbed Euler equations (1D) Genuinely non-linear waves Hugoniot locus Lax condition: Linearly degenerate waves Lax condition: ble ssi i dm ina
18 Shock waves in EQTI fluids Shock wave, perturbed shock velocity, s unperturbed Euler equations (1D) Locally linearly degenerate waves for some Hugoniot locus ib iss m ad in le Liu condition: ina d mis sib le through
19 Shock waves in EQTI fluids Shock wave, perturbed shock velocity, s unperturbed Euler equations (1D) Rankine-Hugoniot conditions One-parameter family of solutions (parameter: )
20 Shock waves in EQTI fluids Shock waves in water specific volume fluid velocity temperature shock velocity
21 Shock waves in EQTI fluids Shock waves in the incompressible limit specific volume fluid velocity temperature shock velocity
22 Rarefaction waves in EQTI fluids Rarefaction wave, perturbed unperturbed Euler equations (1D) The Rarefaction wave is a similarity solution
23 Rarefaction waves in EQTI fluids Rarefaction wave, perturbed unperturbed Euler equations (1D)
24 Rarefaction waves in EQTI fluids Rarefaction wave, perturbed unperturbed Euler equations (1D)
25 Rarefaction waves in EQTI fluids Rarefaction wave, perturbed unperturbed Euler equations (1D) Taking into account that for a EQTI fluid After some manipulations...
26 Rarefaction waves in EQTI fluids Rarefaction wave, perturbed unperturbed Euler equations (1D) The integral curves are defined by
27 Rarefaction waves in EQTI fluids Rarefaction waves in water specific volume fluid velocity temperature 'development' velocity
28 Rarefaction waves in EQTI fluids Rarefaction waves in the incompressible limit specific volume fluid velocity temperature 'development' velocity
29 The Riemann Problem Euler equations (1D) Typical solution: Two nonlinear waves (shock/rarefaction waves) Contact discontinuity
30 The Riemann Problem The knowledge of the Rankine-Hugoniot curves of the shock waves and of the integral curves of the rarefaction waves allow to completely solve the Riemann problem Numerical results
31 References & Acknowledgments References I. Müller, Thermodynamics, Pitman, London (1985) H. Gouin, A. Muracchini, T. Ruggeri, Continuum Mech. Thermodyn. 24, (2012) H. Gouin, T. Ruggeri, Internat. J. Non-Linear Mech. 47, (2012) A. M., T. Ruggeri, Internat. J. Non-Linear Mech. 51, (2013) Acknowledgments Partially supported by GNFM/INdAM Young Researchers Project 2012 Hyperbolic Models for Incompressible Materials
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