Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law

ABSTRACT

The effects of result from the substitution of the classical Fourier law by the non-classical Maxwell-Cattaneo law on the Rayleigh-Bénard Magneto-convection in an electrically conducting micropolar fluid is studied using the Galerkin technique. The eigenvalue is obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal or adiabatic temperature on the spin-vanishing boundaries. The influences of various micropolar fluid parameters are analyzed on the onset of convection. The classical approach predicts an infinite speed for the propagation of heat. The present non-classical theory involves a wave type heat transport (SECOND SOUND) and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. It is found that the results are noteworthy at short times and the critical eigenvalues are less than the classical ones.

The effects of result from the substitution of the classical Fourier law by the non-classical Maxwell-Cattaneo law on the Rayleigh-Bénard Magneto-convection in an electrically conducting micropolar fluid is studied using the Galerkin technique. The eigenvalue is obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal or adiabatic temperature on the spin-vanishing boundaries. The influences of various micropolar fluid parameters are analyzed on the onset of convection. The classical approach predicts an infinite speed for the propagation of heat. The present non-classical theory involves a wave type heat transport (SECOND SOUND) and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. It is found that the results are noteworthy at short times and the critical eigenvalues are less than the classical ones.

Cite this paper

nullS. Pranesh and R. Kiran, "Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 470-480. doi: 10.4236/am.2010.16062.

nullS. Pranesh and R. Kiran, "Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law,"

References

[1] J. C. Maxwell, “On the Dynamical Theory of Gases,” The Philosophical Transactions of the Royal Society, Vol. 157, 1867, pp. 49-88.

[2] C. Cattaneo, “Sulla Condizione Del Calore,” Atti Del Semin. Matem. E Fis. Della Univ. Modena, Vol. 3, 1948, pp. 83-101.

[3] K. A. Lindsay and B. Straughan, “Acceleration Waves and Second Sound in a Perfect Fluid,” Archive for Rational Mechanics and Analysis, Vol.68, 1978, pp 53-87.

[4] B. Straughan and F. Franchi, “Bénard Convection and the Cattaneo Law of Heat Conduction,” Proceedings of the Royal Society of Edinburgh, Vol. 96A, 1984, pp. 175- 178.

[5] G. Lebon and A. Cloot, “A Nonlinear Stability Analysis of the Bénard-Marangoni Problem,” Journal of Fluid Mechanics, Vol. 145, 1984, pp. 447-469.

[6] P. G. Siddheshwar, “Rayleigh Benard Convection in a Second Order Ferromagnetic Fluid with Second Sound,” Proceedings of 8th Asian Congress of Fluid Mechanics, Shenzen, December 6-10, 1999, p. 631.

[7] S. Pranesh, “Effect of Second Sound on the Onset of Rayleigh-Bénard Convection in a Coleman-Noll Fluid,” Mapana Journal of Science, Vol. 7, No. 2, 2008, pp. 1-9.

[8] P. C. Dauby, M. Nelis and G. Lebon, “Generalized Fourier Equations and Thermoconvective Instabilities,” Revista Mexicana de Fisica, Vol. 48, 2001, pp. 57-62.

[9] B. Straughan, “Oscillatory Convection and the Cattaneo Law of Heat Conduction,” Ricerche di Matematica, Vol. 58, No. 2, 2009, pp. 157-162.

[10] P. Puri and P. M. Jordan, “Stokes’ First Problem for a Dipolar Fluid with Nonclassical Heat Conduction,” Journal of Engineering Mathematics, Vol. 36, No. 3, 1999, pp. 219-240.

[11] P. Puri and P. M. Jordan, “Wave Structure in Stokes Second Problem for Dipolar Fluid with Nonclassical Heat Conduction,” Acta Mechanica, Vol. 133, No. 1-4, 1999, pp. 145-160.

[12] P. Puri and P. K. Kythe, “Nonclassical Thermal Effects in Stokes Second Problem,” Acta Mechanica, Vol. 112, No. 1-4, 1995, pp. 1-9.

[13] P. Puri and P. K. Kythe, “Discontinuities in Velocity Gradients and Temperature in the Stokes First Problem with Nonclassical Heat Conduction,” Quarterly of Applied Mathematics, Vol. 55, No. 1, March 1997, pp. 167- 176.

[14] A. C. Eringen, “Simple Microfluids,” International Journal of Engineering Science, Vol. 2, No. 2, 1964, pp. 205-217.

[15] A. C. Eringen, “Theory of Thermomicrofluids,” Journal of Mathematical Analysis and Applications, Vol. 38, No. 2, May 1972, pp. 480-496.

[16] A. C. Eringen, “Microcontinuum Field Theory,” Springer Verlag, New York, 1999.

[17] H. Power, “Bio-Fluid Mechanics, Advances in Fluid Mechanics,” W. I. T. Press, U. K., 1995.

[18] G. Lukaszewicz, “Micropolar Fluid Theory and Applications,” Birkhauser Boston, Boston, 1999.

[19] G. Ahmadi, “Stability of a Micropolar Fluid Layer Heated from Below,” International Journal of Engineering Science, Vol. 14, No. 1, January 1976, pp. 81-89.

[20] Datta and V. U. K. Sastry, “Thermal Instability of a Horizontal Layer of Micropolar Fluid Heated from Below,” International Journal of Engineering Science, Vol. 14, No. 7, 1976, pp. 631-637.

[21] A. Perez Garcia and J. M. Rubi, “On the Possibility of Overstable Motions of Micropolar Fluids Heated from Below,” International Journal of Engineering Science, Vol. 20, No. 7, 1982, pp. 873-878.

[22] L. E. Payne and B. Straughan, “Critical Rayleigh Numbers for Oscillatory and Nonlinear Convection in an Instropic Thermomicropolar Fluid,” International Journal of Engineering Science, Vol. 27, 1989, p. 827.

[23] G. Lebon and C. Perez-Garcia, “Convective Instability of a Micropolar Fluid Layer by the Method of Energy,” International Journal of Engineering Science, Vol. 19, No. 10, 1981, pp. 1321-1329.

[24] S. Pranesh, “Effects of Suction-Injection-Combination (SIC) on the Onset of Rayleigh-Bénard Magnetoconvection in a Fluid with Suspended Particles,” International Journal of Engineering Science, Vol. 41, No. 15, September 2003, pp. 1741-1766.

[25] P. G. Siddheshwar and S. Pranesh, “Effects of a Non- Uniform Basic Temperature Gradient on Rayleigh- Bénard Convection in a Micropolar Fluid,” International Journal of Engineering Science, Vol. 36, No. 11, September 1998, pp. 1183-1196.

[26] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Bénard-Marangoni Convection in a Fluid with Suspended Particles,” Acta Mechanica, Vol. 152, No. 1-4, 2001, pp. 241-252.

[27] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1g and ?G,” In ternational Journal of Aerospace Science and Technology, Vol. 6, No. 2, February 2001, pp. 105-114.

[28] P. G. Siddheshwar and S. Pranesh, “Effect of Temperature/Gravity Modulation on the Onset of Magneto Convection in Weak Electrically Conducting Fluids with Internal Angular Momentum,” Journal of Magnetism and Magnetic Materials, Vol. 192, No. 1, February 1999, pp. 159-176.

[29] Lawrence C. Evans, “Partial Differential Equations (Schaum’s Outline Series),” Tata McGraw Hill, India, 2010.

[1] J. C. Maxwell, “On the Dynamical Theory of Gases,” The Philosophical Transactions of the Royal Society, Vol. 157, 1867, pp. 49-88.

[2] C. Cattaneo, “Sulla Condizione Del Calore,” Atti Del Semin. Matem. E Fis. Della Univ. Modena, Vol. 3, 1948, pp. 83-101.

[3] K. A. Lindsay and B. Straughan, “Acceleration Waves and Second Sound in a Perfect Fluid,” Archive for Rational Mechanics and Analysis, Vol.68, 1978, pp 53-87.

[4] B. Straughan and F. Franchi, “Bénard Convection and the Cattaneo Law of Heat Conduction,” Proceedings of the Royal Society of Edinburgh, Vol. 96A, 1984, pp. 175- 178.

[5] G. Lebon and A. Cloot, “A Nonlinear Stability Analysis of the Bénard-Marangoni Problem,” Journal of Fluid Mechanics, Vol. 145, 1984, pp. 447-469.

[6] P. G. Siddheshwar, “Rayleigh Benard Convection in a Second Order Ferromagnetic Fluid with Second Sound,” Proceedings of 8th Asian Congress of Fluid Mechanics, Shenzen, December 6-10, 1999, p. 631.

[7] S. Pranesh, “Effect of Second Sound on the Onset of Rayleigh-Bénard Convection in a Coleman-Noll Fluid,” Mapana Journal of Science, Vol. 7, No. 2, 2008, pp. 1-9.

[8] P. C. Dauby, M. Nelis and G. Lebon, “Generalized Fourier Equations and Thermoconvective Instabilities,” Revista Mexicana de Fisica, Vol. 48, 2001, pp. 57-62.

[9] B. Straughan, “Oscillatory Convection and the Cattaneo Law of Heat Conduction,” Ricerche di Matematica, Vol. 58, No. 2, 2009, pp. 157-162.

[10] P. Puri and P. M. Jordan, “Stokes’ First Problem for a Dipolar Fluid with Nonclassical Heat Conduction,” Journal of Engineering Mathematics, Vol. 36, No. 3, 1999, pp. 219-240.

[11] P. Puri and P. M. Jordan, “Wave Structure in Stokes Second Problem for Dipolar Fluid with Nonclassical Heat Conduction,” Acta Mechanica, Vol. 133, No. 1-4, 1999, pp. 145-160.

[12] P. Puri and P. K. Kythe, “Nonclassical Thermal Effects in Stokes Second Problem,” Acta Mechanica, Vol. 112, No. 1-4, 1995, pp. 1-9.

[13] P. Puri and P. K. Kythe, “Discontinuities in Velocity Gradients and Temperature in the Stokes First Problem with Nonclassical Heat Conduction,” Quarterly of Applied Mathematics, Vol. 55, No. 1, March 1997, pp. 167- 176.

[14] A. C. Eringen, “Simple Microfluids,” International Journal of Engineering Science, Vol. 2, No. 2, 1964, pp. 205-217.

[15] A. C. Eringen, “Theory of Thermomicrofluids,” Journal of Mathematical Analysis and Applications, Vol. 38, No. 2, May 1972, pp. 480-496.

[16] A. C. Eringen, “Microcontinuum Field Theory,” Springer Verlag, New York, 1999.

[17] H. Power, “Bio-Fluid Mechanics, Advances in Fluid Mechanics,” W. I. T. Press, U. K., 1995.

[18] G. Lukaszewicz, “Micropolar Fluid Theory and Applications,” Birkhauser Boston, Boston, 1999.

[19] G. Ahmadi, “Stability of a Micropolar Fluid Layer Heated from Below,” International Journal of Engineering Science, Vol. 14, No. 1, January 1976, pp. 81-89.

[20] Datta and V. U. K. Sastry, “Thermal Instability of a Horizontal Layer of Micropolar Fluid Heated from Below,” International Journal of Engineering Science, Vol. 14, No. 7, 1976, pp. 631-637.

[21] A. Perez Garcia and J. M. Rubi, “On the Possibility of Overstable Motions of Micropolar Fluids Heated from Below,” International Journal of Engineering Science, Vol. 20, No. 7, 1982, pp. 873-878.

[22] L. E. Payne and B. Straughan, “Critical Rayleigh Numbers for Oscillatory and Nonlinear Convection in an Instropic Thermomicropolar Fluid,” International Journal of Engineering Science, Vol. 27, 1989, p. 827.

[23] G. Lebon and C. Perez-Garcia, “Convective Instability of a Micropolar Fluid Layer by the Method of Energy,” International Journal of Engineering Science, Vol. 19, No. 10, 1981, pp. 1321-1329.

[24] S. Pranesh, “Effects of Suction-Injection-Combination (SIC) on the Onset of Rayleigh-Bénard Magnetoconvection in a Fluid with Suspended Particles,” International Journal of Engineering Science, Vol. 41, No. 15, September 2003, pp. 1741-1766.

[25] P. G. Siddheshwar and S. Pranesh, “Effects of a Non- Uniform Basic Temperature Gradient on Rayleigh- Bénard Convection in a Micropolar Fluid,” International Journal of Engineering Science, Vol. 36, No. 11, September 1998, pp. 1183-1196.

[26] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Bénard-Marangoni Convection in a Fluid with Suspended Particles,” Acta Mechanica, Vol. 152, No. 1-4, 2001, pp. 241-252.

[27] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1g and ?G,” In ternational Journal of Aerospace Science and Technology, Vol. 6, No. 2, February 2001, pp. 105-114.

[28] P. G. Siddheshwar and S. Pranesh, “Effect of Temperature/Gravity Modulation on the Onset of Magneto Convection in Weak Electrically Conducting Fluids with Internal Angular Momentum,” Journal of Magnetism and Magnetic Materials, Vol. 192, No. 1, February 1999, pp. 159-176.

[29] Lawrence C. Evans, “Partial Differential Equations (Schaum’s Outline Series),” Tata McGraw Hill, India, 2010.