Numerical Solution for Accelerated Rotating Disk in a Viscous Fluid

Affiliation(s)

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.

Punjab Higher Education Department, Govt. Degree College Darya Khan, Bhakkar, Pakistan.

Department of Mathematics, Gomal University, Dera Ismail Khan, Pakistan.

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.

Punjab Higher Education Department, Govt. Degree College Darya Khan, Bhakkar, Pakistan.

Department of Mathematics, Gomal University, Dera Ismail Khan, Pakistan.

ABSTRACT

The problem of a disk rotating in a viscous fluid has been investigated. The disk is accelerated with angular velocity proportional to time. Employing suitable similarity transformations the governing partial differential equations are transformed in to ordinary differential form. The resulting equations are solved numerically using SOR method and Simpson’s (1/3) rule. The results have been improved by using Richardson’s extrapolation. The effect of the non-dimensional parameter s which measures unsteadiness is observed on velocity components, skin friction coefficient and torque of the disk.

The problem of a disk rotating in a viscous fluid has been investigated. The disk is accelerated with angular velocity proportional to time. Employing suitable similarity transformations the governing partial differential equations are transformed in to ordinary differential form. The resulting equations are solved numerically using SOR method and Simpson’s (1/3) rule. The results have been improved by using Richardson’s extrapolation. The effect of the non-dimensional parameter s which measures unsteadiness is observed on velocity components, skin friction coefficient and torque of the disk.

Cite this paper

S. Hussain, F. Ahmad, M. Shafique and S. Hussain, "Numerical Solution for Accelerated Rotating Disk in a Viscous Fluid,"*Applied Mathematics*, Vol. 4 No. 6, 2013, pp. 899-902. doi: 10.4236/am.2013.46124.

S. Hussain, F. Ahmad, M. Shafique and S. Hussain, "Numerical Solution for Accelerated Rotating Disk in a Viscous Fluid,"

References

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[2] D. E. Dolidge, “Unsteady Motion of a Viscous Liquid Produced by a Rotating Disk,” Prikladnaya Matematika i Mekhanika, Vol. 18, 1954, pp. 371-378.

[3] E. M. Sparrow and J. L. Gregg, “Flow about an Unsteadily Rotating Disk,” Journal of the Aeronautical Sciences, Vol. 27, No. 4, 1960, pp. 252-257.

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[5] L. T. Watson, K. K. Sanakara and L. C. Mounfield, “Deceleration of a Porous Rotating Disk in a Viscous Fluid,” International Journal of Engineering Science, Vol. 23, 1985, pp. 131-152. doi:10.1016/0020-7225(85)90022-9

[6] E. B. Watson, W. H. H. Banks, M. B. Zaturska and P. G. Drazin, “On Transition to Chaos in Two Dimensional Channel Flow Symmetrically Driven by Accelerating Walls,” Journal of Fluid Mechanics, Vol. 212, 1990, pp. 451-485. doi:10.1017/S0022112090002051

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[9] S. Asghar, K. Hanif, T. Hayat and C. M. Khalique, “MHD non Newtonian Flow Due to Non Coaxial Rotations of an Accelerated Disk and a Fluid at Infinity,” Communication in Nonlinear Science and Numerical Simulation, Vol. 12, No. 4, 2007, pp. 465-485. doi:10.1016/j.cnsns.2005.04.006

[10] C. Y. Wang, “The Squeezing of a Fluid between Two Plates,” Journal of Applied Mechanics, Vol. 43, No. 4, 1976, pp. 579-583. doi:10.1115/1.3423935

[11] S. Ishizawa, “The Unsteady Laminar Flow between Two Parallel Disks with Arbitrarily Varying Gap Width,” Bulletin of JSME, Vol. 9, No. 35, 1966, pp. 533-550. doi:10.1299/jsme1958.9.533

[12] I. Pop, “Unsteady Flow past a Stretching Sheet,” Mechanics Research Communications, Vol. 23, No. 4, 1996, pp. 413-422. doi:10.1016/0093-6413(96)00040-7

[13] R. Nazar, N. Amin and I. Pop, “Unsteady Boundary Layer Flow Due to a Rotating Fluid,” Mechanics Research Communications, Vol. 31, 2004, pp. 121-128. doi:10.1016/j.mechrescom.2003.09.004

[14] H. Xu, S. J. Liao and I. Pop, “Series Solution of Unsteady Three Dimensional MHD Flow and Heat Transfer in Boundary Layer over an Impulsively Stretching Plate,” European Journal of Mechanics, Vol. 26, No. 1, 2007, pp. 15-27. doi:10.1016/j.euromechflu.2005.12.003

[15] L. T. Watson and C. Y. Wang, “Deceleration of a Rotating Disk in a Viscous Fluid,” Physics of Fluids, Vol. 22, No. 12, 1979, pp. 226-2269. doi:10.1063/1.862535

[16] G. D. Smith, “Numerical Solution of Partial Differential Equation,” Clarendon Press, Oxford, 1979.

[17] C. F. Gerald, “Applied Numerical Analysis,” AddisonWesley Publication, New York, 1989.

[18] W. E. Milne, “Numerical Solution of Differential Equation,” Dover Publication, New York, 1970.

[19] R. L. Burden, “Numerical Analysis,” Prindle, Weber & Schmidt, Boston, 1985.

[1] T. Von Karaman, “Uberlaminare und Turbulent Reibung,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 1, 1921, pp. 233-252. doi:10.1002/zamm.19210010401

[2] D. E. Dolidge, “Unsteady Motion of a Viscous Liquid Produced by a Rotating Disk,” Prikladnaya Matematika i Mekhanika, Vol. 18, 1954, pp. 371-378.

[3] E. M. Sparrow and J. L. Gregg, “Flow about an Unsteadily Rotating Disk,” Journal of the Aeronautical Sciences, Vol. 27, No. 4, 1960, pp. 252-257.

[4] E. R. Benton, “On the Flow Due to a Rotating Disk,” Journal of Fluid Mechanics, Vol. 24, No. 4, 1966, pp. 781-800. doi:10.1017/S0022112066001009

[5] L. T. Watson, K. K. Sanakara and L. C. Mounfield, “Deceleration of a Porous Rotating Disk in a Viscous Fluid,” International Journal of Engineering Science, Vol. 23, 1985, pp. 131-152. doi:10.1016/0020-7225(85)90022-9

[6] E. B. Watson, W. H. H. Banks, M. B. Zaturska and P. G. Drazin, “On Transition to Chaos in Two Dimensional Channel Flow Symmetrically Driven by Accelerating Walls,” Journal of Fluid Mechanics, Vol. 212, 1990, pp. 451-485. doi:10.1017/S0022112090002051

[7] P. D. Ariel, “Computation of Flow of a Second Grade Fluid near a Rotating Disk,” International Journal of Engineering Science, Vol. 23, 1997, pp. 1335-1357. doi:10.1016/S0020-7225(97)87427-7

[8] E. A. Hamza, “Unsteady Flow between Two Disks with Heat Transfer in the Presence of a Magnetic Field,” Journal of Physics D: Applied Physics, Vol. 25, No. 10, 1992, pp. 1425-1431. doi:10.1088/0022-3727/25/10/007

[9] S. Asghar, K. Hanif, T. Hayat and C. M. Khalique, “MHD non Newtonian Flow Due to Non Coaxial Rotations of an Accelerated Disk and a Fluid at Infinity,” Communication in Nonlinear Science and Numerical Simulation, Vol. 12, No. 4, 2007, pp. 465-485. doi:10.1016/j.cnsns.2005.04.006

[10] C. Y. Wang, “The Squeezing of a Fluid between Two Plates,” Journal of Applied Mechanics, Vol. 43, No. 4, 1976, pp. 579-583. doi:10.1115/1.3423935

[11] S. Ishizawa, “The Unsteady Laminar Flow between Two Parallel Disks with Arbitrarily Varying Gap Width,” Bulletin of JSME, Vol. 9, No. 35, 1966, pp. 533-550. doi:10.1299/jsme1958.9.533

[12] I. Pop, “Unsteady Flow past a Stretching Sheet,” Mechanics Research Communications, Vol. 23, No. 4, 1996, pp. 413-422. doi:10.1016/0093-6413(96)00040-7

[13] R. Nazar, N. Amin and I. Pop, “Unsteady Boundary Layer Flow Due to a Rotating Fluid,” Mechanics Research Communications, Vol. 31, 2004, pp. 121-128. doi:10.1016/j.mechrescom.2003.09.004

[14] H. Xu, S. J. Liao and I. Pop, “Series Solution of Unsteady Three Dimensional MHD Flow and Heat Transfer in Boundary Layer over an Impulsively Stretching Plate,” European Journal of Mechanics, Vol. 26, No. 1, 2007, pp. 15-27. doi:10.1016/j.euromechflu.2005.12.003

[15] L. T. Watson and C. Y. Wang, “Deceleration of a Rotating Disk in a Viscous Fluid,” Physics of Fluids, Vol. 22, No. 12, 1979, pp. 226-2269. doi:10.1063/1.862535

[16] G. D. Smith, “Numerical Solution of Partial Differential Equation,” Clarendon Press, Oxford, 1979.

[17] C. F. Gerald, “Applied Numerical Analysis,” AddisonWesley Publication, New York, 1989.

[18] W. E. Milne, “Numerical Solution of Differential Equation,” Dover Publication, New York, 1970.

[19] R. L. Burden, “Numerical Analysis,” Prindle, Weber & Schmidt, Boston, 1985.