Research Papers: Fundamental Issues and Canonical Flows

Some Exact Solutions to Equations of Motion of an Incompressible Second Grade Fluid

[+] Author and Article Information
Saif Ullah

Department of Mathematics,
Government College University,
Lahore 54000, Pakistan
e-mail: dr.saifullah@gcu.edu.pk

Irsa Maqbool

Department of Mathematics,
Government College University,
Lahore 54000, Pakistan

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 27, 2013; final manuscript received June 5, 2014; published online September 10, 2014. Editor: Malcolm J. Andrews.

J. Fluids Eng 137(1), 011205 (Sep 10, 2014) (4 pages) Paper No: FE-13-1752; doi: 10.1115/1.4027826 History: Received December 27, 2013; Revised June 05, 2014

In this paper, we derive some exact solutions of the equations governing the steady plane motions of an incompressible second grade fluid. For this purpose, the vorticity and stream functions both are expressed in terms of complex variables and complex functions. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. Some physical features of the derived solutions are also illustrated by their contour plots.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Dunn, J. E., and Fosdick, R. L., 1974, “Thermodynamics, Stability and Boundedness of Fluids of Complexity 2 and Fluids of Second Grade,” Arch. Ration. Mech. Anal., 56, pp. 191–252. [CrossRef]
Kaloni, P. N., and Siddiqui, A. M., 1983, “The Flow of a Second Grade Fluid,” Int. J. Eng. Sci., 21(10), pp. 1157–1169. [CrossRef]
Rajagopal, K. R., 1984, “On the Creeping Flow of the Second-Order Fluid,” J. Non-Newtonian Fluid Mech., 15, pp. 239–246. [CrossRef]
Erdogan, M. E., 1995, “Plane Surface Suddenly Set in Motion in a Non-Newtonian Fluid,” Acta Mech., 108, pp. 179–187. [CrossRef]
Rajagopal, K. R., and Gupta, A. S., 1984, “An Exact Solution for the Flow of a Non-Newtonian Fluid Past an Infinite Porous Plate,” Mechanica, 19, pp. 158–160. [CrossRef]
Labropulu, F., 2000, “Exact Solutions of Non-Newtonian Fluid Flows With Prescribed Vorticity,” Acta Mech., 141, pp. 11–20. [CrossRef]
Tong, D., and Liu, Y., 2005, “Exact Solutions for the Unsteady Rotational Flow of a Non-Newtonian Fluid in an Annular Pipe,” Int. J. Eng. Sci., 43, pp. 281–289. [CrossRef]
Tan, W., Xian, F., and Wei, L., 2002, “Exact Solution for the Unsteady Couette Flow of the Generalized Second Grade Fluid,” Chin. Sci. Bull., 47, pp. 1226–1228. [CrossRef]
Rivlin, R. S., and Ericksen, J. L., 1955, “Stress-Deformation Relations for Isotropic Materials,” J. Ration. Mech. Anal., 4, pp. 323–425.
Dunn, J. E., and Rajagopal, K. R., 1995, “Fluids of Differential Type: Critical Review and Thermodynamic Analysis,” Int. J. Eng. Sci., 33, pp. 689–729. [CrossRef]
Stallybrass, M. P., 1983, “A Class of Exact Solutions of the Navier–Stokes Equations. Plane Unsteady Flow,” Lett. Appl. Eng. Sci.21(2), pp. 179–186. [CrossRef]


Grahic Jump Location
Fig. 1

Contour plot of ψ when I = (a1 + ıa2)ln z

Grahic Jump Location
Fig. 2

Contour plot of ψ when I = (b1 + ıb2)z2




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In