Abstract
The classical heat law of Fourier associates an infinite speed of propagation to a thermal disturbance in a material body. Such behavior is a violation of the causality principle. In recent years, several modifications of Fourier's heat law have been proposed. In this work a modification of Fourier's heat law based on the Maxwell-Cattaneo-Fox (MCF) model is used to describe the influence of heat conduction at low temperatures and/or high heat-flux conditions on Stokes' first problem for a dipolar fluid. The effects of discontinuous boundary data and a finite propagation speed of thermal waves on the velocity and stress fields are investigated. In addition, special and limiting cases of the material constants are examined. Lastly, results for the special case of equal dipolar constants are compared to the corresponding results found using Fourier's heat law.
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References
J. C. Maxwell, On the dynamical theory of gases. Phil. Trans. R. Soc. London 157 (1867) 49-88.
V. Peshkov, “Second sound” in Helium II. J. Phys. (U.S.S.R.) 8 (1944) 381.
C. Cattaneo, Sullacondizione del Calore. Atti del Semin. Matem. e Fis. della Univ. Modena. 3 (1948) 83-101.
D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics. 2nd ed. Berlin: Springer-Verlag (1996) 383 pp.
M. Chester, Second sound in solids. Phys. Rev. 131 (1963) 2013-2015.
C. C. Ackerman, B. Bertman, H. A. Fairbank and R. A. Guyer, Second sound in solid helium. Phys. Rev. Letters. 16 (1966) 789-791.
B. Straughan and F. Franchi, Bénard convection and the Cattaneo law of heat conduction. Proc. R. Soc. Edin. 96A (1984) 175-178.
C. L. McTaggart and K. A. Lindsay, Nonclassical effects in the Bénard problem. SIAM J. Appl. Math. 45 (1985) 70-92.
P. Puri and P. K. Kythe, Nonclassical thermal effects in Stokes' second problem. Acta Mech. 112 (1995) 1-9.
P. Puri and P. K. Kythe, Discontinuities in velocity gradients and temperature in the Stokes' first problem with nonclassical heat conduction. Quart. Appl. Math. 55 (1997) 167-176.
P. Puri, Plane waves in generalized thermoelasticity. Int. J. Eng. Sci. 11 (1973) 35-744; 13 (1975) 339–340.
D. D. Joseph and L. Preziosi, Heat waves. Rev. Modern Phys. 61 (1989) 41-73.
D. D. Joseph and L. Preziosi, Addendum to the paper “Heat waves”. Rev. Modern Phys. 62 (1990) 375-391.
W. Dreyer and H. Struchtrup, Heat pulse experiments revisited. Continuum Mech. Thermodyn. 5 (1993) 3-50.
D. S. Chandrasekharaiah, Thermoelasticity with second sound: A review. Appl. Mech. Rev. 39 (1986) 355-376.
B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals. Arch. Rational Mech. Anal. 80 (1982) 135-158.
A. Morro and T. Ruggeri, Non-equilibrium properties of solids obtained from second-sound measurements. J. Phys. C: Solid State Phys. 21 (1988) 1743-1752.
T. Ruggeri, A. Muracchini and L. Seccia, Continuum approach to phonon gas and shape changes of second sound via shock waves theory. Il Nuovo Cimento 16D (1994) 15-44.
J. L. Martinez, F. J. Bermejo, M. García-Hernández, F. J. Mompeán, E. Enciso, and D. Martín, J. Phys.: Condens. Matter 3 (1991) 4075-4087.
S. C. Cowin, The theory of polar fluids. Advs. Appl Mech. 14 (1974) 279-347.
M. E. Erdogan, Dynamics of polar fluids. Acta Mech. 15 (1972) 33-253.
J. L. Bleustein and A. E. Green, Dipolar fluids. Int. J. Eng. Sci. 5 (1967) 323-340.
A. E. Green and P. M. Naghdi, A Note on dipolar inertia. Quart. Appl. Math. 28 (1970) 458-460.
T. Ariman, M. A. Turk and N. D. Sylvester, Microcontinuum fluid mechanics-A review. Int. J. Eng. Sci. 11 (1973) 905-930.
G. S. Guram, Flow of a dipolar fluid due to suddenly accelerated flat plate. Acta Mech. 49 (1983) 133-138.
H. Schlichting, Boundary Layer Theory. 4th ed., New York: McGraw-Hill (1960) 647 pp.
K. S. Sran, Heat transfer in a dipolar flow through a porous channel. J. Franklin Inst. 324 (1987) 303-317.
B. Straughan, Stability of a layer of dipolar fluid heated from below. Math. Meth. Appl. Sci. 9 (1987) 5-45.
P. M. Jordan, Nonclassical Thermal Effects on Dipolar Fluid Flow. Masters of Science thesis, Department of Mathematics, University of New Orleans, New Orleans, Louisiana, USA (1996) 64 pp.
P. Puri and P. M. Jordan, Wave structure in Stokes' second problem for a dipolar fluid with nonclassical heat conduction. To appear in Acta Mech.
I. Müller and T. Ruggeri, Extended thermodynamics. In: C. Truesdell (ed.) Springer Tracts in Natural Philosophy, vol. 37. New York: Springer-Verlag (1993) 230 pp.
A. Erdélyi, (ed.) Tables of Laplace transforms. In: Bateman Manuscripts Project, vol. I, New York: McGraw-Hill (1954) 391 pp.
B. A. Boley, Discontinuities in integral-transform solutions. Quart. Appl. Math. 19 (1962) 273-284.
P. Chadwick and B. Powdrill, Applications of the Laplace transform method to wave motions involving strong discontinuities. Proc. Camb. Phil. Soc. 60 (1964) 313-324.
C. Truesdell and R. A. Toupin, The classical field theories. In: S. Flügge (ed.), Handbuch der Physik, vol. III/l. Berlin: Springer-Verlag (1960) pp. 226-793.
W. Prager, Discontinuous fields of plastic stress and flow. In: Proc. 2nd U.S. Cong. Appl. Mech. Ann Arbor, Michigan, USA (1954) pp. 21-32.
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Puri, P., Jordan, P.M. Stokes' first problem for a dipolar fluid with nonclassical heat conduction. Journal of Engineering Mathematics 36, 219–240 (1999). https://doi.org/10.1023/A:1004454227370
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DOI: https://doi.org/10.1023/A:1004454227370