Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

Dostalík, Mark; Průša, Vít; Tůma, Karel

doi:10.3390/e21121219

Cited by 10 publications

(7 citation statements)

References 130 publications

(164 reference statements)

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“…The reader interested in more involved applications of the proposed construction is kindly referred to Appendix A , where we discuss the stability the spatially inhomogeneous steady temperature field in a rigid body wherein the heat conductivity is a function of the temperature. Application to the flows of polymeric fluids is discussed in Dostalík et al [ 50 ], where the authors analyse the stability of the Taylor–Couette type flow of the Giesekus fluid. Finally, Dostalík and Průša [ 51 ] use the proposed method in a coupled thermal convection/conduction setting.…”

Section: Discussionmentioning

confidence: 99%

Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems

Bulíček

Málek

Průša

2019

Entropy

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Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium steady states in thermodynamically isolated systems. Unfortunately, the limitation to thermodynamically isolated systems is essential, and standard arguments are not applicable even for some very simple thermodynamically open systems.On the other hand, the nonlinear stability of thermodynamically open systems is usually investigated using the so-called energy method. Unfortunately, the designation "energy method" is clearly a misnomer. The mathematical quantity that is traditionally referred to as the "energy" is by no means linked to the energy in the physical sense of the word. Consequently, it would seem that genuine thermodynamical concepts are of no use in the nonlinear stability analysis of thermodynamically open systems.We show that this is not true. In particular, we propose a construction that in the case of simple heat conduction problem leads to a physically well-motivated Lyapunov functional, which effectively replaces the artificial Lyapunov functional used in the standard energy method. The proposed construction seems to be general enough to be applied in complex thermomechanical settings.

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Section: Discussionmentioning

confidence: 99%

Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems

Bulíček

Málek

Průša

2019

Entropy

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“…However, we recall that we have not discussed the relation between the functional and a metric structure on the state space; hence, the functional does not, at the moment, deserve to be referred to as the Lyapunov functional. See also Dostalík et al [50,51] for a similar application in the case of macroscopic viscoelastic rate-type models, where the authors have actually found a relation between the proposed Lyapunov type functionals and a metric on the state space. Inspecting (119) and (118), we also note that the time derivative of V meq is proportional to the net entropy production that vanishes at the stationary spatially homogeneous equilibrium state (81).…”

Section: Solvent Part: Noble-abel Stiffened-gas Equation Of Statementioning

confidence: 91%

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

Dostalík

Málek

Průša

et al. 2020

Fluids

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We revisit some classical models for dilute polymeric fluids, and we show that thermodynamically consistent models for non-isothermal flows of these fluids can be derived in a very elementary manner. Our approach is based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid of interest, which, in turn, leads to explicit formulae for the Cauchy stress tensor and for all of the fluxes involved. Having identified these mechanisms and derived the governing equations, we document the potential use of the thermodynamic basis of the model in a rudimentary stability analysis. In particular, we focus on finite amplitude (nonlinear) stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

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“…(The metric is constructed using the Bures-Wasserstein distance on the set of positive definite matrices, see Bhatia et al [15].) For details regarding this concept we refer the interested reader to Dostalík et al [16].…”

Section: Decay Of Perturbations -Mechanical Quantitiesmentioning

confidence: 99%

Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

Dostalík,

Průša,

Stein

2020

Preprint

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We investigate finite amplitude stability of spatially inhomogeneous steady state of an incompressible viscoelastic fluid which occupies a mechanically isolated vessel with walls kept at spatially non-uniform temperature. For a wide class of incompressible viscoelastic models including the Oldroyd-B model, the Giesekus model, the FENE-P model, the Johnson-Segalman model, and the Phan-Thien-Tanner model we prove that the steady state is stable subject to any finite perturbation.

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Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

Cited by 10 publications

References 130 publications

Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems

Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

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