Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition
<p>Solution of the macroscopic heat conduction given by Equation (<a href="#FD9-entropy-20-00126" class="html-disp-formula">9</a>) as a function of the dimensionless coordinate <math display="inline"> <semantics> <mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mo>=</mo> <mi>x</mi> <mi>k</mi> </mrow> </semantics> </math> and time <math display="inline"> <semantics> <mrow> <msup> <mi>t</mi> <mo>*</mo> </msup> <mo>=</mo> <mi>t</mi> <mi>c</mi> <mi>k</mi> </mrow> </semantics> </math>.</p> "> Figure 2
<p>Comparison of the fast, advective mode given by Equation (49a) for different Knudsen numbers with the slow, diffusive mode, that is, Macro, given by Equation (49b). Only the leading order of the expansion of the characteristic frequencies in Equations (48a) and (48b) is shown. The dimensionless time is <math display="inline"> <semantics> <mrow> <msup> <mi>t</mi> <mo>*</mo> </msup> <mo>=</mo> <mi>t</mi> <mi>c</mi> <mi>k</mi> </mrow> </semantics> </math>.</p> "> Figure 3
<p>Comparison of the (single-mode) mesoscopic solution given by Equation (<a href="#FD61-entropy-20-00126" class="html-disp-formula">61</a>) and the macroscopic solution given by Equation (<a href="#FD9-entropy-20-00126" class="html-disp-formula">9</a>). The two analyzed cases are (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mi>ϵ</mi> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo stretchy="false">/</mo> <mi>c</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </semantics> </math> and (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mi>ϵ</mi> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo stretchy="false">/</mo> <mi>c</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </semantics> </math>. The dimensionless time is <math display="inline"> <semantics> <mrow> <msup> <mi>t</mi> <mo>*</mo> </msup> <mo>=</mo> <mi>t</mi> <mi>c</mi> <mi>k</mi> </mrow> </semantics> </math>.</p> ">
Abstract
:1. Introduction
2. Macroscopic Description
2.1. Parabolic Heat Conduction: Infinite Harmonics
2.2. Parabolic Heat Conduction: Finite Harmonics
3. Mesoscopic Description
3.1. Modeling Approach and Physical Background
- Hydrodynamic regime for : This can be described using macroscopic continuum models, for example, the Navier–Stokes–Fourier (NSF) system of equations. With regard to heat transfer, the standard applications in this regime involve the thermal analysis of macroscopic engineering devices.
- Slip-flow regime for : this can still be described using the NSF system, but additional boundary conditions must be taken into account to correctly describe the slip velocity and temperature jumps at the interfaces. In particular, example applications of temperature jumps (or Kapitza discontinuities) involve the analysis of the heat transfer in nano-particle suspensions at the interface between a solvent and solute.
- Transition regime for : the NSF equations are no longer valid, and a more complete approach must be used to correctly describe the fluid flow. In this regime, extended equations for higher-order hydrodynamics must be adopted, and the thermal transport mechanisms transition from diffusive, , to ballistic, . Example applications of thermal transport in this regime include hyperbolic heat transfer, for which there is a finite speed of energy transfer.
- Free molecular flow for : this is dominated by particle-wall collisions and must be described using a molecular level of detail. Typical applications in this regime include the analysis of phonon transport in nano-structures and -aggregates.
3.2. First Mesoscopic System: Two-Moment Hyperbolic Equations
3.3. Second Mesoscopic System: Switched Two-Moment Hyperbolic Equations
3.4. Third Mesoscopic System: Three-Moment Hyperbolic Equations
3.5. Recovering the Cattaneo Equation
4. Solution Analysis
4.1. Two-Equation Systems: MESO1 and MESO2
4.2. Three-Equation System: MESO3
5. Slow- and Fast-Mode Decomposition
5.1. Case 1: < 1/2
5.2. Case 2: > 1/2
5.3. Recovering the Single-Mode Solution
6. Discussion
7. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Bergamasco, L.; Alberghini, M.; Fasano, M.; Cardellini, A.; Chiavazzo, E.; Asinari, P. Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition. Entropy 2018, 20, 126. https://doi.org/10.3390/e20020126
Bergamasco L, Alberghini M, Fasano M, Cardellini A, Chiavazzo E, Asinari P. Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition. Entropy. 2018; 20(2):126. https://doi.org/10.3390/e20020126
Chicago/Turabian StyleBergamasco, Luca, Matteo Alberghini, Matteo Fasano, Annalisa Cardellini, Eliodoro Chiavazzo, and Pietro Asinari. 2018. "Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition" Entropy 20, no. 2: 126. https://doi.org/10.3390/e20020126
APA StyleBergamasco, L., Alberghini, M., Fasano, M., Cardellini, A., Chiavazzo, E., & Asinari, P. (2018). Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition. Entropy, 20(2), 126. https://doi.org/10.3390/e20020126