Kinetic theory is a discipline introduced by Boltzmann in the late 19th century tostudy monatomic gas at the microscopic level, introducing a distribution function,which depends on macroscopic variables, such as time and position of particles inspace and microscopic variables such as particle velocity. The evolution of the distributionfunction is defined by the Boltzmann equation from which the macroscopicequations of Euler and NavierStokes can be derived.The theory of Rational Extended Thermodynamics, formulated by Ingo Müller andTommaso Ruggeri in the last century, deals with the study of non equilibrium phenomenaat a macroscopic level, such as shock waves, micro and nanoflows, secondsounds, light scattering, rarefied gases and so on. It consists of a hierarchy of balancelaws, where dissipative fluxes are assumed as field variables. The same hierarchy ofequations is found in the moment systems of kinetic theory by truncating at an arbitraryorder of moments.There are various points of contact between the two theories. Extended thermodynamicspostulates the existence of a law of entropy that imposes conditions of nonnegativityfor the production of entropy, and kinetic theory also places conditions onthe sign of entropic dissipation through the H theorem. Grad’s 13moment theoryprovides the same phenomenological equations as extended thermodynamics andrepresents a theoretical validation. It is therefore interesting to study a phenomenonof unbalance for gases from the perspective of kinetic theory and from the point ofview of extended thermodynamics, then making comparisons between the resultsprovided by the two theories.viiiIn this thesis, a study is conducted on monatomic granular gases, particles thatare subject to inelastic collisions, in which there is no energy conservation. The firstcase study concerns dilute granular gases, characterized by very spaced particles,where the centers of two colliding particles coincide and in the collision of two particlesthe effect of nearby particles is neglected. The study is done first by consideringa model of differential equations for 13 moments, of which the terms of productionare calculated with the method of Grad moments, typical of kinetic theory. Thenthe same model is derived using the theory of extended thermodynamics and someinvestigations are conducted on the hyperbolicity region of the system and the convexityof entropy. In addition, spatially homogeneous solutions are studied in theonedimensional case, comparing in particular the decay of the temperature of thegas to the Haff law. Stationary solutions are also determined in the onedimensionalcase.A more complex case to be considered from the point of view of kinetic theory andextended thermodynamics concerns dense granular gases: particles interact in sucha way that in binary collision the effect of other nearby particles cannot be neglected,and the centers of two colliding particles are distinct. A nearly linear model of differentialequations for 14 moments is presented and flows and production termsare determined through kinetic theory. The model is then derived in the contextof extended thermodynamics for moderately dense gases. This thesis also deepenssome biological applications through Extended Thermodynamics. A model of14 moments is proposed for the study of blood, thought as a mixture, formed byplasma, red blood cells and white blood cells, of which solutions are determined inthe linear case, in plane symmetry and cylindrical symmetry. Another biological applicationregards the evolution of a chronic wasting disease through the definition ofa hyperbolic system that predicts finite wave speeds. The linear stability of solutionsand the behavior of acceleration waves are investigated.
(2024). Mathematical modelling of granular gases in the context of Grad’s Theory and Rational Extended Thermodynamics.
Mathematical modelling of granular gases in the context of Grad’s Theory and Rational Extended Thermodynamics
POLLINO, Anna Maria
20240101
Abstract
Kinetic theory is a discipline introduced by Boltzmann in the late 19th century tostudy monatomic gas at the microscopic level, introducing a distribution function,which depends on macroscopic variables, such as time and position of particles inspace and microscopic variables such as particle velocity. The evolution of the distributionfunction is defined by the Boltzmann equation from which the macroscopicequations of Euler and NavierStokes can be derived.The theory of Rational Extended Thermodynamics, formulated by Ingo Müller andTommaso Ruggeri in the last century, deals with the study of non equilibrium phenomenaat a macroscopic level, such as shock waves, micro and nanoflows, secondsounds, light scattering, rarefied gases and so on. It consists of a hierarchy of balancelaws, where dissipative fluxes are assumed as field variables. The same hierarchy ofequations is found in the moment systems of kinetic theory by truncating at an arbitraryorder of moments.There are various points of contact between the two theories. Extended thermodynamicspostulates the existence of a law of entropy that imposes conditions of nonnegativityfor the production of entropy, and kinetic theory also places conditions onthe sign of entropic dissipation through the H theorem. Grad’s 13moment theoryprovides the same phenomenological equations as extended thermodynamics andrepresents a theoretical validation. It is therefore interesting to study a phenomenonof unbalance for gases from the perspective of kinetic theory and from the point ofview of extended thermodynamics, then making comparisons between the resultsprovided by the two theories.viiiIn this thesis, a study is conducted on monatomic granular gases, particles thatare subject to inelastic collisions, in which there is no energy conservation. The firstcase study concerns dilute granular gases, characterized by very spaced particles,where the centers of two colliding particles coincide and in the collision of two particlesthe effect of nearby particles is neglected. The study is done first by consideringa model of differential equations for 13 moments, of which the terms of productionare calculated with the method of Grad moments, typical of kinetic theory. Thenthe same model is derived using the theory of extended thermodynamics and someinvestigations are conducted on the hyperbolicity region of the system and the convexityof entropy. In addition, spatially homogeneous solutions are studied in theonedimensional case, comparing in particular the decay of the temperature of thegas to the Haff law. Stationary solutions are also determined in the onedimensionalcase.A more complex case to be considered from the point of view of kinetic theory andextended thermodynamics concerns dense granular gases: particles interact in sucha way that in binary collision the effect of other nearby particles cannot be neglected,and the centers of two colliding particles are distinct. A nearly linear model of differentialequations for 14 moments is presented and flows and production termsare determined through kinetic theory. The model is then derived in the contextof extended thermodynamics for moderately dense gases. This thesis also deepenssome biological applications through Extended Thermodynamics. A model of14 moments is proposed for the study of blood, thought as a mixture, formed byplasma, red blood cells and white blood cells, of which solutions are determined inthe linear case, in plane symmetry and cylindrical symmetry. Another biological applicationregards the evolution of a chronic wasting disease through the definition ofa hyperbolic system that predicts finite wave speeds. The linear stability of solutionsand the behavior of acceleration waves are investigated.File  Dimensione  Formato  

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