Hyperbolicity of the heat equation

Cassol, Guilherme Ozorio; Dubljević, Stevan

doi:10.1016/j.ifacol.2019.07.011

Cited by 7 publications

(15 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same results are shown for the hyperbolic PDE in As expected, the PDEs converge to the same steady-state profile, although the apparent difference is seen in the initial dynamic response. This difference also shown in Cassol et al, 23 which showed the distinct slope in the profiles of the hyperbolic and parabolic PDEs.…”

Section: Analytic Solutionssupporting

confidence: 68%

“…The velocities profiles are initially different, as the parabolic presents a smooth profile, while the hyperbolic presents sudden changes.As expected, the PDEs converge to the same steady-state profile, although the apparent difference is seen in the initial dynamic response. This difference also shown in Cassol et al,23 which showed the distinct slope in the profiles of the hyperbolic and parabolic PDEs.Next, a comparison for different initial conditions for the ∂ t T(ζ, t), as one might argue that the previous results show a distinguished difference between the hyperbolic and parabolic because the initial condition was ∂ t T(ζ, 0) = 0. The system responses, only considering the second set of boundary conditions, for different initial conditions of ∂ t T(ζ, 0), but with the same T(ζ, 0), are displayed in Figure 4.…”

supporting

confidence: 68%

See 1 more Smart Citation

Hyperbolicity of reaction‐transport processes

Cassol

Dubljević

2021

AIChE Journal

Self Cite

View full text Add to dashboard Cite

In this manuscript, common chemical engineering transport-reaction systems are modeled by taking into consideration the inertia in the transport phenomena, leading to hyperbolic partial differential equations (PDEs), which have finite speed of propagation. The findings provide the derivation of the hyperbolic transport-reaction PDEs and a direct comparison with their corresponding dissipative parabolic PDEs. The stability analysis of the dynamical systems is considered and complemented by numerical simulations for typical processes, such as the heat diffusion in a slab, transport-reaction inside a chemical reactor, and a phase change problem. The results demonstrate the effects associated with the modeling and/or limits of the approximation used in the derivation and application of the hyperbolic PDE for transportreaction systems, and the difference between the dynamics of this type of equation and the generally used parabolic PDEs.

show abstract

Section: Analytic Solutionssupporting

confidence: 68%

supporting

confidence: 68%

Hyperbolicity of reaction‐transport processes

Cassol

Dubljević

2021

AIChE Journal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The CVe arises when there is a time delay τ between the application of a temperature gradient and the ensuing heat flux q( r,t). In this case the heat flux is [1][2][3][4] , q( r,t) + τ ∂ q( r,t) ∂t = −κ∇T ( r,t),…”

Section: Thermal Waves Cattaneo-vernotte Equationmentioning

confidence: 99%

“…The constant amplitudes A and B in Eq. (4) and Eq. (6) are determined from the boundary conditions that are the continuity (i) of the temperature and (ii) of the heat flux, both are consequences of assuming perfect thermal contact.…”

Section: Thermal Waves Cattaneo-vernotte Equationmentioning

confidence: 99%

“…The sudden heating of materials leads to a deviation from the usual Fourier law of heat conduction due to a time delay between the temperature gradient and the heat flux. The time delay implies a finite velocity of heat propagation and the partial differential equation for the temperature is no longer a diffusion equation but a wave equation (hyperbolic partial differential equation) known as the Cattaneo-Vernotte equation (CVe) [1][2][3][4] . The wave-like nature of the temperature and heat flux derived by Cattaneo's equation opens new possibilities for thermal management and can find analogies to wave phenomena in other areas of physics.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Thermal Bragg Shields for Thermal Waves.

Rosa¹,

Becerril²,

Gómez-Farfán³

et al. 2021

Preprint

View full text Add to dashboard Cite

Ultrafast heating processes do not follow Fourier's heat conduction law, but rather the proposed Cattaneo-Vernotte equation (CVe) which has wave-like solutions that have some important differences from other wave phenomenon. In a periodic system made of materials with different thermal conductivities, solutions of the CVe lead to a band-like structure in the dispersion relation. In this work, we show that highly reflective Bragg mirrors for thermal waves can be designed. Even for a mirrors with a few layers a very high reflectance is achieved (>90%). The mirrors are made of materials with large thermal response times, where thermal waves have been measured. A second alternative consists of adding a thin metallic film which also leads to an efficient thermal Bragg mirror. Finally, the role of defects in opening new thermal-stop bands is demonstrated.

show abstract

Time-Harmonic Photothermal Heating by Nanoparticles in a Non-Fourier Medium

et al. 2021

View full text Add to dashboard Cite

We analyze the time-harmonic heating of a non-Fourier medium by spherical nanoparticles via the photothermal effect. The nanoparticle is embedded in a medium with thermal properties similar to those reported for organic tissue that does not obey Fourier’s law of heat conduction but rather the Cattaneo–Vernotte equation. By assuming the nanoparticle is illuminated with an intensity-modulated laser, we show that the temperature profile outside the nanoparticle oscillates and, at specific separations, can have a temperature 16% lower than predicted using Fourier’s law of heat conduction.

show abstract

Hyperbolicity of the heat equation

Cited by 7 publications

References 3 publications

Hyperbolicity of reaction‐transport processes

Hyperbolicity of reaction‐transport processes

Thermal Bragg Shields for Thermal Waves.

Time-Harmonic Photothermal Heating by Nanoparticles in a Non-Fourier Medium

Contact Info

Product

Resources

About