# American Institute of Mathematical Sciences

September  2014, 19(7): 2133-2143. doi: 10.3934/dcdsb.2014.19.2133

## On the Green-Naghdi Type III heat conduction model

 1 DICATAM, Università di Brescia, Via Valotti, 9 - 25133 Brescia 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria 3 Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received  November 2012 Revised  May 2013 Published  August 2014

In this work, we compare different constitutive models of heat flux in a rigid heat conductor. In particular, we investigate the relation between the solutions of the Green-Naghdi type III equation and those of the classical Fourier heat equation. The latter is often referred to as a limit case of the former one, as (formally) obtained by letting certain small positive parameter $\epsilon$ vanish. In presence of steady heat sources, we prove that the type III equation may be considered as a perturbation of the Fourier one only if the solutions are compared on a finite time interval of order $1/\epsilon$, whereas significant differences occur in the longterm. Moreover, for a bar with finite length and prescribed heat flux at its ends, the solutions to the type III equation do not converge asymptotically in time to the steady solutions to the corresponding Fourier model. This suggests that the Green-Naghdi type III theory is not to be viewed as comprehensive of the Fourier theory, at least when either asymptotic or stationary phenomena are involved.
Citation: Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133
##### References:
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##### References:
 [1] S. Bargmann, P. Steinmann and P. Jordan, On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory, Phys. Lett. A, 372 (2008), 4418-4424. doi: 10.1016/j.physleta.2008.04.010. [2] C. C. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Comm., 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003. [3] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912. [4] M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana U. Math. J., 55 (2006), 169-216. doi: 10.1512/iumj.2006.55.2661. [5] C. Giorgi, V. Pata and A. Marzocchi, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math., 58 (2000), 661-683. [6] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, Nonlinear Differential Equations and Applications, 8 (2001), 157-171. doi: 10.1007/PL00001443. [7] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., Birkhäuser Basel, (Eds. A. Lorenzi and B. Ruf), 50 (2002), 155-178. [8] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. London Ser. A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012. [9] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. [10] P. Podio-Guidugli, A virtual power format for thermomechanics, Continuum Mech. Thermodyn., 20 (2009), 479-487. doi: 10.1007/s00161-009-0093-5. [11] R. Quintanilla and B. Straughan, Green-Naghdi type III viscous fluids, Int. J. Heat Mass Tran., 55 (2012), 710-714. doi: 10.1016/j.ijheatmasstransfer.2011.10.039. [12] B. Straughan, Heat Waves, Springer {New York}, 2011. doi: 10.1007/978-1-4614-0493-4. [13] J. J. Vadasz, S. Govender and P. Vadasz, Heat transfer enhancement in nano-fluids suspensions: Possible mechanisms and explanations, Int. J. Heat Mass Tran., 48 (2005), 2673-2683. doi: 10.1016/j.ijheatmasstransfer.2005.01.023.
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