Hyperbolic heat/mass transport and stochastic modelling - Three simple problems

Massimiliano Giona; Luigi Pucci; Massimiliano Giona; Luigi Pucci

doi:10.3934/mine.2019.2.224

Mathematics in Engineering

2019, Volume 1, Issue 2: 224-251. doi: 10.3934/mine.2019.2.224

Previous Article Next Article

Research article

Hyperbolic heat/mass transport and stochastic modelling - Three simple problems

Massimiliano Giona ^,,
Luigi Pucci

Dipartimento di Ingegneria Chimica DICMA Facoltà di Ingegneria, La Sapienza Università di Roma via Eudossiana 18, 00184, Roma, Italy

Received: 06 October 2018 Accepted: 27 January 2019 Published: 14 February 2019

Starting from the correspondence between the Cattaneo hyperbolic heat equation and the stochastic formulation based on Poisson-Kac processes, that holds solely for one-dimensional spatial models, this article analyzes three paradigmatic problems in the hyperbolic theory of heat and mass transport. The problems considered involve unbounded, semi-bounded and bounded domains, and are aimed at : (i) highlighting analogies and differences between the two approaches (Cattaneo vs Poisson-Kac), (ii) addressing the role of a bounded propagation velocity in order to regularize the properties of the solutions of heat/mass transport problems. A typical example of the latter phenomenology is expressed by boundary-layer regulatization of interfacial fluxes. The case of transport in bounded domains permits to pinpoint unambiguously the need of a stochastic interpretation of the transport equation in order to unveil the occurrence of physical inconsistencies that may occur in the linear Cattaneo hyperbolic model in some range of parameter values.
- hyperbolic transport models,
- Poisson-Kac processes,
- Cattaneo equation,
- stochastic dynamics,
- positivity requirement
Citation: Massimiliano Giona, Luigi Pucci. Hyperbolic heat/mass transport and stochastic modelling - Three simple problems[J]. Mathematics in Engineering, 2019, 1(2): 224-251. doi: 10.3934/mine.2019.2.224

Related Papers:

Abstract

Starting from the correspondence between the Cattaneo hyperbolic heat equation and the stochastic formulation based on Poisson-Kac processes, that holds solely for one-dimensional spatial models, this article analyzes three paradigmatic problems in the hyperbolic theory of heat and mass transport. The problems considered involve unbounded, semi-bounded and bounded domains, and are aimed at : (i) highlighting analogies and differences between the two approaches (Cattaneo vs Poisson-Kac), (ii) addressing the role of a bounded propagation velocity in order to regularize the properties of the solutions of heat/mass transport problems. A typical example of the latter phenomenology is expressed by boundary-layer regulatization of interfacial fluxes. The case of transport in bounded domains permits to pinpoint unambiguously the need of a stochastic interpretation of the transport equation in order to unveil the occurrence of physical inconsistencies that may occur in the linear Cattaneo hyperbolic model in some range of parameter values.

References

[1]	Wang L, Li B (2008) Phononics gets hot. Phys World 21: 27-29.
[2]	Maldovan M (2013) Sound and heat revolutions in phononics. Nature 503: 209-217. doi: 10.1038/nature12608
[3]	Li N, Ren J, Wang L, et al. (2012) Colloquium: Phononics: manipulating heat flow with electronic analogs and beyond. Rev Mod Phys 84: 1045-1066. doi: 10.1103/RevModPhys.84.1045
[4]	Straughan B (2011) Heat waves, Springer, New York.
[5]	Sellitto A, Cimmelli VA, Jou D (2016) Mesoscopic Theories of Heat Transport in Nanosystems, Springer, Heidelberg.
[6]	Cattaneo C (1948) Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena 3: 83-101.
[7]	Cattaneo C (1958) Sur une forme de l'equation de la chaleur eliminant le paradoxe s'une propagation instantanee. Compt Rendu Acad Sci 247: 431-433.
[8]	Bubnov VA (1976) Wave concepts in the theory of heat. Int J Heat Mass Tran 19: 175-184. doi: 10.1016/0017-9310(76)90110-1
[9]	Glass DE, Özisik MN, Vick B (1985) Hyperbolic heat conduction with surface radiation. Int J Heat mass Tran 28: 1823-1830. doi: 10.1016/0017-9310(85)90204-2
[10]	Joseph DD, Preziosi L (1989) Heat waves, Rev Mod Phys 61: 41-72.
[11]	Guyer RA, Krumhansl JA (1966) Solution of the Linearized Phonon Boltzmann Equation. Phys Rev 148: 766-778. doi: 10.1103/PhysRev.148.766
[12]	Guyer RA, Krumhansl JA (1966) Thermal Conductivity, Second Sound, and Phonon Hydrodynamic phenomena in Nonmetallic Crystals. Phys Rev 148: 778-788. doi: 10.1103/PhysRev.148.778
[13]	Jou D, Cimmelli VA (2016) Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview. Communications in Applied and Industrial Mathematics 7: 196-222. doi: 10.1515/caim-2016-0014
[14]	Jou D, Cimmelli VA, Sellitto A (2012) Nonlocal heat transport with phonons and electrons: Application to metallic nanowires. Int J Heat Mass Tran 55: 2338-2344. doi: 10.1016/j.ijheatmasstransfer.2012.01.033
[15]	Guo Y, Jou D, Wang M (2017) Macroscopic heat transport equations and heat waves in nonequilibrium states. Physica D: Nonlinear Phenomena 342: 24-31. doi: 10.1016/j.physd.2016.10.005
[16]	Camera-Roda G, Sarti GC (1990) Mass Transport with Relaxation in Polymers. AIChE J 36: 851-860. doi: 10.1002/aic.690360606
[17]	Edwards DA, Cohen DS (1995) A Mathematical Model for a Dissolving Polymer. AIChE J 41: 2345-2355. doi: 10.1002/aic.690411102
[18]	Ferreira JA, de Oliveira P, da Silva PM, et al. (2014) Molecular Transport in Viscoelastic Materials: Mechanistic Properties and Chemical Affinities. SIAM J Appl Math 74: 1598-1614. doi: 10.1137/140954027
[19]	Ferreira JA, Grassi M, Gudino E, et al. (2015) A new look to non-Fickian diffusion. Appl Math Model 39: 194-204. doi: 10.1016/j.apm.2014.05.030
[20]	Ferreira JA, Naghipoor J, de Oliveira P (2016) A coupled non-Fickian model of a cardiovascular drug delivery system. Math Med Biol 33: 329-357. doi: 10.1093/imammb/dqv023
[21]	Müller I, Ruggeri T (1993) Extended Thermodynamics, Springer-Verlag, New York.
[22]	de Groot SR, Mazur P (1984) Non-Equilibrium Thermodynamics, Dover Publ. Inc. New York.
[23]	Jou D, Casas-Vazquez J, Lebon G (2010) Extended Irreversible Thermodynamics, Springer, New York.
[24]	Jou D, Casas-Vazquez J, Lebon G (1988) Extended Irreversible Thermodynamics, Rep Prog Phys 51: 1105-1179.
[25]	Jou D, Casas-Vazquez J, Lebon G (1999) Extended Irreversible Thermodynamics revisited (1988-1998). Rep Prog Phys 62: 1035-1142. doi: 10.1088/0034-4885/62/7/201
[26]	Körner C, Bergmann HW (1998) The physical defects of the hyperbolic heat conduction equation. Appl Phys A 67: 397-401. doi: 10.1007/s003390050792
[27]	Struchtrup H (2005) Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin.
[28]	Giona M, Brasiello A, Crescitelli S (2016) Generalized Poisson-Kac processes: basic properties and implications in extended thermodynamics and transport. J Non-Equil Thermody 41: 107-114.
[29]	Giona M, Brasiello A, Crescitelli S (2017) Stochastic foundations of undulatory transport phenomena: generalized Poisson-Kac processes - part I basic theory. J Phys A 50: 335002. doi: 10.1088/1751-8121/aa79d4
[30]	Giona M, Brasiello A, Crescitelli S (2017) Stochastic foundations of undulatory transport phenomena: generalized Poisson-Kac processes - part II Irreversibility, norms and entropies. J Phys A 50: 335003. doi: 10.1088/1751-8121/aa79c5
[31]	Giona M, Brasiello A, Crescitelli S (2017) Stochastic foundations of undulatory transport phenomena: generalized Poisson-Kac processes - part III extensions and applications to kinetic theory and transport. J Phys A 50: 335004. doi: 10.1088/1751-8121/aa79d6
[32]	Kac M (1956) Some Stochastic Problems in Physics and Mathematics, Socony Mobil Oil Company Colloquium Lectures.
[33]	Kac M (1974) A stochastic model related to the telegrapher's equation. Rocky MT J Math 4: 497-510.
[34]	Gardiner CW (1997) Handbook of Stochastic Methods, Springer-Verlag, Berlin.
[35]	Falconer K (2003) Fractal Geometry, J Wiley & Sons, Chichister.
[36]	Giona M (2018) Covariance and spinorial statistical description of simple relativistic stochastic kinematics, in preparation.
[37]	Rosenau P (1993) Random walker and the telegrapher's equation: A paradigm of a generalized hydrodynamics. Phys Rev E 48: R655-R657. doi: 10.1103/PhysRevE.48.R655
[38]	Brasiello A, Crescitelli S, Giona M (2016) One-dimensional hyperbolic transport: positivity and admissible boundary conditions derived from the wave formulation. Physica A 449: 176-191. doi: 10.1016/j.physa.2015.12.111
[39]	Zhukovsky K (2016) Violation of the maximum principle and negative solutions with pulse propagation in Guyer-Krumhansl model. Int J Heat Mass Tran 98: 523-529. doi: 10.1016/j.ijheatmasstransfer.2016.03.021
[40]	Zhukovsky K (2017) Exact negative solutions for Guyer-Krumhansl type equation and the maximum principle violation. Entropy 19: 440. doi: 10.3390/e19090440
[41]	Giona M (2017) Relativistic analysis of stochastic kinematics. Phys Rev E 96: 042133. doi: 10.1103/PhysRevE.96.042133
[42]	Weiss G (1994) Aspects and Applications of the Random Walk, North-Holland, Amsterdam.
[43]	Metzler R, Klafter J (2000) The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339: 1-77. doi: 10.1016/S0370-1573(00)00070-3
[44]	Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous diffusion by fractional dynamics. J Phys A 37: R161-R208. doi: 10.1088/0305-4470/37/31/R01
[45]	Oldham KB, Spanier J (2006) The Fractional Calculus, Dover Publ. Inc., Mineola.
[46]	Criado-Sancho M, Lebot JE (1993) On the admissible values of the heat flux in hyperbolic heat transport. Phys Lett A 177: 323-326. doi: 10.1016/0375-9601(93)90008-N
[47]	Camacho J (1995) Third law of thermodynamics and maximum heat flux. Phys Lett A 202: 88-90. doi: 10.1016/0375-9601(95)00325-W
[48]	Polyanin AD (2002) Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton.
[49]	Kolesnik AD, Ratanov N (2013) Telegraph Processes and Option Pricing, Springer, Heidelberg.
[50]	Ghizzetti A, Ossicini A (1971) Trasformate di Laplace e Calcolo Simbolico, UTET, Torino.
[51]	Leveque A (1928) Les lois de la transmission de chaleur par convection. Annales des Mines 13: 201-299.
[52]	Giona M, Adrover A, Cerbelli S, et al. (2009) Laminar dispersion at high éclet numbers in finite-length channels: Effects of the near-wall velocity profile and connection
[53]	Marin E, Vaca-Oyola LS, Delgado-Vasallo O (2016) On thermal waves' velocity: some open questions in thermal waves' physics. Rev Mex Fis E 62: 1-4.
[54]	Kolesnik AD, Turbin AF (1998) The equation of symmetric Markovian random evolution in a plane, Stoch Proc Appl 75: 67-87.
[55]	Kolesnik AD (2001) Weak Convergence of a Planar Random Evolution to the Wiener process. J Theor Probab 14: 485-494. doi: 10.1023/A:1011167815206
[56]	Kolesnik AD, Pinsky MA (2011) Random Evolutions are Driven by the Hyperparabolic Operators. J Stat Phys 142: 828-846. doi: 10.1007/s10955-011-0131-0

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)