Uniform Solution on the Effect of Internal Heat Generation on Rayleigh-Benard Convection in Micropolar Fluid
The effect of internal heat generation is applied to the Rayleigh-Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be rigid-free, rigid-rigid and free-free with the combination of isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number decreases as the value of internal heat generation increase and hence destabilize the system.
<p>[1] A. C. Eringen, “Simple microfluids,” Int. J. Engng. Science, vol. 2,
1964, pp. 205–217.
[2] A. C. Eringen, “Micropolar fluids with stretch,” Int. J. Engng. Sci., vol.
7, 1969, pp. 115–127.
[3] A. C. Eringen, “Theory of anistropic micropolar fluids,” Int. J. Engng.
Sci., vol. 18, 1980, pp. 5–17.
[4] A. C. Eringen, “Continuum theory of dense rigid suspensions,” Rheol.
Acta, vol. 30, 1991, pp. 23–32.
[5] A. C. Eringen, “Theory of micropolar fluids,” J. Math. Mech., vol. 16,
1966b, pp. 1–18.
[6] A. C. Eringen, “Theory of thermomicrofluids,” J. Math. Analyt. Appl.,
vol. 38, 1972, pp. 480–496.
[7] H. Benard, “Les tourbillions cellulaires dans une nappe liquide,” Revue
generale des Sciences pures et appliquees, vol. 11, 1900, pp. 1261–
1271.
[8] H. Benard, “Les tourbillions cellulaires dans une nappe liquid
transportant de la chaleur en regime permanent,” Ann. Chem. Phys., vol.
23, 1901, pp. 62–144.
[9] L. Rayleigh, “On convection currents in a horizontal layer of fluid, when
the higher temperature is on the under side,” Phil. Mag., vol. 32, 1916,
pp. 529–546.
[10] U. Walzer, “Convective instability of a micropolar fluid layer,” Ger.
Beitr. Geophysik. Leipzig, vol. 85, 1976, pp. 137-143.
[11] K. V. R. Rao, “Numerical solution of the thermal instability in a
micropolar fluid layer between rigid boundaries,” Acta Mech., vol. 32,
1979, pp. 79-88.
[12] V. U. K. Sastry, and V. R. Rao, “Numerical solution of thermal
instability of a rotating micropolar fluid layer,” Int. J. Engng. Sci., vol.
21, 1983, pp. 449-461.
[13] Y. Qin, and P. N. Kaloni, “A thermal instability problem in a rotating
micropolar fluid,” Int. J. Engng. Sci., vol. 30, 1992, pp. 1117-1126.
[14] G. Ahmadi, and M. Shahinpoor, “Universal stability of magneto
micropolar fluid motions,” Int. J. Engng. Sci., vol.12, 1974, pp. 657-663.
[15] Y. N. Murty, and V. V. R. Rao, “Effect of throughflow on Marangoni
convection in micropolar fluids,” Acta Mech., vol. 138, 1999, pp. 211-
217.
[16] P. G. Siddeshwar, and S. Pranesh, “Magnetoconvection in fluids with
suspended particles under 1 g and μ g ,” Aero. Sci. Tech., vol. 6, 2002,
pp. 105-114.
[17] A. Vidal, and A. Acrivos, “Nature of the Neutral State in Surface
Tension driven convection,” Phys. Fluids, vol. 9, 1966, pp. 615-616.
[18] W. R. Debler, and L. F. Wolf, “The effect of Gravity and Surface
Tension Gradients on cellular convection in fluid layers with parabolic
temperature profiles,” J. Heat Transfer, vol. 92, 1970, pp. 351-358.
[19] D. A. Nield, “The onset of transient convective instability,” J. Fluid
Mech., vol. 71, 1975, pp. 441-454.
[20] N. Rudraiah, and P. G. Siddeshwar, “Effects of non-uniform temperature
gradient on the onset of Marangoni convection in a fluid with suspended
particles,” Aerosp. Sci. Technol., vol. 4, 2000, pp. 517-523.
[21] N. Rudraiah, “The onset of transient Marangoni convection in a liquid
layer subjected to rotation about a vertical axis,” Mater. Sci. Bull. Indian
Acad. Sci., vol. 4, 1982, pp. 297-316.
[22] R. Friedrich, and N. Rudraiah, “Marangoni convection in a rotating fluid
layer with non-uniform temperature gradient,” Int. J. Heat Mass
Transfer, vol. 27, 1984, pp. 443-449.
[23] G. Lebon, and A. Cloot, “Effects of non-uniform temperature gradients
on Benard-Marangoni’s instability,” J. Non-Equilib. Thermodyn., vol. 6,
1981, pp. 37-50.
[24] N. Rudraiah, and V. Ramachandramurthy, “Effects of non-uniform
temperature gradient and coriolis force on Benard-Marangoni’s
instability,” Acta Mech., vol. 61, 1986, pp. 37-50.
[25] P. G. Siddeshwar, and S. Pranesh, “Effect of non-uniform basic
temperature gradient on Rayeligh-Benard convection in a micropolar
fluid,” Int. J. Engng. Sci., vol. 36, 1998, pp. 1183-1196.
[26] N. Rudraiah, V. Ramachandramurty, and O. P. Chandna, “Effects of
magnetic field and non-uniform temperature gradient on Marangoni
convection,” Int. J. Heat Mass Trans., vol. 28, 1985, pp. 1621-1624.
[27] N. Rudraiah, O. P. Chandna, and M. R. Garg, “Effects of non-uniform
temperature gradient on magneto-convection driven by surface tension
and buoyancy,” Indian J. Tech., vol. 24, 1986, pp. 279-284.
[28] M. J. Fu, N. M. Arifin, M. N. Saad, and R. M. Nazar, “Effects of nonuniform
temperature gradient on Marangoni convection in a micropolar
fluid,” Euro. J. Sci. R., vol. 70, 2009, pp. 612-620.
[29] R. Idris, H. Othman, and I. Hashim, “On effect of non-uniform basic
temperature gradient on Benard-Marangoni convection in micropolar
fluid,” Int. Commun. Heat Mass Transfer, vol. 36, 2009, pp. 192-203.
[30] Z. Alloui, and P. Vasseur, “Onset of Benard-Marangoni convection in a
micropolar fluid,” Int. J. Heat Mass Trans., vol. 54 2011, pp. 2765-2773.
[31] E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson, “Thermal instability
in a horizontal fluid layer: effect of boundary conditions and non-linear
temperature profile,” J. Fluid Mech., vol. 18, 1964, pp. 513-528.
[32] P. H. Roberts, “Convection in horizontal layers with internal heat
generation theory,” J. Fluid Mech., vol. 30, 1967, pp. 33-49.
[33] M. I. Chiang, and K. T. Chiang, “Stability analysis of Benard-
Marangoni convection in fluid with internal heat generation,” Phys. D:
Appl. Phys., vol. 27, 1994, pp. 748-755.
[34] S. K. Wilson, “The effect of uniform internal heat generation on the
onset of steady Marangoni convection in a horizontal layer of fluid,”
Acta Mechanica, vol. 124, 1997, pp. 63-78.
[35] N. Bachok, and N. M. Arifin, “Feedback control of the Marangoni-
Benard convection in a horizontal fluid layer with internal heat
generation,” Microgravity Sci. Technol., vol. 22, 2010, pp. 97-105.</p>
<p>[1] A. C. Eringen, “Simple microfluids,” Int. J. Engng. Science, vol. 2,
1964, pp. 205–217.
[2] A. C. Eringen, “Micropolar fluids with stretch,” Int. J. Engng. Sci., vol.
7, 1969, pp. 115–127.
[3] A. C. Eringen, “Theory of anistropic micropolar fluids,” Int. J. Engng.
Sci., vol. 18, 1980, pp. 5–17.
[4] A. C. Eringen, “Continuum theory of dense rigid suspensions,” Rheol.
Acta, vol. 30, 1991, pp. 23–32.
[5] A. C. Eringen, “Theory of micropolar fluids,” J. Math. Mech., vol. 16,
1966b, pp. 1–18.
[6] A. C. Eringen, “Theory of thermomicrofluids,” J. Math. Analyt. Appl.,
vol. 38, 1972, pp. 480–496.
[7] H. Benard, “Les tourbillions cellulaires dans une nappe liquide,” Revue
generale des Sciences pures et appliquees, vol. 11, 1900, pp. 1261–
1271.
[8] H. Benard, “Les tourbillions cellulaires dans une nappe liquid
transportant de la chaleur en regime permanent,” Ann. Chem. Phys., vol.
23, 1901, pp. 62–144.
[9] L. Rayleigh, “On convection currents in a horizontal layer of fluid, when
the higher temperature is on the under side,” Phil. Mag., vol. 32, 1916,
pp. 529–546.
[10] U. Walzer, “Convective instability of a micropolar fluid layer,” Ger.
Beitr. Geophysik. Leipzig, vol. 85, 1976, pp. 137-143.
[11] K. V. R. Rao, “Numerical solution of the thermal instability in a
micropolar fluid layer between rigid boundaries,” Acta Mech., vol. 32,
1979, pp. 79-88.
[12] V. U. K. Sastry, and V. R. Rao, “Numerical solution of thermal
instability of a rotating micropolar fluid layer,” Int. J. Engng. Sci., vol.
21, 1983, pp. 449-461.
[13] Y. Qin, and P. N. Kaloni, “A thermal instability problem in a rotating
micropolar fluid,” Int. J. Engng. Sci., vol. 30, 1992, pp. 1117-1126.
[14] G. Ahmadi, and M. Shahinpoor, “Universal stability of magneto
micropolar fluid motions,” Int. J. Engng. Sci., vol.12, 1974, pp. 657-663.
[15] Y. N. Murty, and V. V. R. Rao, “Effect of throughflow on Marangoni
convection in micropolar fluids,” Acta Mech., vol. 138, 1999, pp. 211-
217.
[16] P. G. Siddeshwar, and S. Pranesh, “Magnetoconvection in fluids with
suspended particles under 1 g and μ g ,” Aero. Sci. Tech., vol. 6, 2002,
pp. 105-114.
[17] A. Vidal, and A. Acrivos, “Nature of the Neutral State in Surface
Tension driven convection,” Phys. Fluids, vol. 9, 1966, pp. 615-616.
[18] W. R. Debler, and L. F. Wolf, “The effect of Gravity and Surface
Tension Gradients on cellular convection in fluid layers with parabolic
temperature profiles,” J. Heat Transfer, vol. 92, 1970, pp. 351-358.
[19] D. A. Nield, “The onset of transient convective instability,” J. Fluid
Mech., vol. 71, 1975, pp. 441-454.
[20] N. Rudraiah, and P. G. Siddeshwar, “Effects of non-uniform temperature
gradient on the onset of Marangoni convection in a fluid with suspended
particles,” Aerosp. Sci. Technol., vol. 4, 2000, pp. 517-523.
[21] N. Rudraiah, “The onset of transient Marangoni convection in a liquid
layer subjected to rotation about a vertical axis,” Mater. Sci. Bull. Indian
Acad. Sci., vol. 4, 1982, pp. 297-316.
[22] R. Friedrich, and N. Rudraiah, “Marangoni convection in a rotating fluid
layer with non-uniform temperature gradient,” Int. J. Heat Mass
Transfer, vol. 27, 1984, pp. 443-449.
[23] G. Lebon, and A. Cloot, “Effects of non-uniform temperature gradients
on Benard-Marangoni’s instability,” J. Non-Equilib. Thermodyn., vol. 6,
1981, pp. 37-50.
[24] N. Rudraiah, and V. Ramachandramurthy, “Effects of non-uniform
temperature gradient and coriolis force on Benard-Marangoni’s
instability,” Acta Mech., vol. 61, 1986, pp. 37-50.
[25] P. G. Siddeshwar, and S. Pranesh, “Effect of non-uniform basic
temperature gradient on Rayeligh-Benard convection in a micropolar
fluid,” Int. J. Engng. Sci., vol. 36, 1998, pp. 1183-1196.
[26] N. Rudraiah, V. Ramachandramurty, and O. P. Chandna, “Effects of
magnetic field and non-uniform temperature gradient on Marangoni
convection,” Int. J. Heat Mass Trans., vol. 28, 1985, pp. 1621-1624.
[27] N. Rudraiah, O. P. Chandna, and M. R. Garg, “Effects of non-uniform
temperature gradient on magneto-convection driven by surface tension
and buoyancy,” Indian J. Tech., vol. 24, 1986, pp. 279-284.
[28] M. J. Fu, N. M. Arifin, M. N. Saad, and R. M. Nazar, “Effects of nonuniform
temperature gradient on Marangoni convection in a micropolar
fluid,” Euro. J. Sci. R., vol. 70, 2009, pp. 612-620.
[29] R. Idris, H. Othman, and I. Hashim, “On effect of non-uniform basic
temperature gradient on Benard-Marangoni convection in micropolar
fluid,” Int. Commun. Heat Mass Transfer, vol. 36, 2009, pp. 192-203.
[30] Z. Alloui, and P. Vasseur, “Onset of Benard-Marangoni convection in a
micropolar fluid,” Int. J. Heat Mass Trans., vol. 54 2011, pp. 2765-2773.
[31] E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson, “Thermal instability
in a horizontal fluid layer: effect of boundary conditions and non-linear
temperature profile,” J. Fluid Mech., vol. 18, 1964, pp. 513-528.
[32] P. H. Roberts, “Convection in horizontal layers with internal heat
generation theory,” J. Fluid Mech., vol. 30, 1967, pp. 33-49.
[33] M. I. Chiang, and K. T. Chiang, “Stability analysis of Benard-
Marangoni convection in fluid with internal heat generation,” Phys. D:
Appl. Phys., vol. 27, 1994, pp. 748-755.
[34] S. K. Wilson, “The effect of uniform internal heat generation on the
onset of steady Marangoni convection in a horizontal layer of fluid,”
Acta Mechanica, vol. 124, 1997, pp. 63-78.
[35] N. Bachok, and N. M. Arifin, “Feedback control of the Marangoni-
Benard convection in a horizontal fluid layer with internal heat
generation,” Microgravity Sci. Technol., vol. 22, 2010, pp. 97-105.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65130", author = "Izzati K. Khalid and Nor Fadzillah M. Mokhtar and Norihan Md. Arifin", title = "Uniform Solution on the Effect of Internal Heat Generation on Rayleigh-Benard Convection in Micropolar Fluid", abstract = "The effect of internal heat generation is applied to the Rayleigh-Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be rigid-free, rigid-rigid and free-free with the combination of isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number decreases as the value of internal heat generation increase and hence destabilize the system.
", keywords = "Internal heat generation, micropolar fluid, rayleighbenard
convection.", volume = "7", number = "3", pages = "409-6", }
