Self-Similar Solution of Heat and Mass Transfer of Unsteady Mixed Convection Flow on a Rotating Cone Embedded in a Porous Medium Saturated with a Rotating Fluid

Author(s)
Saleh M. Al-Harbi

ABSTRACT

A self-similar solution of unsteady mixed convection flow on a rotating cone embedded in a porous medium saturated with a rotating fluid in the presence of the first and second orders resistances has been obtained. It has been shown that a self-similar solution is possible when the free stream angular velocity and the angular velocity of the cone vary inversely as a linear function of time. The system of ordinary differential equations governing the flow has been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. Both prescribe wall temperature and prescribed heat flux conditions are considered. Numerical results are reported for the skin friction coefficients, Nusselt number and Sherwood number. The effect of various parameters on the velocity, temperature and concentration profiles are also presented here.

A self-similar solution of unsteady mixed convection flow on a rotating cone embedded in a porous medium saturated with a rotating fluid in the presence of the first and second orders resistances has been obtained. It has been shown that a self-similar solution is possible when the free stream angular velocity and the angular velocity of the cone vary inversely as a linear function of time. The system of ordinary differential equations governing the flow has been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. Both prescribe wall temperature and prescribed heat flux conditions are considered. Numerical results are reported for the skin friction coefficients, Nusselt number and Sherwood number. The effect of various parameters on the velocity, temperature and concentration profiles are also presented here.

KEYWORDS

Unsteady Mixed Convection, Heat and Mass Transfer, Rotating Cone, Rotating Fluid, Porous Media, Self-Similar Solution

Unsteady Mixed Convection, Heat and Mass Transfer, Rotating Cone, Rotating Fluid, Porous Media, Self-Similar Solution

Cite this paper

nullS. Al-Harbi, "Self-Similar Solution of Heat and Mass Transfer of Unsteady Mixed Convection Flow on a Rotating Cone Embedded in a Porous Medium Saturated with a Rotating Fluid,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1196-1203. doi: 10.4236/am.2011.210166.

nullS. Al-Harbi, "Self-Similar Solution of Heat and Mass Transfer of Unsteady Mixed Convection Flow on a Rotating Cone Embedded in a Porous Medium Saturated with a Rotating Fluid,"

References

[1] J. P. Hartnett and E. C. Deland, “The Influence of Prandtl Number on the Heat Transfer from Rotating Non-Isothermal Disks and Cones,” ASME Journal of Heat Transfer, Vol. 83, 1961, pp. 95-96.

[2] R. G. Hering and R. J. Grosh, “Laminar Free Convection from a Non-Isothermal Cone,” International Journal of Heat and Mass Transfer, Vol. 5, No. 11, 1962, pp. 1059-1068. doi:10.1016/0017-9310(62)90059-5

[3] C. L. Tien and I. J. Tsuji, “A Theoretical Analysis of Laminar Forced Flow and Heat Transfer about a Rotating Cone,” ASME Journal of Heat Transfer, Vol. 87, 1965, pp. 184-190.

[4] R. G. Hering and R. J. Grosh, “Laminar Free Convection from a Non-Isothermal Cone at Low Prandtl Number,” International Journal of Heat and Mass Transfer, Vol. 8, No. 10, 1965. pp. 1333-1337. doi:10.1016/0017-9310(65)90059-1

[5] K. Himasekhar, P. K. Sarma and K. Janardhan, “Laminar Mixed Convec-tion from Vertical Rotating Cone,” International Com-munications in Heat and Mass Transfer, Vol. 16, No. 1, 1989, pp. 99-106. doi:10.1016/0735-1933(89)90045-6

[6] C. Y. Wang, “Boundary Layers on Rotating Cones, Discs and Axi-symmetric Surfaces with a Concentrated Heat Source,” Acta Mechanica, Vol. 81, No. 3-4, 1990, pp. 245-251. doi:10.1007/BF01176992

[7] K. A. Yih, “Mixed Con-vection about a Cone in a Porous Medium: The Entire Regime,” International Communications in Heat and Mass Transfer, Vol. 26, No. 7, 1999, pp. 1041-1050. doi:10.1016/S0735-1933(99)00093-7

[8] S. M. Al-Harbi, “Numerical Study of Natural Convection Heat Transfer with Variable Viscosity and Thermal Radiation from a Cone and Wedge in Porous Media,” Applied Ma-thematics and Computation, Vol. 170, 2005, pp. 64-75. doi:10.1016/j.amc.2004.10.093

[9] H. S. Takhar, A. Chamkha and G. Nath, “Unsteady Mixed Convention Flow from a Rotating Vertical Cone with a Magnetic Field,” Heat and Mass Transfer, Vol. 39, No. 4, 2003, pp. 297-304.

[10] D. Anilkumar and S. Roy, “Unsteady Mixed Convection Flow on a Rotating Cone in a Rotating Fluid,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 545-561. doi:10.1016/S0096-3003(03)00799-9

[11] I. Pop and D. B. Ingham, “Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media,” Pergamon, Oxford, 2001.

[12] D. A. Nield and A. Bejan, “Convection in Porous Media,” Springer-Verlag, New York, 2005.

[13] I. A. Hassanien, F. S. Ibrahim and Gh. M. Omer, “Unsteady Free Convection Flow in the Stagnation-Point Region of a Rotating Sphere Embedded in a Porous Medium,” Mechanical Engineering, Vol. 7, 2004, pp. 89-98.

[14] I. A. Hassanien, F. S. Ibrahim and Gh. M. Omer, “Unsteady Flow and Heat Transfer of a Viscous Fluid in the Stagnation Region of a Three-Dimensional Body Embedded in a Porous Me-dium,” Journal of Porous Media, Vol. 9. No. 4, 2006, pp. 357-372. doi:10.1615/JPorMedia.v9.i4.60

[15] F. S. Ibrahim, A. M. Elaiw and A. A. Bakr, “Effect of Chemical Reaction and Radiation Absorption on the Un-steady MHD Free Convection Flow Past a Semi Infinite Vertical Permeable Moving Plate with Heat Source and Suction,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 6, 2008, pp. 1056- 1066. doi:10.1016/j.cnsns.2006.09.007

[16] S. Roy, P. Datta and N. C. Mahanti, “Non-Similar Solution of an Unsteady Mixed Convection Flow over a Vertical Cone with Suction or Injection,” International Journal of Heat and Mass Transfer, Vol. 50, 2007, pp. 181- 187. doi:10.1016/j.ijheatmasstransfer.2006.06.024

[17] E. Osalusi, J. Side, R. Harris and P. Clark, “The Effect of Combined Viscous Dissipation and Joule Heating on Un-steady Mixed Convection MHD Flow on a Rotating Cone in a Rotating Fluid with Variable Properties in the Pres-ence of Hall and Ion-Slip Currents,” International Com-munications in Heat and Mass Transfer, Vol. 35, No. 4, 2008, pp. 413-429. doi:10.1016/j.icheatmasstransfer.2007.09.002

[18] K. Inouye and A. Tate, “Finite Difference Version Quasili-nearization Applied to Boundary Layer Equations,” AIAA Journal, Vol. 12, No. 4, 1974, pp. 558-560. doi:10.2514/3.49286

[1] J. P. Hartnett and E. C. Deland, “The Influence of Prandtl Number on the Heat Transfer from Rotating Non-Isothermal Disks and Cones,” ASME Journal of Heat Transfer, Vol. 83, 1961, pp. 95-96.

[2] R. G. Hering and R. J. Grosh, “Laminar Free Convection from a Non-Isothermal Cone,” International Journal of Heat and Mass Transfer, Vol. 5, No. 11, 1962, pp. 1059-1068. doi:10.1016/0017-9310(62)90059-5

[3] C. L. Tien and I. J. Tsuji, “A Theoretical Analysis of Laminar Forced Flow and Heat Transfer about a Rotating Cone,” ASME Journal of Heat Transfer, Vol. 87, 1965, pp. 184-190.

[4] R. G. Hering and R. J. Grosh, “Laminar Free Convection from a Non-Isothermal Cone at Low Prandtl Number,” International Journal of Heat and Mass Transfer, Vol. 8, No. 10, 1965. pp. 1333-1337. doi:10.1016/0017-9310(65)90059-1

[5] K. Himasekhar, P. K. Sarma and K. Janardhan, “Laminar Mixed Convec-tion from Vertical Rotating Cone,” International Com-munications in Heat and Mass Transfer, Vol. 16, No. 1, 1989, pp. 99-106. doi:10.1016/0735-1933(89)90045-6

[6] C. Y. Wang, “Boundary Layers on Rotating Cones, Discs and Axi-symmetric Surfaces with a Concentrated Heat Source,” Acta Mechanica, Vol. 81, No. 3-4, 1990, pp. 245-251. doi:10.1007/BF01176992

[7] K. A. Yih, “Mixed Con-vection about a Cone in a Porous Medium: The Entire Regime,” International Communications in Heat and Mass Transfer, Vol. 26, No. 7, 1999, pp. 1041-1050. doi:10.1016/S0735-1933(99)00093-7

[8] S. M. Al-Harbi, “Numerical Study of Natural Convection Heat Transfer with Variable Viscosity and Thermal Radiation from a Cone and Wedge in Porous Media,” Applied Ma-thematics and Computation, Vol. 170, 2005, pp. 64-75. doi:10.1016/j.amc.2004.10.093

[9] H. S. Takhar, A. Chamkha and G. Nath, “Unsteady Mixed Convention Flow from a Rotating Vertical Cone with a Magnetic Field,” Heat and Mass Transfer, Vol. 39, No. 4, 2003, pp. 297-304.

[10] D. Anilkumar and S. Roy, “Unsteady Mixed Convection Flow on a Rotating Cone in a Rotating Fluid,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 545-561. doi:10.1016/S0096-3003(03)00799-9

[11] I. Pop and D. B. Ingham, “Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media,” Pergamon, Oxford, 2001.

[12] D. A. Nield and A. Bejan, “Convection in Porous Media,” Springer-Verlag, New York, 2005.

[13] I. A. Hassanien, F. S. Ibrahim and Gh. M. Omer, “Unsteady Free Convection Flow in the Stagnation-Point Region of a Rotating Sphere Embedded in a Porous Medium,” Mechanical Engineering, Vol. 7, 2004, pp. 89-98.

[14] I. A. Hassanien, F. S. Ibrahim and Gh. M. Omer, “Unsteady Flow and Heat Transfer of a Viscous Fluid in the Stagnation Region of a Three-Dimensional Body Embedded in a Porous Me-dium,” Journal of Porous Media, Vol. 9. No. 4, 2006, pp. 357-372. doi:10.1615/JPorMedia.v9.i4.60

[15] F. S. Ibrahim, A. M. Elaiw and A. A. Bakr, “Effect of Chemical Reaction and Radiation Absorption on the Un-steady MHD Free Convection Flow Past a Semi Infinite Vertical Permeable Moving Plate with Heat Source and Suction,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 6, 2008, pp. 1056- 1066. doi:10.1016/j.cnsns.2006.09.007

[16] S. Roy, P. Datta and N. C. Mahanti, “Non-Similar Solution of an Unsteady Mixed Convection Flow over a Vertical Cone with Suction or Injection,” International Journal of Heat and Mass Transfer, Vol. 50, 2007, pp. 181- 187. doi:10.1016/j.ijheatmasstransfer.2006.06.024

[17] E. Osalusi, J. Side, R. Harris and P. Clark, “The Effect of Combined Viscous Dissipation and Joule Heating on Un-steady Mixed Convection MHD Flow on a Rotating Cone in a Rotating Fluid with Variable Properties in the Pres-ence of Hall and Ion-Slip Currents,” International Com-munications in Heat and Mass Transfer, Vol. 35, No. 4, 2008, pp. 413-429. doi:10.1016/j.icheatmasstransfer.2007.09.002

[18] K. Inouye and A. Tate, “Finite Difference Version Quasili-nearization Applied to Boundary Layer Equations,” AIAA Journal, Vol. 12, No. 4, 1974, pp. 558-560. doi:10.2514/3.49286