Effect of Non-Uniform Basic Concentration Gradient on the Onset of Double-Diffusive Convection in Micropolar Fluid

Affiliation(s)

Department of Mathematics, Christ University, Bangalore, India.

Department of Mathematics, Faculty of Engineering, Christ University, Bangalore, India.

Department of Mathematics, Christ University, Bangalore, India.

Department of Mathematics, Faculty of Engineering, Christ University, Bangalore, India.

ABSTRACT

The effect of non-uniform basic concentration gradient on the onset of double diffusive convection in a micropolar fluid layer heated and saluted from below and cooled from above has been studied. The linear stability analysis is performed. The eigen value of the problem is obtained using Galerkian method. The eigen values are obtained for 1) free-free 2) rigid-free 3) rigid-rigid velocity boundary combination with isothermal temperature condition on spin-vanishing permeable boundaries. The influence of various micropolar parameters on the onset of convection has been analyzed. One linear and five non linear concentration profiles are considered and their comparative influence on onset is discussed and results are depicted graphically. It is observed that fluid layer with suspended particles heated and soluted from below is more stable compare to the classical fluid without suspended particles.

The effect of non-uniform basic concentration gradient on the onset of double diffusive convection in a micropolar fluid layer heated and saluted from below and cooled from above has been studied. The linear stability analysis is performed. The eigen value of the problem is obtained using Galerkian method. The eigen values are obtained for 1) free-free 2) rigid-free 3) rigid-rigid velocity boundary combination with isothermal temperature condition on spin-vanishing permeable boundaries. The influence of various micropolar parameters on the onset of convection has been analyzed. One linear and five non linear concentration profiles are considered and their comparative influence on onset is discussed and results are depicted graphically. It is observed that fluid layer with suspended particles heated and soluted from below is more stable compare to the classical fluid without suspended particles.

Cite this paper

S. Pranesh and A. Narayanappa, "Effect of Non-Uniform Basic Concentration Gradient on the Onset of Double-Diffusive Convection in Micropolar Fluid,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 417-424. doi: 10.4236/am.2012.35064.

S. Pranesh and A. Narayanappa, "Effect of Non-Uniform Basic Concentration Gradient on the Onset of Double-Diffusive Convection in Micropolar Fluid,"

References

[1] A. Mojtabi and M. C. Charrier-Mojtabi, “Double Diffusive Convection in Porous Media,” Handbook of Porous Media, Marcel Dekker, New York, 2005, pp. 269-320.

[2] M. E. Stern, “The Salt Fountain and Thermohaline Convection,” Tellus, Vol. 12, No. 2, 1960, pp. 172-175. doi:10.1111/j.2153-3490.1960.tb01295.x

[3] M. E. Stern, “Collective Instability of Salt Fingers,” Journal of Fluid Mechanics, Vol. 35, No. 2, 1969, pp. 209218. doi:10.1017/S0022112069001066

[4] J. S. Turner, “Buoyancy Effects in Fluids,” Cambridge University Press, Cambridge, 1973.

[5] S. Chen, J. Tolke and M. Krafczyk, “Numerical Investigation of Double-Diffusive (Natural) Convection in Vertical Annuluses with Opposing Temperature and Concentration Gradients,” International Journal of Heat and Fluid Flow, Vol. 31, No. 2, 2010, pp. 217-226. doi:10.1016/j.ijheatfluidflow.2009.12.013

[6] M. S. Malashetty and B. S. Biradar, “The Onset of Double Diffusive Convection in a Binary Maxwell Fluid Saturated Porous Layer with Cross Diffusion Effects,” Physics of Fluids, Vol. 23, No. 6, 2011, p. 13. doi:10.1063/1.3601482

[7] A. C. Eringen, “Micropolar Theory of Liquid Crystals,” In: J. F. Johnson and R. S. Porter, Eds., Liquid Crystals and Ordered Fluids, Vol. 3, Plenum Publishing, New York, 1978.

[8] A. C. Eringen, “Theory of Micropolar Fluids,” Journal of Mathematics and Mechanics, Vol. 16, 1966, pp. 1-18.

[9] A. C. Eringen, “Micro Continuum Field Theory,” Springer Verlag, New York, 1999.

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

[11] H. Power, “Bio-Fluid Mechanics, Advances in Fluid Mechanics,” W.I.T. Press, UK, 1995.

[12] A. B. 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. doi:10.1016/0020-7225(76)90005-7

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

[14] K. V. Rama Rao, “Thermal Instability in a Micropolar Fluid Layer Subject to Magnetic Field,” International Journal of Engineering Science, Vol. 18, No. 5, 1980, pp. 741-750. doi:10.1016/0020-7225(80)90107-X

[15] C. 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. doi:10.1016/0020-7225(82)90009-X

[16] S. P. Bhattacharyya and S. K. Jena, “On the Stability of Hot Layer of Micropolar Fluid,” International Journal of Engineering Science, Vol. 21, No. 9, 1983, pp. 1019-1024. doi:10.1016/0020-7225(83)90043-5

[17] P. G. Siddheshwar and S. Pranesh, “Magneto Convection in a Micropolar Fluid,” International Journal of Engineering Science, Vol. 36, No. 10, 1998, pp. 1173-1181. doi:10.1016/S0020-7225(98)00013-5

[18] P. G. Siddheshwar and S. Pranesh, “Linear and Weakly Non-Linear Analyses of Convection in a Micropolar Fluid,” Hydrodynamics VI-Theory and Applications, 2005, pp. 487-493.

[19] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1g and ?g,” Aerospace Science and Technology, Vol. 6, No. 2, 2001, pp. 105-114. doi:10.1016/S1270-9638(01)01144-0

[20] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Benard-Marangoni Convection in a Fluid with Suspended Articles,” Acta Mechanica, Vol. 152, No. 1-4, 2001, pp. 241-252. doi:10.1007/BF01176958

[21] Y. N. Murthy and V. V. Ramana Rao, “Effect of Through Flow on Marangoni Convection in Micropolar Fluids,” Acta Mechanica, Vol. 138, No. 3-4, 1999, pp. 211-217, doi:10.1007/BF01291845

[22] S. Pranesh and R. V. Kiran, “Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law,” Applied Mathematics, Vol. 1, 2010, pp. 470-480, doi:10.4236/am.2010.16062

[23] Z. Alloui, H. Beji and P. Vasseur, “Double-Diffusive and Soret-Induced Convection of a Micropolar Fluid in a Vertical Channel,” Computers & Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 725-736. doi:10.1016/j.camwa.2011.05.053

[1] A. Mojtabi and M. C. Charrier-Mojtabi, “Double Diffusive Convection in Porous Media,” Handbook of Porous Media, Marcel Dekker, New York, 2005, pp. 269-320.

[2] M. E. Stern, “The Salt Fountain and Thermohaline Convection,” Tellus, Vol. 12, No. 2, 1960, pp. 172-175. doi:10.1111/j.2153-3490.1960.tb01295.x

[3] M. E. Stern, “Collective Instability of Salt Fingers,” Journal of Fluid Mechanics, Vol. 35, No. 2, 1969, pp. 209218. doi:10.1017/S0022112069001066

[4] J. S. Turner, “Buoyancy Effects in Fluids,” Cambridge University Press, Cambridge, 1973.

[5] S. Chen, J. Tolke and M. Krafczyk, “Numerical Investigation of Double-Diffusive (Natural) Convection in Vertical Annuluses with Opposing Temperature and Concentration Gradients,” International Journal of Heat and Fluid Flow, Vol. 31, No. 2, 2010, pp. 217-226. doi:10.1016/j.ijheatfluidflow.2009.12.013

[6] M. S. Malashetty and B. S. Biradar, “The Onset of Double Diffusive Convection in a Binary Maxwell Fluid Saturated Porous Layer with Cross Diffusion Effects,” Physics of Fluids, Vol. 23, No. 6, 2011, p. 13. doi:10.1063/1.3601482

[7] A. C. Eringen, “Micropolar Theory of Liquid Crystals,” In: J. F. Johnson and R. S. Porter, Eds., Liquid Crystals and Ordered Fluids, Vol. 3, Plenum Publishing, New York, 1978.

[8] A. C. Eringen, “Theory of Micropolar Fluids,” Journal of Mathematics and Mechanics, Vol. 16, 1966, pp. 1-18.

[9] A. C. Eringen, “Micro Continuum Field Theory,” Springer Verlag, New York, 1999.

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

[11] H. Power, “Bio-Fluid Mechanics, Advances in Fluid Mechanics,” W.I.T. Press, UK, 1995.

[12] A. B. 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. doi:10.1016/0020-7225(76)90005-7

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

[14] K. V. Rama Rao, “Thermal Instability in a Micropolar Fluid Layer Subject to Magnetic Field,” International Journal of Engineering Science, Vol. 18, No. 5, 1980, pp. 741-750. doi:10.1016/0020-7225(80)90107-X

[15] C. 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. doi:10.1016/0020-7225(82)90009-X

[16] S. P. Bhattacharyya and S. K. Jena, “On the Stability of Hot Layer of Micropolar Fluid,” International Journal of Engineering Science, Vol. 21, No. 9, 1983, pp. 1019-1024. doi:10.1016/0020-7225(83)90043-5

[17] P. G. Siddheshwar and S. Pranesh, “Magneto Convection in a Micropolar Fluid,” International Journal of Engineering Science, Vol. 36, No. 10, 1998, pp. 1173-1181. doi:10.1016/S0020-7225(98)00013-5

[18] P. G. Siddheshwar and S. Pranesh, “Linear and Weakly Non-Linear Analyses of Convection in a Micropolar Fluid,” Hydrodynamics VI-Theory and Applications, 2005, pp. 487-493.

[19] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1g and ?g,” Aerospace Science and Technology, Vol. 6, No. 2, 2001, pp. 105-114. doi:10.1016/S1270-9638(01)01144-0

[20] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Benard-Marangoni Convection in a Fluid with Suspended Articles,” Acta Mechanica, Vol. 152, No. 1-4, 2001, pp. 241-252. doi:10.1007/BF01176958

[21] Y. N. Murthy and V. V. Ramana Rao, “Effect of Through Flow on Marangoni Convection in Micropolar Fluids,” Acta Mechanica, Vol. 138, No. 3-4, 1999, pp. 211-217, doi:10.1007/BF01291845

[22] S. Pranesh and R. V. Kiran, “Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law,” Applied Mathematics, Vol. 1, 2010, pp. 470-480, doi:10.4236/am.2010.16062

[23] Z. Alloui, H. Beji and P. Vasseur, “Double-Diffusive and Soret-Induced Convection of a Micropolar Fluid in a Vertical Channel,” Computers & Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 725-736. doi:10.1016/j.camwa.2011.05.053