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 and Continuous Dynamical Systems - B, 2011, 15 (4) : 991-998. doi: 10.3934/dcdsb.2011.15.991
References:
[1]

M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: doi:10.1103/PhysRev.131.2013.

[2]

C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Comm., 36 (2009), 481-486. doi: doi:10.1016/j.mechrescom.2008.11.003.

[3]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. doi: doi:10.1103/PhysRevLett.94.154301.

[4]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticty with Finite Wave Speeds," Oxford Mathematical Monographs, Oxford University Press, 2009.

[5]

L. E. Malvern, "Introduction to the Mechanics of a Continuous Medium," Prentice-Hall, N. J., 1969.

[6]

J. C. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. A, 200 (1950), 523-541. doi: doi:10.1098/rspa.1950.0035.

[7]

M. Ostoja-Starzewski, A derivation of the Maxwell-Cattaneo equation from the free energy and dissipation potentials, Int. J. Eng. Sci., 47 (2009), 807-810. doi: doi:10.1016/j.ijengsci.2009.03.002.

[8]

V. Peshkov, "Second sound" in helium II, J. Phys. USSR, 8 (1944), 131.

[9]

E. Schrödinger, "Space-Time Structure," Cambridge University Press, 1950.

show all references

References:
[1]

M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: doi:10.1103/PhysRev.131.2013.

[2]

C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Comm., 36 (2009), 481-486. doi: doi:10.1016/j.mechrescom.2008.11.003.

[3]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. doi: doi:10.1103/PhysRevLett.94.154301.

[4]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticty with Finite Wave Speeds," Oxford Mathematical Monographs, Oxford University Press, 2009.

[5]

L. E. Malvern, "Introduction to the Mechanics of a Continuous Medium," Prentice-Hall, N. J., 1969.

[6]

J. C. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. A, 200 (1950), 523-541. doi: doi:10.1098/rspa.1950.0035.

[7]

M. Ostoja-Starzewski, A derivation of the Maxwell-Cattaneo equation from the free energy and dissipation potentials, Int. J. Eng. Sci., 47 (2009), 807-810. doi: doi:10.1016/j.ijengsci.2009.03.002.

[8]

V. Peshkov, "Second sound" in helium II, J. Phys. USSR, 8 (1944), 131.

[9]

E. Schrödinger, "Space-Time Structure," Cambridge University Press, 1950.

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