# American Institute of Mathematical Sciences

June  2011, 15(4): 991-998. doi: 10.3934/dcdsb.2011.15.991

## On the objective rate of heat and stress fluxes. Connection with micro/nano-scale heat convection

 1 Department of Mechanical and Materials Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9, Canada 2 Department of Mechanical and Industrial Engineering, 1206 W. Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2906

Received  January 2010 Revised  June 2010 Published  March 2011

In this paper, the derivation of the convected derivatives for the heat flux and stress tensor is revisited. A kinematic approach is adopted based on material invariance. These upper-convected derivatives are used in the literature to generalize Newton's law of viscosity and Fourier's heat law of heat. The former constitutive law represents the behaviour of a viscoelastic fluid of the Boger type obeying the Oldroyd-B model, and the latter represents fluids obeying the Maxwell-Cattaneo's heat equation. The invariance of the derivatives under orthogonal transformation is also shown. Although the presentation here is limited to the derivatives of vector and second-rank tensor fluxes, the formulation can be generalized to generate the convected derivative of a tensor flux of arbitrary rank. Finally, the connection with micro- or nano-channel flow is noted.
Citation: Roger E. Khayat, Martin Ostoja-Starzewski. On the objective rate of heat and stress fluxes. Connection with micro/nano-scale heat convection. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 991-998. doi: 10.3934/dcdsb.2011.15.991
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##### References:
 [1] M. Chester, Second sound in solids,, Phys. Rev., 131 (1963), 2013. doi: doi:10.1103/PhysRev.131.2013. Google Scholar [2] C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction,, Mech. Res. Comm., 36 (2009), 481. doi: doi:10.1016/j.mechrescom.2008.11.003. Google Scholar [3] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second sound propagation in moving media,, Phys. Rev. Lett., 94 (2005). doi: doi:10.1103/PhysRevLett.94.154301. Google Scholar [4] J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticty with Finite Wave Speeds,", Oxford Mathematical Monographs, (2009). Google Scholar [5] L. E. Malvern, "Introduction to the Mechanics of a Continuous Medium,", Prentice-Hall, (1969). Google Scholar [6] J. C. Oldroyd, On the formulation of rheological equations of state,, Proc. Roy. Soc. A, 200 (1950), 523. doi: doi:10.1098/rspa.1950.0035. Google Scholar [7] M. Ostoja-Starzewski, A derivation of the Maxwell-Cattaneo equation from the free energy and dissipation potentials,, Int. J. Eng. Sci., 47 (2009), 807. doi: doi:10.1016/j.ijengsci.2009.03.002. Google Scholar [8] V. Peshkov, "Second sound" in helium II,, J. Phys. USSR, 8 (1944). Google Scholar [9] E. Schrödinger, "Space-Time Structure,", Cambridge University Press, (1950). Google Scholar
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