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We investigate Rayleigh–Bénard convection problem in an extended Boussinesq approximation suitable for conditions in the Earths mantle. The aim is to evaluate the influence of depth-dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection, and we characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly nonlinear analysis. We have found that the character of convection differs substantially from the standard case of Rayleigh–Bénard convection.
Mark Dostalík; Ctirad Matyska; Vít Průša. Weakly nonlinear analysis of Rayleigh–Bénard convection problem in extended Boussinesq approximation. Applied Mathematics and Computation 2021, 408, 126374 .
AMA StyleMark Dostalík, Ctirad Matyska, Vít Průša. Weakly nonlinear analysis of Rayleigh–Bénard convection problem in extended Boussinesq approximation. Applied Mathematics and Computation. 2021; 408 ():126374.
Chicago/Turabian StyleMark Dostalík; Ctirad Matyska; Vít Průša. 2021. "Weakly nonlinear analysis of Rayleigh–Bénard convection problem in extended Boussinesq approximation." Applied Mathematics and Computation 408, no. : 126374.
Using the finite element method we quantitatively analyse temperature field evolution in a viscoelastic solid undergoing a loading–unloading process. In particular we investigate the temperature field inside a Kelvin–Voigt type viscoelastic body with a thin cutout. We find that the viscosity significantly contributes to the temperature field changes, and that the temperature field changes initiated by the loading–unloading process are strongly concentrated at the tip of the thin cutout. The predicted temperature field qualitatively corresponds to the temperature field observed in experiments focused on simultaneous heat and strain measurements at the crack tip inside materials such as the filled rubber.
Vít Průša; Karel Tůma. Temperature field and heat generation at the tip of a cutout in a viscoelastic solid body undergoing loading. Applications in Engineering Science 2021, 6, 100054 .
AMA StyleVít Průša, Karel Tůma. Temperature field and heat generation at the tip of a cutout in a viscoelastic solid body undergoing loading. Applications in Engineering Science. 2021; 6 ():100054.
Chicago/Turabian StyleVít Průša; Karel Tůma. 2021. "Temperature field and heat generation at the tip of a cutout in a viscoelastic solid body undergoing loading." Applications in Engineering Science 6, no. : 100054.
Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of the elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy–Green tensor is expressed as a function of stress, is of interest. We show how to use this new perspective in the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.
Vít Průša; K. Rajagopal. Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids. Fluids 2021, 6, 131 .
AMA StyleVít Průša, K. Rajagopal. Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids. Fluids. 2021; 6 (3):131.
Chicago/Turabian StyleVít Průša; K. Rajagopal. 2021. "Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids." Fluids 6, no. 3: 131.
A fluid occupying a mechanically isolated vessel with walls kept at spatially non-uniform temperature is in the long run expected to reach the spatially inhomogeneous steady state. Irrespective of the initial conditions the velocity field is expected to vanish, and the temperature field is expected to be fully determined by the steady heat equation. This simple observation is however difficult to prove using the corresponding governing equations. The main difficulties are the presence of the dissipative heating term in the evolution equation for temperature and the lack of control on the heat fluxes through the boundary. Using thermodynamical-based arguments, it is shown that these difficulties in the proof can be overcome, and it is proved that the velocity and temperature perturbations to the steady state actually vanish as the time goes to infinity.
M. Dostalík; V. Průša; K. R. Rajagopal. Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature. Continuum Mechanics and Thermodynamics 2020, 33, 515 -543.
AMA StyleM. Dostalík, V. Průša, K. R. Rajagopal. Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature. Continuum Mechanics and Thermodynamics. 2020; 33 (2):515-543.
Chicago/Turabian StyleM. Dostalík; V. Průša; K. R. Rajagopal. 2020. "Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature." Continuum Mechanics and Thermodynamics 33, no. 2: 515-543.
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.
Mark Dostalík; Josef Málek; Vít Průša; Endre Süli. A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids. Fluids 2020, 5, 133 .
AMA StyleMark Dostalík, Josef Málek, Vít Průša, Endre Süli. A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids. Fluids. 2020; 5 (3):133.
Chicago/Turabian StyleMark Dostalík; Josef Málek; Vít Průša; Endre Süli. 2020. "A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids." Fluids 5, no. 3: 133.
Implicit rate-type constitutive relations utilising discontinuous functions provide a novel approach to the purely phenomenological description of the inelastic response of solids undergoing finite deformation. However, this type of constitutive relation has so far been considered only in the purely mechanical setting, and the complete thermodynamic basis is largely missing. We address this issue, and we develop a thermodynamic basis for such constitutive relations. In particular, we focus on the thermodynamic basis for the classical elastic–perfectly plastic response, but the framework is flexible enough to describe other types of inelastic response as well.
David Cichra; Vít Průša. A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation. Mathematics and Mechanics of Solids 2020, 25, 2222 -2230.
AMA StyleDavid Cichra, Vít Průša. A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation. Mathematics and Mechanics of Solids. 2020; 25 (12):2222-2230.
Chicago/Turabian StyleDavid Cichra; Vít Průša. 2020. "A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation." Mathematics and Mechanics of Solids 25, no. 12: 2222-2230.
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.
Mark Dostalík; Vít Průša; Judith Stein. Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature. Mathematics and Computers in Simulation 2020, 189, 5 -20.
AMA StyleMark Dostalík, Vít Průša, Judith Stein. Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature. Mathematics and Computers in Simulation. 2020; 189 ():5-20.
Chicago/Turabian StyleMark Dostalík; Vít Průša; Judith Stein. 2020. "Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature." Mathematics and Computers in Simulation 189, no. : 5-20.
The so-called smart structures are frequently inspired by the art of origami, and they are in many cases considered to be manufactured of smart materials such as shape memory materials. We propose a mathematical model for such origami-like structures made of light activated shape memory polymers, and we develop a reference code for computer simulations of such structures. The proposed mathematical model is based on an existing macroscopic phenomenological model by (Sodhi & Rao 2010, Int. J. Eng. Sci. 48, 1576), which has been developed in order to describe full three-dimensional deformation of light activated shape memory polymers. This model is suitably modified for the reduced representation of origami-like structures which is based on the bar-and-hinge approach. The numerical solution of the corresponding governing equations is implemented using a transparent modification of a state-of-the-art code for numerical simulation of purely elastic origami-like structures developed by (Liu & Paulino 2017, Proc. R. Soc. A: Math. Phys. Eng. Sci. 473, 20170348).
Jakub Cehula; Vít Průša. Computer modelling of origami-like structures made of light activated shape memory polymers. International Journal of Engineering Science 2020, 150, 103235 .
AMA StyleJakub Cehula, Vít Průša. Computer modelling of origami-like structures made of light activated shape memory polymers. International Journal of Engineering Science. 2020; 150 ():103235.
Chicago/Turabian StyleJakub Cehula; Vít Průša. 2020. "Computer modelling of origami-like structures made of light activated shape memory polymers." International Journal of Engineering Science 150, no. : 103235.
We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner.
Vít Průša; K.R. Rajagopal; Karel Tůma. Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids. International Journal of Non-Linear Mechanics 2020, 121, 103433 .
AMA StyleVít Průša, K.R. Rajagopal, Karel Tůma. Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids. International Journal of Non-Linear Mechanics. 2020; 121 ():103433.
Chicago/Turabian StyleVít Průša; K.R. Rajagopal; Karel Tůma. 2020. "Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids." International Journal of Non-Linear Mechanics 121, no. : 103433.
Using a Lyapunov type functional constructed on the basis of thermodynamical arguments, we investigate the finite amplitude stability of internal steady flows of viscoelastic fluids described by the Giesekus model. Using the functional, we derive bounds on the Reynolds and the Weissenberg number that guarantee the unconditional asymptotic stability of the corresponding steady internal flow, wherein the distance between the steady flow field and the perturbed flow field is measured with the help of the Bures–Wasserstein distance between positive definite matrices. The application of the theoretical results is documented in the finite amplitude stability analysis of Taylor–Couette flow.
Mark Dostalík; Vít Průša; Karel Tůma. Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid. Entropy 2019, 21, 1219 .
AMA StyleMark Dostalík, Vít Průša, Karel Tůma. Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid. Entropy. 2019; 21 (12):1219.
Chicago/Turabian StyleMark Dostalík; Vít Průša; Karel Tůma. 2019. "Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid." Entropy 21, no. 12: 1219.
Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium states in thermodynamically isolated systems. Unfortunately, the limitation to 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. The mathematical quantity that is referred to as the “energy” is, however, in most cases not linked to the energy in the physical sense of the word. Consequently, it would seem that genuine thermo-dynamical concepts are of no use in the nonlinear stability analysis of thermodynamically open systems. We show that this is not the case. In particular, we propose a construction that in the case of a simple heat conduction problem leads to a physically well-motivated Lyapunov type 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.
Miroslav Bulíček; Josef Málek; Vít Průša. Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems. Entropy 2019, 21, 704 .
AMA StyleMiroslav Bulíček, Josef Málek, Vít Průša. Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems. Entropy. 2019; 21 (7):704.
Chicago/Turabian StyleMiroslav Bulíček; Josef Málek; Vít Průša. 2019. "Thermodynamics and Stability of Non-Equilibrium Steady States in Open Systems." Entropy 21, no. 7: 704.
We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are reformulated as a system for the triple pressure–velocity–apparent viscosity, where the apparent viscosity is given by a scalar implicit equation. We prove that the proposed numerical scheme has—on the discrete level—a solution, and using the proposed scheme, we numerically solve several flow problems.
Adam Janečka; Josef Malek; Vít Průša; Giordano Tierra. Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor. Acta Mechanica 2019, 230, 729 -747.
AMA StyleAdam Janečka, Josef Malek, Vít Průša, Giordano Tierra. Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor. Acta Mechanica. 2019; 230 (3):729-747.
Chicago/Turabian StyleAdam Janečka; Josef Malek; Vít Průša; Giordano Tierra. 2019. "Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor." Acta Mechanica 230, no. 3: 729-747.
Karel Tůma; Judith Stein; Vít Průša; Elfriede Friedmann. Motion of the vitreous humour in a deforming eye–fluid-structure interaction between a nonlinear elastic solid and viscoleastic fluid. Applied Mathematics and Computation 2018, 335, 50 -64.
AMA StyleKarel Tůma, Judith Stein, Vít Průša, Elfriede Friedmann. Motion of the vitreous humour in a deforming eye–fluid-structure interaction between a nonlinear elastic solid and viscoleastic fluid. Applied Mathematics and Computation. 2018; 335 ():50-64.
Chicago/Turabian StyleKarel Tůma; Judith Stein; Vít Průša; Elfriede Friedmann. 2018. "Motion of the vitreous humour in a deforming eye–fluid-structure interaction between a nonlinear elastic solid and viscoleastic fluid." Applied Mathematics and Computation 335, no. : 50-64.
Mark Dostalík; Vít Průša; Karel Tůma. Finite amplitude stability of internal steady flows of the Giesekus viscoelastic rate-type fluid. 2018, 1 .
AMA StyleMark Dostalík, Vít Průša, Karel Tůma. Finite amplitude stability of internal steady flows of the Giesekus viscoelastic rate-type fluid. . 2018; ():1.
Chicago/Turabian StyleMark Dostalík; Vít Průša; Karel Tůma. 2018. "Finite amplitude stability of internal steady flows of the Giesekus viscoelastic rate-type fluid." , no. : 1.
The chapter starts with overview of the derivation of the balance equations for mass, momentum, angular momentum, and total energy, which is followed by a detailed discussion of the concept of entropy and entropy production. While the balance laws are universal for any continuous medium, the particular behavior of the material of interest must be described by an extra set of material-specific equations. These equations relating, for example, the Cauchy stress tensor and the kinematical quantities are called the constitutive relations. The core part of the chapter is devoted to the presentation of a modern thermodynamically based phenomenological theory of constitutive relations. The key feature of the theory is that the constitutive relations stem from the choice of two scalar quantities, the internal energy and the entropy production. This is tantamount to the proposition that the material behavior is fully characterized by the way it stores the energy and produces the entropy. The general theory is documented by several examples of increasing complexity. It is shown how to derive the constitutive relations for compressible and incompressible viscous heat-conducting fluids (Navier-Stokes-Fourier fluid), Korteweg fluids, and compressible and incompressible heat-conducting viscoelastic fluids (Oldroyd-B and Maxwell fluid).
Josef Málek; Vít Průša. Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids 2018, 3 -72.
AMA StyleJosef Málek, Vít Průša. Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. 2018; ():3-72.
Chicago/Turabian StyleJosef Málek; Vít Průša. 2018. "Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids." Handbook of Mathematical Analysis in Mechanics of Viscous Fluids , no. : 3-72.
We propose a new class of models for electrorheological fluids. While the standard constitutive relations for electrorheological fluids are based on the assumption that the stress is a function of the symmetric part of the velocity gradient and the intensity of the electric field, we formulate constitutive relations in an implicit way. The stress, the symmetric part of the velocity gradient and the intensity of the electric field are linked via a tensorial implicit equation. The potential benefit of the new class of models is investigated by the analysis of a simple shear flow in a transverse electric field.
Vít Průša; Kumbakonam R. Rajagopal. A New Class of Models to Describe the Response of Electrorheological and Other Field Dependent Fluids. Advanced Structured Materials 2018, 655 -673.
AMA StyleVít Průša, Kumbakonam R. Rajagopal. A New Class of Models to Describe the Response of Electrorheological and Other Field Dependent Fluids. Advanced Structured Materials. 2018; ():655-673.
Chicago/Turabian StyleVít Průša; Kumbakonam R. Rajagopal. 2018. "A New Class of Models to Describe the Response of Electrorheological and Other Field Dependent Fluids." Advanced Structured Materials , no. : 655-673.
We study the motion of vitreous humour in a deforming eyeball. From the mechanical and computational perspective this is a task to solve a fluid-structure interaction problem between a complex viscoelastic fluid (vitreour humour) and a nonlinear elastic solid (sclera and lens). We propose a numerical methodology capable of handling the fluid-structure interaction problem, and we demonstrate its applicability via solving the corresponding governing equations in a realistic geometrical setting and for realistic parameter values. It is shown that the choice of the rheological model for the vitreous humour has a negligible influence on the overall flow pattern in the domain of interest, whilst it is has a significant impact on the mechanical stress distribution in the domain of interest.
Karel Tůma; Judith Stein; Vít Průša; Elfriede Friedmann. Motion of the vitreous humour in a deforming eye -- fluid-structure interaction between a nonlinear elastic solid and a nonlinear viscoleastic fluid. 2018, 1 .
AMA StyleKarel Tůma, Judith Stein, Vít Průša, Elfriede Friedmann. Motion of the vitreous humour in a deforming eye -- fluid-structure interaction between a nonlinear elastic solid and a nonlinear viscoleastic fluid. . 2018; ():1.
Chicago/Turabian StyleKarel Tůma; Judith Stein; Vít Průša; Elfriede Friedmann. 2018. "Motion of the vitreous humour in a deforming eye -- fluid-structure interaction between a nonlinear elastic solid and a nonlinear viscoleastic fluid." , no. : 1.
We propose thermodynamically consistent models for viscoelastic fluids with a stress diffusion term. In particular, we derive variants of compressible/incompressible Maxwell/Oldroyd-B models with a stress diffusion term in the evolution equation for the extra stress tensor. It is shown that the stress diffusion term can be interpreted either as a consequence of a nonlocal energy storage mechanism or as a consequence of a nonlocal entropy production mechanism, while different interpretations of the stress diffusion mechanism lead to different evolution equations for the temperature. The benefits of the knowledge of the thermodynamical background of the derived models are documented in the study of nonlinear stability of equilibrium rest states. The derived models open up the possibility to study fully coupled thermomechanical problems involving viscoelastic rate-type fluids with stress diffusion.
Josef Málek; Vít Průša; Tomáš Skřivan; Endre Süli. Thermodynamics of viscoelastic rate-type fluids with stress diffusion. Physics of Fluids 2018, 30, 023101 .
AMA StyleJosef Málek, Vít Průša, Tomáš Skřivan, Endre Süli. Thermodynamics of viscoelastic rate-type fluids with stress diffusion. Physics of Fluids. 2018; 30 (2):023101.
Chicago/Turabian StyleJosef Málek; Vít Průša; Tomáš Skřivan; Endre Süli. 2018. "Thermodynamics of viscoelastic rate-type fluids with stress diffusion." Physics of Fluids 30, no. 2: 023101.
Miroslav Bulíček; Josef Malek; Vít Průša; Endre Süli. PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion. Contemporary Mathematics 2018, 710, 25 -51.
AMA StyleMiroslav Bulíček, Josef Malek, Vít Průša, Endre Süli. PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion. Contemporary Mathematics. 2018; 710 ():25-51.
Chicago/Turabian StyleMiroslav Bulíček; Josef Malek; Vít Průša; Endre Süli. 2018. "PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion." Contemporary Mathematics 710, no. : 25-51.
We derive a class of thermodynamically consistent variants of Maxwell/Oldroyd-B type models for viscoelastic fluids. In particular, we study the models that allow one to consider temperature dependent material coefficients. This naturally calls for the formulation of a temperature evolution equation that would accompany the evolution equations for the mechanical quantities. The evolution equation for the temperature is explicitly formulated, and it is shown to be consistent with the laws of thermodynamics and the evolution equations for the mechanical quantities. The temperature evolution equation contains terms that are ignored or even not thought of in most of the works dealing with this class of fluids. The impact of the additional terms in the temperature evolution equation on the flow dynamics is documented by the solution of simple initial/boundary value problems.Comment: Introduction, Page 2 and Page 3: We have been referred to relevant papers by I. J. Rao, K. Kannan and K. R. Rajagopal, who have considered viscoelastic rate type fluids with temperature dependent material coefficients. The comment on the relation between their work and our work has been added to the introductio
Jaroslav Hron; Vojtěch Miloš; Vít Průša; Ondřej Souček; Karel Tůma. On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients. International Journal of Non-Linear Mechanics 2017, 95, 193 -208.
AMA StyleJaroslav Hron, Vojtěch Miloš, Vít Průša, Ondřej Souček, Karel Tůma. On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients. International Journal of Non-Linear Mechanics. 2017; 95 ():193-208.
Chicago/Turabian StyleJaroslav Hron; Vojtěch Miloš; Vít Průša; Ondřej Souček; Karel Tůma. 2017. "On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients." International Journal of Non-Linear Mechanics 95, no. : 193-208.