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Volume 10 (2019): Issue 2 (January 2019)
Special Issue on Mathematical Models and Methods in Biology, Medicine and Physiology. Guest Editors: Michele Piana, Luigi Preziosi

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Special Issue on Mathematical modelling for complex systems: multi-agents methods. Guest Editor: Elena De Angelis

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Volume 7 (2016): Issue 3 (September 2016)
"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

Volume 7 (2016): Issue 2 (June 2016)
Special Issue on Constitutive Equations for Heat Conduction in Nanosystems and Non-equilibrium Processes. Guest Editors: Vito Antonio Cimmelli and David Jou

Volume 7 (2016): Issue 1 (January 2016)
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Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

Volume 7 (2016): Issue 2 (June 2016)
Special Issue on Constitutive Equations for Heat Conduction in Nanosystems and Non-equilibrium Processes. Guest Editors: Vito Antonio Cimmelli and David Jou

Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

14 Articles

Editorial

access type Open Access

Editor’s Note

Published Online: 20 May 2016
Page range: 1 - 3

Abstract

Guest Editorial

access type Open Access

Foreword

Published Online: 20 May 2016
Page range: 4 - 7

Abstract

Research Papers

access type Open Access

Electron transport in silicon nanowires having different cross-sections

Published Online: 20 May 2016
Page range: 8 - 25

Abstract

Abstract

Transport phenomena in silicon nanowires with different cross-section are investigated using an Extended Hydrodynamic model, coupled to the Schrödinger-Poisson system. The model has been formulated by closing the moment system derived from the Boltzmann equation on the basis of the maximum entropy principle of Extended Thermodynamics, obtaining explicit closure relations for the high-order fluxes and the production terms. Scattering of electrons with acoustic and non polar optical phonons have been taken into account. The bulk mobility is evaluated for square and equilateral triangle cross-sections of the wire.

Keywords

  • Quantum wires
  • Semiconductors
  • Kinetic theory of gases
access type Open Access

Thermal rectification based on phonon hydrodynamics and thermomass theory

Published Online: 20 May 2016
Page range: 26 - 38

Abstract

Abstract

The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.

Keywords

  • thermal rectification
  • thermomass theory
  • phonon hydrodynamics
  • phonon Boltzmann equation
access type Open Access

Heat-pulse propagation along nonequilibrium nanowires in thermomass theory

Published Online: 20 May 2016
Page range: 39 - 55

Abstract

Abstract

We analyze the consequences of the nonlinear terms in the heat-transport equation of the thermomass theory on heat pulses propagating in a nanowire in nonequilibrium situations. As a consequence of the temperature dependence of the speeds of propagation, in temperature ranges wherein the specific heat shows negligible variations, heat pulses will shrink (or extend) spatially, and will increase (or decrease) their average temperature when propagating along a temperature gradient. A comparison with the results predicted by a different theoretical proposal on the shape of a propagating heat pulse is made, too.

Keywords

  • heat-waves propagation
  • nonlinear effects
  • energy-pulse propagation
  • thermomass theory
access type Open Access

Role of thermodynamics in extensions of mesoscopic dynamical theories

Published Online: 20 May 2016
Page range: 56 - 80

Abstract

Abstract

Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a universal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details).

Keywords

  • Contact geometry
  • non-equilibrium thermodynamics
  • Hamiltonian evolution
  • dissipation potential
  • non-local effects
access type Open Access

Non-equilibrium temperatures and heat transport in nanosystems with defects, described by a tensorial internal variable

Published Online: 20 May 2016
Page range: 81 - 97

Abstract

Abstract

The paper deals with the meaning of non-equilibrium temperatures in nanosystems with an internal variable, describing defects inside them, and implications on heat transport. In equilibrium all definitions of temperature lead to the same value, but in nonequilibrium steady states they lead to different values, giving information on different degrees of freedom. We discuss the caloric and entropic non-equilibrium temperatures and the relations among them, in defective nanosystems (crystals with dislocations or porous channels, carbon nanotubes in a solid matrix and so on), crossed by an external energy flux. Here, we present a model for nanocrystals with dislocation defects submitted to an external energy flux. The dislocations may have a strong influence on the effective thermal conductivity, and their own dynamics may be coupled in relevant way to the heat flux dynamics. In the linear case the constitutive relations, the rate equations for the internal variable and the heat flux are worked out and a generalized telegraphic heat equation is derived in the anisotropic and isotropic case, describing the thermal disturbances with finite velocity.

Keywords

  • Non-equilibrium Thermodynamics
  • internal variables
  • defects of dislocation
  • nanosystems
access type Open Access

Optimization of supercooling effect in nanoscaled thermoelectric layers

Published Online: 20 May 2016
Page range: 98 - 110

Abstract

Abstract

In this paper we address the problem of optimization of the so called supercooling effect in thermoelectric nanoscaled layers. The effect arises when a short term electric pulse is applied to the layer. The analysis is based on constitutive equations of the Maxwell-Cattaneo type describing the time evolution of dissipative flows with the thermal and electric conductivities depending on the width of the layer. This introduces memory and nonlocal effects and consequently a wave-like behaviour of system’s temperature. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses of the form ta with a a power going from 0 to 10. Pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaled devices equals the best of the microscaled ones. We use the spectral methods of solution which assure a well representation of wave behaviour of heat and electric charge in short time scales given their spectral convergence.

Keywords

  • Heat Transfer
  • Supercooling Effect
  • Nanoscaled Thermoelectric Layers
access type Open Access

Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

Published Online: 20 May 2016
Page range: 111 - 129

Abstract

Abstract

In this paper we extend previous results on the effective thermal conductivity of liquid helium II in cylindrical channels to rectangular channels with high aspect ratio. The aim is to compare the results in the laminar regime, the turbulent regime and the ballistic regime, all of them obtained within a single mesoscopic formalism of heat transport, with heat flux as an independent variable.

Keywords

  • Thermal conductivity
  • Liquid helium
  • Quantum turbulence
  • Quantized vortices
access type Open Access

Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

Published Online: 20 May 2016
Page range: 130 - 149

Abstract

Abstract

We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

Keywords

  • Quantum turbulence
  • quantized vortices
  • heat transfer
  • inhomogeneous vortex tangle
  • vortex diffusion
access type Open Access

Theories and heat pulse experiments of non-Fourier heat conduction

Published Online: 20 May 2016
Page range: 150 - 166

Abstract

Abstract

The experimental basis and theoretical background of non-Fourier heat conduction is shortly reviewed from the point of view of non-equilibrium thermodynamics. The performance of different theories is compared in case of heat pulse experiments.

Keywords

  • ballistic propagation
  • second sound
  • non-equilibrium thermodynamics
  • kinetic theory
access type Open Access

Thermodynamic framework for a generalized heat transport equation

Published Online: 20 May 2016
Page range: 167 - 176

Abstract

Abstract

In this paper, a generalized heat transport equation including relaxational, nonlocal and nonlinear effects is provided, which contains diverse previous phenomenological models as particular cases. The aim of the present work is to establish an extended irreversible thermodynamic framework, with generalized expressions of entropy and entropy flux. Nonlinear thermodynamic force-flux relation is proposed as an extension of the usual linear one, giving rise to the nonlinear terms in the heat transport equation and ensuring compatibility with the second law. Several previous results are recovered in the linear case, and some additional results related to nonlinear terms are also obtained.

Keywords

  • generalized heat transport equation
  • extended irreversible thermodynamics
  • generalized entropy flux
  • nanoscale heat transport
access type Open Access

Heat transfer at nanometric scales described by extended irreversible thermodynamics

Published Online: 20 May 2016
Page range: 177 - 195

Abstract

Abstract

The purpose of this work is to present a study on heat conduction in systems that are composed out of spherical and cylindrical micro- and nanoparticles dispersed in a bulk matrix. Special emphasis is put on the dependence of the effective heat conductivity on various selected parameters as particle size and also its shape, surface specularity and density, including particle-matrix interaction. The heat transfer at nanometric scales is modelled using extended irreversible thermodynamics, whose main feature is to elevate the heat flux vector to the status of independent variable. The model is illustrated by a Copper-Silicium (Cu-Si) system. It is shown that all the investigated parameters have a considerable influence, the particle size being especially useful to either increase or decrease the effective thermal conductivity.

Keywords

  • nanometric heat transfer
  • thermal conductivity
  • extended irreversible thermodynamics
  • spherical and cylindrical nanocomposites
access type Open Access

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

Published Online: 20 May 2016
Page range: 196 - 222

Abstract

Abstract

We provide an overview on the problem of modeling heat transport at nanoscale and in far-from-equilibrium processes. A survey of recent results is summarized, and a conceptual discussion of them in the framework of Extended Irreversible Thermodynamics is developed.

Keywords

  • Fourier law
  • heat transport equations
  • nanosystems
  • nonlocal effects
  • nonlinear effects
  • thermoelectric effects
  • nonequilibrium processes
14 Articles

Editorial

access type Open Access

Editor’s Note

Published Online: 20 May 2016
Page range: 1 - 3

Abstract

Guest Editorial

access type Open Access

Foreword

Published Online: 20 May 2016
Page range: 4 - 7

Abstract

Research Papers

access type Open Access

Electron transport in silicon nanowires having different cross-sections

Published Online: 20 May 2016
Page range: 8 - 25

Abstract

Abstract

Transport phenomena in silicon nanowires with different cross-section are investigated using an Extended Hydrodynamic model, coupled to the Schrödinger-Poisson system. The model has been formulated by closing the moment system derived from the Boltzmann equation on the basis of the maximum entropy principle of Extended Thermodynamics, obtaining explicit closure relations for the high-order fluxes and the production terms. Scattering of electrons with acoustic and non polar optical phonons have been taken into account. The bulk mobility is evaluated for square and equilateral triangle cross-sections of the wire.

Keywords

  • Quantum wires
  • Semiconductors
  • Kinetic theory of gases
access type Open Access

Thermal rectification based on phonon hydrodynamics and thermomass theory

Published Online: 20 May 2016
Page range: 26 - 38

Abstract

Abstract

The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.

Keywords

  • thermal rectification
  • thermomass theory
  • phonon hydrodynamics
  • phonon Boltzmann equation
access type Open Access

Heat-pulse propagation along nonequilibrium nanowires in thermomass theory

Published Online: 20 May 2016
Page range: 39 - 55

Abstract

Abstract

We analyze the consequences of the nonlinear terms in the heat-transport equation of the thermomass theory on heat pulses propagating in a nanowire in nonequilibrium situations. As a consequence of the temperature dependence of the speeds of propagation, in temperature ranges wherein the specific heat shows negligible variations, heat pulses will shrink (or extend) spatially, and will increase (or decrease) their average temperature when propagating along a temperature gradient. A comparison with the results predicted by a different theoretical proposal on the shape of a propagating heat pulse is made, too.

Keywords

  • heat-waves propagation
  • nonlinear effects
  • energy-pulse propagation
  • thermomass theory
access type Open Access

Role of thermodynamics in extensions of mesoscopic dynamical theories

Published Online: 20 May 2016
Page range: 56 - 80

Abstract

Abstract

Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a universal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details).

Keywords

  • Contact geometry
  • non-equilibrium thermodynamics
  • Hamiltonian evolution
  • dissipation potential
  • non-local effects
access type Open Access

Non-equilibrium temperatures and heat transport in nanosystems with defects, described by a tensorial internal variable

Published Online: 20 May 2016
Page range: 81 - 97

Abstract

Abstract

The paper deals with the meaning of non-equilibrium temperatures in nanosystems with an internal variable, describing defects inside them, and implications on heat transport. In equilibrium all definitions of temperature lead to the same value, but in nonequilibrium steady states they lead to different values, giving information on different degrees of freedom. We discuss the caloric and entropic non-equilibrium temperatures and the relations among them, in defective nanosystems (crystals with dislocations or porous channels, carbon nanotubes in a solid matrix and so on), crossed by an external energy flux. Here, we present a model for nanocrystals with dislocation defects submitted to an external energy flux. The dislocations may have a strong influence on the effective thermal conductivity, and their own dynamics may be coupled in relevant way to the heat flux dynamics. In the linear case the constitutive relations, the rate equations for the internal variable and the heat flux are worked out and a generalized telegraphic heat equation is derived in the anisotropic and isotropic case, describing the thermal disturbances with finite velocity.

Keywords

  • Non-equilibrium Thermodynamics
  • internal variables
  • defects of dislocation
  • nanosystems
access type Open Access

Optimization of supercooling effect in nanoscaled thermoelectric layers

Published Online: 20 May 2016
Page range: 98 - 110

Abstract

Abstract

In this paper we address the problem of optimization of the so called supercooling effect in thermoelectric nanoscaled layers. The effect arises when a short term electric pulse is applied to the layer. The analysis is based on constitutive equations of the Maxwell-Cattaneo type describing the time evolution of dissipative flows with the thermal and electric conductivities depending on the width of the layer. This introduces memory and nonlocal effects and consequently a wave-like behaviour of system’s temperature. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses of the form ta with a a power going from 0 to 10. Pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaled devices equals the best of the microscaled ones. We use the spectral methods of solution which assure a well representation of wave behaviour of heat and electric charge in short time scales given their spectral convergence.

Keywords

  • Heat Transfer
  • Supercooling Effect
  • Nanoscaled Thermoelectric Layers
access type Open Access

Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

Published Online: 20 May 2016
Page range: 111 - 129

Abstract

Abstract

In this paper we extend previous results on the effective thermal conductivity of liquid helium II in cylindrical channels to rectangular channels with high aspect ratio. The aim is to compare the results in the laminar regime, the turbulent regime and the ballistic regime, all of them obtained within a single mesoscopic formalism of heat transport, with heat flux as an independent variable.

Keywords

  • Thermal conductivity
  • Liquid helium
  • Quantum turbulence
  • Quantized vortices
access type Open Access

Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

Published Online: 20 May 2016
Page range: 130 - 149

Abstract

Abstract

We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

Keywords

  • Quantum turbulence
  • quantized vortices
  • heat transfer
  • inhomogeneous vortex tangle
  • vortex diffusion
access type Open Access

Theories and heat pulse experiments of non-Fourier heat conduction

Published Online: 20 May 2016
Page range: 150 - 166

Abstract

Abstract

The experimental basis and theoretical background of non-Fourier heat conduction is shortly reviewed from the point of view of non-equilibrium thermodynamics. The performance of different theories is compared in case of heat pulse experiments.

Keywords

  • ballistic propagation
  • second sound
  • non-equilibrium thermodynamics
  • kinetic theory
access type Open Access

Thermodynamic framework for a generalized heat transport equation

Published Online: 20 May 2016
Page range: 167 - 176

Abstract

Abstract

In this paper, a generalized heat transport equation including relaxational, nonlocal and nonlinear effects is provided, which contains diverse previous phenomenological models as particular cases. The aim of the present work is to establish an extended irreversible thermodynamic framework, with generalized expressions of entropy and entropy flux. Nonlinear thermodynamic force-flux relation is proposed as an extension of the usual linear one, giving rise to the nonlinear terms in the heat transport equation and ensuring compatibility with the second law. Several previous results are recovered in the linear case, and some additional results related to nonlinear terms are also obtained.

Keywords

  • generalized heat transport equation
  • extended irreversible thermodynamics
  • generalized entropy flux
  • nanoscale heat transport
access type Open Access

Heat transfer at nanometric scales described by extended irreversible thermodynamics

Published Online: 20 May 2016
Page range: 177 - 195

Abstract

Abstract

The purpose of this work is to present a study on heat conduction in systems that are composed out of spherical and cylindrical micro- and nanoparticles dispersed in a bulk matrix. Special emphasis is put on the dependence of the effective heat conductivity on various selected parameters as particle size and also its shape, surface specularity and density, including particle-matrix interaction. The heat transfer at nanometric scales is modelled using extended irreversible thermodynamics, whose main feature is to elevate the heat flux vector to the status of independent variable. The model is illustrated by a Copper-Silicium (Cu-Si) system. It is shown that all the investigated parameters have a considerable influence, the particle size being especially useful to either increase or decrease the effective thermal conductivity.

Keywords

  • nanometric heat transfer
  • thermal conductivity
  • extended irreversible thermodynamics
  • spherical and cylindrical nanocomposites
access type Open Access

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

Published Online: 20 May 2016
Page range: 196 - 222

Abstract

Abstract

We provide an overview on the problem of modeling heat transport at nanoscale and in far-from-equilibrium processes. A survey of recent results is summarized, and a conceptual discussion of them in the framework of Extended Irreversible Thermodynamics is developed.

Keywords

  • Fourier law
  • heat transport equations
  • nanosystems
  • nonlocal effects
  • nonlinear effects
  • thermoelectric effects
  • nonequilibrium processes

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