An Unsteady Mixed Convection in a Driven Cavity Filled with Nanofluids Using an Externally Oscillating Lid

Affiliation(s)

Mechanical Engineering Department, University College of Engineering, University of Tehran, Tehran, Iran.

Mechanical Engineering Department, University College of Engineering, University of Tehran, Tehran, Iran.

ABSTRACT

A numerical investigation of an unsteady, periodic, laminar mixed-convection in a cavity utilized with copper-water nanofluid is presented. In this study both top and bottom walls are assumed to be isolated, meanwhile sidewalls are considered under constant temperature condition. We consider a time-dependent oscillating wall on top to fulfill a periodic mixed-convection inside the cavity. In this type of problems both Grashof and Reynolds numbers play a great role in flow pattern and heat transfer characteristics, so we focus our study on four major parameters that can be crucial such as Grashof and Reynoldsnumbers, solid volume fraction and the non-dimensional lid frequency . The obtained results show that the augmentation of Reynolds number and Grashof number would enhance the average Nusselt number. It is also found that unlike steady state condition, at high Reynolds numbers, as lid is moving in the negative direction the average Nusselt number on the hot wall becomes higher in respect to the case that lid is moving in the positive direction due to thermal boundary layer disturbance. Lid frequency does not have a significant effect on thermal characteristics at low Reynolds numbers, meanwhile at higher Reynolds numbers, increment of lid frequency results in heat transfer reduction. Moreover, solid volume fraction is found to have better efficiency at higher Grashof numbers.

A numerical investigation of an unsteady, periodic, laminar mixed-convection in a cavity utilized with copper-water nanofluid is presented. In this study both top and bottom walls are assumed to be isolated, meanwhile sidewalls are considered under constant temperature condition. We consider a time-dependent oscillating wall on top to fulfill a periodic mixed-convection inside the cavity. In this type of problems both Grashof and Reynolds numbers play a great role in flow pattern and heat transfer characteristics, so we focus our study on four major parameters that can be crucial such as Grashof and Reynoldsnumbers, solid volume fraction and the non-dimensional lid frequency . The obtained results show that the augmentation of Reynolds number and Grashof number would enhance the average Nusselt number. It is also found that unlike steady state condition, at high Reynolds numbers, as lid is moving in the negative direction the average Nusselt number on the hot wall becomes higher in respect to the case that lid is moving in the positive direction due to thermal boundary layer disturbance. Lid frequency does not have a significant effect on thermal characteristics at low Reynolds numbers, meanwhile at higher Reynolds numbers, increment of lid frequency results in heat transfer reduction. Moreover, solid volume fraction is found to have better efficiency at higher Grashof numbers.

Cite this paper

Jafari, A. , Rahimian, M. and Saeedmanesh, A. (2013) An Unsteady Mixed Convection in a Driven Cavity Filled with Nanofluids Using an Externally Oscillating Lid.*Journal of Electronics Cooling and Thermal Control*, **3**, 58-73. doi: 10.4236/jectc.2013.32008.

Jafari, A. , Rahimian, M. and Saeedmanesh, A. (2013) An Unsteady Mixed Convection in a Driven Cavity Filled with Nanofluids Using an Externally Oscillating Lid.

References

[1] [1] K. Torrance, R. Davis, K. Eike, D. Gill, D. Gutman, A. Hsui, S. Lyons and H. Zien, “Cavity Flows Driven by Buoyancy and Shear,” Journal of Fluid Mechanic, Vol. 51, No. 2, 1972, pp. 221-231. doi:10.1017/S0022112072001181

[2] M. K. Moallemi and K. S. Jang, “Prandtl Number Effects on Laminar Mixed Convection Heat Transfer in a Lid-Driven Cavity,” International Journal of Heat and Mass Transfer, Vol. 35, No. 8, 1992, pp. 1881-1892. doi:10.1016/0017-9310(92)90191-T

[3] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Mixed Convection in a Driven Cavity Witha Stable Vertical Temperature Gradient,” International Journal of Heat and Mass Transfer, Vol. 36, No. 6, 1993, pp. 1601-1608. doi:10.1016/S0017-9310(05)80069-9

[4] R. Iwatsu and J. M. Hyun, “Three-Dimensional Driven-Cavity Flows with a Vertical Temperature Gradient,” International Journal of Heat and Mass Transfer, Vol. 38, No. 18, 1995, pp. 3319-3328. doi:10.1016/0017-9310(95)00080-S

[5] L. Martinez-Suastegui, C. Trevino and F. Mendez, “Natural Convection in a Vertical Strip Immersed in a Porous Medium,” European Journal of Mechanics B/Fluids, Vol. 22, No. 6, 2003, pp. 545-553. doi:10.1016/j.euromechflu.2003.08.003

[6] Q. G. Xiong, B. Li, F. G. Chen, J. S. Ma, W. Ge and J. H. Li, “Direct Numerical Simulation of Sub-Grid Structures in Gas-Solid Flow-GPU Implementation of Macro-Scale Peudo-Particle Modeling,” Chemical Engineering Science, Vol. 65, No. 19, 2010, pp. 5356-5365. doi:10.1016/j.ces.2010.06.035

[7] Q. G. Xiong, B. Li and J. Xu, “GPU-Accelerated Adaptive Particle Splitting and Merging in SPH,” Computer Physics Communications, Vol. 184, No. 7, 2013, pp. 1701-1707. doi:10.1016/j.cpc.2013.02.021

[8] Q. G. Xiong, B. Li, G. F. Zhou, X. J. Fang, J. Xu, J. W. Wang, X. F. He, X. W. Wang, L. M. Wang, W. Ge and J. H. Li, “Large-Scale DNS of Gas-Solid Flows on Mole-8.5,” Chemical Engineering Science, Vol. 71, 2012, pp. 422-430. doi:10.1016/j.ces.2011.10.059

[9] H. Shokouhmand, S. M. A. Noori Rahim Abadi and A. Jafari, “The Effect of the Horizontal Vibrations on Natural Heat Transfer from an Isothermal Array of Cylinders,” International Journal of Mechanics and Materials in Design, Vol. 7, No. 4, 2011, pp. 313-326.

[10] H. Shokouhmand, S. M. A. Noori Rahim Abadi and A. Jafari, “Finite Element Analysis of Heat Transfer within a Square Cavity with Uniform and Nonuniform Boundary Heating,” Heat Transfer Research, Vol. 42, No. 4, 2011, pp. 337-358.

[11] F. J. K. Ideriah, “Prediction of Turbulent Cavity Flow Driven by Buoyancy and Shear,” Journal of Mechanical Engineering Science, Vol. 22, No. 6, 1980, pp. 287-295. doi:10.1243/JMES_JOUR_1980_022_054_02

[12] C. K. Cha and Y. Jaluria, “Recirculating Mixed Convection Flow for Energy Extraction,” International Journal of Heat and Mass Transfer, Vol. 27, No. 10, 1984, pp. 1801-1810. doi:10.1016/0017-9310(84)90162-5

[13] G. De Vahl Davis, “Natural Convection in Air in a Square Cavity: A Bench Mark Numerical Solution,” International Journal for Numerical Methods in Fluids, Vol. 3, No. 3, 1983, pp. 249-264. doi:10.1002/fld.1650030305

[14] T. Fusegi, J. M. Hyun and K. Kuwahara, “Numerical Study of Natural Convection in a Differentially Heated Cavity with Internal Heat Generation: Effects of the Aspect Ratio,” Journal of Heat Transfer, Vol. 114, No. 3, 1992, pp. 773-778. doi:10.1115/1.2911350

[15] H. F. Oztop, E. Abu-Nada, Y. Varol and A. Chamkha, “Natural Convection in Wavy Enclosures with Volumetric Heat Sources,” International Journal of Thermal Sciences, Vol. 50, No. 4, 2010, pp. 502-514.

[16] M. K. Moallemi and K. S. Jang, “Prandtl Number Effects on Laminar Mixed Convection Heat Transfer in a Lid-Driven Cavity,” International Journal of Heat and Mass Transfer, Vol. 35, No. 8, 1992, pp. 1881-1892. doi:10.1016/0017-9310(92)90191-T

[17] M. A. R. Sharif, “Laminar Mixed Convection in Shallow Inclined Driven Cavities with Hot Moving Lid on Top and Cooled From Bottom,” Applied Thermal Engineering, Vol. 27, No. 5-6, 2007, pp. 1036-1042. doi:10.1016/j.applthermaleng.2006.07.035

[18] H. F. Oztop and I. Dagtekin, “Mixed Convection in Two-Sided Lid-Driven Differentially Heated Square Cavity,” International Journal of Heat and Mass Transfer, Vol. 47, No. 8-9, 2004, pp. 1761-1769. doi:10.1016/j.ijheatmasstransfer.2003.10.016

[19] Y. Varol, H. F. Oztop and I. Pop, “Natural Convection in a Diagonally Divided Square Cavity Filled with a Porous Medium,” International Journal of Thermal Sciences, Vol. 48, No. 7, 2009, pp. 1405-1415. doi:10.1016/j.ijthermalsci.2008.12.015

[20] S. Sivasankaran, V. Sivakumar and P. Prakash, “Numerical Study on Mixed Convection in a Lid-Driven Cavity with Non-Uniformheating on Both Sidewalls,” International Journal of Heat and Mass Transfer, Vol. 53, No. 19-20, 2010, pp. 4304-4315. doi:10.1016/j.ijheatmasstransfer.2010.05.059

[21] T. Nishimura and K. Kunitsugu, “Fluid Mixing and Mass Transfer in Two-Dimensional Cavities with Time-Periodic Lid Velocity,” International Journal of Heat and Fluid Flow, Vol. 18, No. 5, 1997, pp. 497-506.

[22] W. H. Soh and J. W. Goodrich, “Unsteady Solution of Incompressible Navier-Stokes Equations,” Journal of Computational Physics, Vol. 79, No. 1, 1988, pp. 113-134. doi:10.1016/0021-9991(88)90007-1

[23] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Numerical Simulation of Flows Driven by a Torsionally Oscillating Lid in a Square Cavity,” Journal of Fluids Engineering, Vol. 114, No. 2, 1992, pp. 143-151. doi:10.1115/1.2910008

[24] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Convection in a Differentially-Heated Square Cavity with Atorsionally-Oscillating Lid,” International Journal of Heat and Mass Transfer, Vol. 35, No. 5, 1992, pp. 1069-1076. doi:10.1016/0017-9310(92)90167-Q

[25] K. M. Khanafer, A. M. Al-Amiri and I. Pop, “Numerical Simulation of Unsteady Mixed Convection in a Driven Cavity Using an Externally Excited Sliding Lid,” European Journal of Mechanics B/Fluids, Vol. 26, No. 5, 2007, pp. 669-687. doi:10.1016/j.euromechflu.2006.06.006

[26] H. T. Zhu, Y. S. Lin and Y. S. Yin, “A Novel One-Step Chemical Method for Preparation of Copper Nanofluids,” Journal of Colloid and Interface Science, Vol. 277, No. 1, 2004, pp. 100-103. doi:10.1016/j.jcis.2004.04.026

[27] S. M. Aminossadati and B. Ghasemi, “Natural Convection Cooling of a Localized Heat Source at the Bottom of a Nanofluid-Filled Enclosure,” European Journal of Mechanics B/Fluids, Vol. 28, No. 5, 2009, pp. 630-640. doi:10.1016/j.euromechflu.2009.05.006

[28] X. Wang, X. Xu and S. U. S. Choi, “Thermal Conductivity of Nanoparticle-Fluid Mixture,” Journal of Thermophysics and Heat Transfer, Vol. 13, No. 4, 1999, pp. 474-480. doi:10.2514/2.6486

[29] K. Khanafer, K. Vafai and M. Lightstone, “Buoyancy-Driven Heat Transfer Enhance Mentin a Two-Dimensional Enclosure Utilizing Nanofluids,” International Journal of Heat and Mass Transfer, Vol. 46, No. 19, 2003, pp. 3639-3653. doi:10.1016/S0017-9310(03)00156-X

[30] A. H. Mahmoudi, M. Shahi, A. M. Shahedin and N. Hemati, “Numerical Modeling of Natural Convection in an Open Cavity with Two Vertical Thin Heat Sources Subjected to a Nanofluid,” International Communications in Heat and Mass Transfer, Vol. 38, No. 1, 2010, pp. 110-118. doi:10.1016/j.icheatmasstransfer.2010.09.009

[31] H. C. Brinkman, “The Viscosity of Concentrated Suspensions and Solutions,” The Journal of Chemical Physics, Vol. 20, No. 4, 1952, pp. 571-581.

[32] H. E. Patel, T. Pradeep, T. Sundararajan, A. Dasgupta, N. Dasgupta and S. K. Das, “A Micro Convection Model for Thermal Conductivity of Nanofluid, Pramana,” Journal of Physics, Vol. 65, No. 5, 2005, pp. 863-869.

[33] H. Kumar, D. Patel, E. Hrishikesh, V. R. Rajeev Kumar, T. Sundararajan, T. Pradeep and S. K. Das, “Model for Heat Conduction in Nanofluids,” Physical Review Letters, Vol. 93, No. 14, 2004, pp. 144301-1-4

[34] P. Keblinski, S. R. Phillpot, S. U. S. Choi and J. A. Eastman, “Mechanisms of Heat Flow in Suspensions of Nano-Sized Particles (Nanofluids),” International Journal of Heat and Mass Transfer, Vol. 45, No. 4, 2002, pp. 855-863. doi:10.1016/S0017-9310(01)00175-2

[35] N. Massarotti, P. Nithiarasu and O. C. Zienkiewicz, “Characteristic Based Split (CBS) Algorithmfor Incompressible Flow Problems with Heat Transfer,” International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 8, No. 8, 1998, pp. 969-990. doi:10.1108/09615539810244067

[36] F. Talebi, A. H. Mahmoudi and M. Shahi, “Numerical Study of Mixed Convection Flows in a Square Lid-Driven Cavity Utilizing Nanofluid,” International Communications in Heat and Mass Transfer, Vol. 37, No. 1, 2010, pp. 79-90. doi:10.1016/j.icheatmasstransfer.2009.08.013

[1] [1] K. Torrance, R. Davis, K. Eike, D. Gill, D. Gutman, A. Hsui, S. Lyons and H. Zien, “Cavity Flows Driven by Buoyancy and Shear,” Journal of Fluid Mechanic, Vol. 51, No. 2, 1972, pp. 221-231. doi:10.1017/S0022112072001181

[2] M. K. Moallemi and K. S. Jang, “Prandtl Number Effects on Laminar Mixed Convection Heat Transfer in a Lid-Driven Cavity,” International Journal of Heat and Mass Transfer, Vol. 35, No. 8, 1992, pp. 1881-1892. doi:10.1016/0017-9310(92)90191-T

[3] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Mixed Convection in a Driven Cavity Witha Stable Vertical Temperature Gradient,” International Journal of Heat and Mass Transfer, Vol. 36, No. 6, 1993, pp. 1601-1608. doi:10.1016/S0017-9310(05)80069-9

[4] R. Iwatsu and J. M. Hyun, “Three-Dimensional Driven-Cavity Flows with a Vertical Temperature Gradient,” International Journal of Heat and Mass Transfer, Vol. 38, No. 18, 1995, pp. 3319-3328. doi:10.1016/0017-9310(95)00080-S

[5] L. Martinez-Suastegui, C. Trevino and F. Mendez, “Natural Convection in a Vertical Strip Immersed in a Porous Medium,” European Journal of Mechanics B/Fluids, Vol. 22, No. 6, 2003, pp. 545-553. doi:10.1016/j.euromechflu.2003.08.003

[6] Q. G. Xiong, B. Li, F. G. Chen, J. S. Ma, W. Ge and J. H. Li, “Direct Numerical Simulation of Sub-Grid Structures in Gas-Solid Flow-GPU Implementation of Macro-Scale Peudo-Particle Modeling,” Chemical Engineering Science, Vol. 65, No. 19, 2010, pp. 5356-5365. doi:10.1016/j.ces.2010.06.035

[7] Q. G. Xiong, B. Li and J. Xu, “GPU-Accelerated Adaptive Particle Splitting and Merging in SPH,” Computer Physics Communications, Vol. 184, No. 7, 2013, pp. 1701-1707. doi:10.1016/j.cpc.2013.02.021

[8] Q. G. Xiong, B. Li, G. F. Zhou, X. J. Fang, J. Xu, J. W. Wang, X. F. He, X. W. Wang, L. M. Wang, W. Ge and J. H. Li, “Large-Scale DNS of Gas-Solid Flows on Mole-8.5,” Chemical Engineering Science, Vol. 71, 2012, pp. 422-430. doi:10.1016/j.ces.2011.10.059

[9] H. Shokouhmand, S. M. A. Noori Rahim Abadi and A. Jafari, “The Effect of the Horizontal Vibrations on Natural Heat Transfer from an Isothermal Array of Cylinders,” International Journal of Mechanics and Materials in Design, Vol. 7, No. 4, 2011, pp. 313-326.

[10] H. Shokouhmand, S. M. A. Noori Rahim Abadi and A. Jafari, “Finite Element Analysis of Heat Transfer within a Square Cavity with Uniform and Nonuniform Boundary Heating,” Heat Transfer Research, Vol. 42, No. 4, 2011, pp. 337-358.

[11] F. J. K. Ideriah, “Prediction of Turbulent Cavity Flow Driven by Buoyancy and Shear,” Journal of Mechanical Engineering Science, Vol. 22, No. 6, 1980, pp. 287-295. doi:10.1243/JMES_JOUR_1980_022_054_02

[12] C. K. Cha and Y. Jaluria, “Recirculating Mixed Convection Flow for Energy Extraction,” International Journal of Heat and Mass Transfer, Vol. 27, No. 10, 1984, pp. 1801-1810. doi:10.1016/0017-9310(84)90162-5

[13] G. De Vahl Davis, “Natural Convection in Air in a Square Cavity: A Bench Mark Numerical Solution,” International Journal for Numerical Methods in Fluids, Vol. 3, No. 3, 1983, pp. 249-264. doi:10.1002/fld.1650030305

[14] T. Fusegi, J. M. Hyun and K. Kuwahara, “Numerical Study of Natural Convection in a Differentially Heated Cavity with Internal Heat Generation: Effects of the Aspect Ratio,” Journal of Heat Transfer, Vol. 114, No. 3, 1992, pp. 773-778. doi:10.1115/1.2911350

[15] H. F. Oztop, E. Abu-Nada, Y. Varol and A. Chamkha, “Natural Convection in Wavy Enclosures with Volumetric Heat Sources,” International Journal of Thermal Sciences, Vol. 50, No. 4, 2010, pp. 502-514.

[16] M. K. Moallemi and K. S. Jang, “Prandtl Number Effects on Laminar Mixed Convection Heat Transfer in a Lid-Driven Cavity,” International Journal of Heat and Mass Transfer, Vol. 35, No. 8, 1992, pp. 1881-1892. doi:10.1016/0017-9310(92)90191-T

[17] M. A. R. Sharif, “Laminar Mixed Convection in Shallow Inclined Driven Cavities with Hot Moving Lid on Top and Cooled From Bottom,” Applied Thermal Engineering, Vol. 27, No. 5-6, 2007, pp. 1036-1042. doi:10.1016/j.applthermaleng.2006.07.035

[18] H. F. Oztop and I. Dagtekin, “Mixed Convection in Two-Sided Lid-Driven Differentially Heated Square Cavity,” International Journal of Heat and Mass Transfer, Vol. 47, No. 8-9, 2004, pp. 1761-1769. doi:10.1016/j.ijheatmasstransfer.2003.10.016

[19] Y. Varol, H. F. Oztop and I. Pop, “Natural Convection in a Diagonally Divided Square Cavity Filled with a Porous Medium,” International Journal of Thermal Sciences, Vol. 48, No. 7, 2009, pp. 1405-1415. doi:10.1016/j.ijthermalsci.2008.12.015

[20] S. Sivasankaran, V. Sivakumar and P. Prakash, “Numerical Study on Mixed Convection in a Lid-Driven Cavity with Non-Uniformheating on Both Sidewalls,” International Journal of Heat and Mass Transfer, Vol. 53, No. 19-20, 2010, pp. 4304-4315. doi:10.1016/j.ijheatmasstransfer.2010.05.059

[21] T. Nishimura and K. Kunitsugu, “Fluid Mixing and Mass Transfer in Two-Dimensional Cavities with Time-Periodic Lid Velocity,” International Journal of Heat and Fluid Flow, Vol. 18, No. 5, 1997, pp. 497-506.

[22] W. H. Soh and J. W. Goodrich, “Unsteady Solution of Incompressible Navier-Stokes Equations,” Journal of Computational Physics, Vol. 79, No. 1, 1988, pp. 113-134. doi:10.1016/0021-9991(88)90007-1

[23] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Numerical Simulation of Flows Driven by a Torsionally Oscillating Lid in a Square Cavity,” Journal of Fluids Engineering, Vol. 114, No. 2, 1992, pp. 143-151. doi:10.1115/1.2910008

[24] R. Iwatsu, J. M. Hyun and K. Kuwahara, “Convection in a Differentially-Heated Square Cavity with Atorsionally-Oscillating Lid,” International Journal of Heat and Mass Transfer, Vol. 35, No. 5, 1992, pp. 1069-1076. doi:10.1016/0017-9310(92)90167-Q

[25] K. M. Khanafer, A. M. Al-Amiri and I. Pop, “Numerical Simulation of Unsteady Mixed Convection in a Driven Cavity Using an Externally Excited Sliding Lid,” European Journal of Mechanics B/Fluids, Vol. 26, No. 5, 2007, pp. 669-687. doi:10.1016/j.euromechflu.2006.06.006

[26] H. T. Zhu, Y. S. Lin and Y. S. Yin, “A Novel One-Step Chemical Method for Preparation of Copper Nanofluids,” Journal of Colloid and Interface Science, Vol. 277, No. 1, 2004, pp. 100-103. doi:10.1016/j.jcis.2004.04.026

[27] S. M. Aminossadati and B. Ghasemi, “Natural Convection Cooling of a Localized Heat Source at the Bottom of a Nanofluid-Filled Enclosure,” European Journal of Mechanics B/Fluids, Vol. 28, No. 5, 2009, pp. 630-640. doi:10.1016/j.euromechflu.2009.05.006

[28] X. Wang, X. Xu and S. U. S. Choi, “Thermal Conductivity of Nanoparticle-Fluid Mixture,” Journal of Thermophysics and Heat Transfer, Vol. 13, No. 4, 1999, pp. 474-480. doi:10.2514/2.6486

[29] K. Khanafer, K. Vafai and M. Lightstone, “Buoyancy-Driven Heat Transfer Enhance Mentin a Two-Dimensional Enclosure Utilizing Nanofluids,” International Journal of Heat and Mass Transfer, Vol. 46, No. 19, 2003, pp. 3639-3653. doi:10.1016/S0017-9310(03)00156-X

[30] A. H. Mahmoudi, M. Shahi, A. M. Shahedin and N. Hemati, “Numerical Modeling of Natural Convection in an Open Cavity with Two Vertical Thin Heat Sources Subjected to a Nanofluid,” International Communications in Heat and Mass Transfer, Vol. 38, No. 1, 2010, pp. 110-118. doi:10.1016/j.icheatmasstransfer.2010.09.009

[31] H. C. Brinkman, “The Viscosity of Concentrated Suspensions and Solutions,” The Journal of Chemical Physics, Vol. 20, No. 4, 1952, pp. 571-581.

[32] H. E. Patel, T. Pradeep, T. Sundararajan, A. Dasgupta, N. Dasgupta and S. K. Das, “A Micro Convection Model for Thermal Conductivity of Nanofluid, Pramana,” Journal of Physics, Vol. 65, No. 5, 2005, pp. 863-869.

[33] H. Kumar, D. Patel, E. Hrishikesh, V. R. Rajeev Kumar, T. Sundararajan, T. Pradeep and S. K. Das, “Model for Heat Conduction in Nanofluids,” Physical Review Letters, Vol. 93, No. 14, 2004, pp. 144301-1-4

[34] P. Keblinski, S. R. Phillpot, S. U. S. Choi and J. A. Eastman, “Mechanisms of Heat Flow in Suspensions of Nano-Sized Particles (Nanofluids),” International Journal of Heat and Mass Transfer, Vol. 45, No. 4, 2002, pp. 855-863. doi:10.1016/S0017-9310(01)00175-2

[35] N. Massarotti, P. Nithiarasu and O. C. Zienkiewicz, “Characteristic Based Split (CBS) Algorithmfor Incompressible Flow Problems with Heat Transfer,” International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 8, No. 8, 1998, pp. 969-990. doi:10.1108/09615539810244067

[36] F. Talebi, A. H. Mahmoudi and M. Shahi, “Numerical Study of Mixed Convection Flows in a Square Lid-Driven Cavity Utilizing Nanofluid,” International Communications in Heat and Mass Transfer, Vol. 37, No. 1, 2010, pp. 79-90. doi:10.1016/j.icheatmasstransfer.2009.08.013