Numerical Solution for a Similar Flow between Two Disks in the Presence of a Magnetic Field

Sajjad Hussain, Muhammad Anwar Kamal, Farooq Ahmad, Muhammad Ali, Muhammad Shafique, Sifat Hussain

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.
Department of Mathematics, Gomal University, D. I. Khan, Pakistan.
Department of Mathematics, King Suleman Bin Abdulaziz University, Al-Kharj, Saudi Arabia.
Department of Mathematics, Minhaj University, Lahore, Pakistan.

DOI: 10.4236/am.2013.48155                

Abstract

Numerical solutions are obtained for non-steady, incompressible fluid flow between two parallel disks which at time t are separated by a distance H(1－αt)^1/2 and a magnetic field proportional to B₀(1－αt) ^-1/2 is applied perpendicular to the disks where H denotes a representative length, B_O denotes a representative magnetic field and α^-1 denotes a representative time. Similarity transformations are used to convert the governing partial differential equations of motion in to ordinary differential form. The resulting ordinary differential equations are solved numerically using SOR method, Richardson extrapolation and Simpson’s (1/3) Rule. Our numerical scheme is straightforward, efficient and easy to program.

Keywords

Numerical Solution; Similar Flow; Newtonian Fluid; Magnetic Field; Reynold’s Number

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	S. Ishizawa, “The Unsteady Laminar Flow between Two Parallel Disks with Arbitrarily Varying Gap Width,” Bul letin of JSME, Vol. 9, No. 35, 1966, pp. 533-550. doi:10.1299/jsme1958.9.533
[3]	J. A. Tichy and P. Bourgin, “A Similarity Solution for Flow in a Narrow Channel of Varying Gap,” ASME Jour nal of Applied Mechanics, Vol. 53, No. 4, 1986, pp. 943-946. doi:10.1115/1.3171885
[4]	K. S. Bhupendera, K. J. Abhay and R. C. Chaudhary, “MHD Forced Flow of a Conducting Viscous Fluid through a Porous Medium Induced by an Impervious Ro tating Disk Ram,” Journal of Physics, Vol. 52, No. 1, 2007, pp. 73-84.
[5]	K. B. Pavlov, “Magnetohydrodynamic Flow of an In compressible, Viscous Fluid Caused by the Deformation of a Plane Surface,” Magnitnaya Gidrodinamika, Vol. 4, 1974, pp. 146-147.
[6]	M. Guria, B. K. Das, R. N. Jana and C. E. Imark, “Hy deromagnetic Flow between Two Porous Disks Rotating about Non Coincident Axes,” Acta Mechanica Sinica, Vol. 24, No. 5, 2008, pp. 489-496. doi:10.1007/s10409-008-0158-x
[7]	H. A. Attia, “On the Effectiveness of the Ion Slip on the Steady Flow of a Conducting Fluid Due to a Porous Ro tating Disk with Heat Transfer,” Tamkang Journal of Science and Engineering, Vol. 9, No. 3, 2006, pp. 185-213.
[8]	M. Sajid, T. Javed and T. Hayat, “MHD Rotating Flow of a Viscous Fluid over a Shrinking Surface,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 259-265. doi:10.1007/s11071-007-9208-3
[9]	B. El-Asir, A. Mansoor, S. Bataineh and R. Arar, “A New Hybrid Analytical Analysis of the Magnetohydrodynamic Flow over a Rotating Disk under Uniform Suction,” Jour nal of Applied Sciences, Vol. 6, No. 5, 2006, pp. 1059-1065. doi:10.3923/jas.2006.1059.1065
[10]	R. Usha and S. Vasudevan, “A Similar Flow between Two Rotating Disks in the Presence of a Magnetic Field,” Journal of Applied Mechanics, Vol. 60, No. 3, 1993, pp. 707-714. doi:10.1115/1.2900862
[11]	G. D. Smith, “Numerical Solution of Partial Differential Equation,” Clarendon Press, Oxford, 1979.
[12]	C. F. Gerald, “Applied Numerical Analysis,” Addison Wesley Pub., New York, 1989.
[13]	W. E. Milne, “Numerical Solution of Differential Equa tion,” Dover Pub., Dover, 1970.
[14]	R. L. Burden, “Numerical Analysis,” Prindle, Weber & Schmidt, Boston, 1985.
[15]	E. A. Hamza, “A Similar Flow between Two Disks in the Presence of a Magnetic Field,” IC/87/287, Trieste, 1987.