Acta Mechanica 129, 263-271 (1998) ACTA MECHANICA 9 Springer-Verlag 1998 Note Certain inverse solutions of the plane creeping flow of a second-order fluid A. M. Siddiqui, York, Pennsylvania (Received June ll, 1995; revised May 15, 1997) Summary. Inverse solutions for the plane steady creeping flow of a second-order fluid are obtained by assuming certain forms of vorticity and its Laplacian. Expressions for streamlines are given explicitly in each case. 1 Introduction In the case ofnon-Newtonian fluids, namely homogeneous incompressible Rivlin-Ericksen fluids of second order [1], [2], it is found that non-lineafities occur not only in the interim part but also in the viscosity part of the governing equations. As a result, the number of exact solutions becomes much smaller as compared to the exact solutions of Navier-Stokes equations. However, many flow situations of interest are such that a number of terms in the equations of motion either may be neglected or disappear automatially, and the resulting equations reduce to a form that can be readily solved. It is well known [6], [7] that the equation governing the creeping flow of a second-order fluid is an order higher than the equation governing Stokes flow, and to solve a well posed problem one needs additional boundary conditions over the usual adherence condition. However, for special classes of solutions in unbounded domains it may not be necessary to impose additional conditions. One common way of simplifying the equations of motion of second order fluid is by complete or partial omission of the inertial terms. Complete omission of the inertial terms results in the so called creeping motion. Tanner [3] showed that "any plane creeping Newtonian velocity field is also a solution for second-order fluid under identical velocity boundary conditons". Hnilgol [4] proved that under certain conditions the plane creeping Newtonian velocity field is the unique solution to the steady creeping flow of an incompressible second order fluid. These results were extended by Fosdick and Rajagopal [5] for the creeping three dimensional flows of a second order fluid. On the other hand, Rajagopal [6] presented a steady creeping flow solution to the boundary value problem for a second order fluid for which there is no corresponding Newtonian solution. Recently Kaloni [7] showed that the Newtonian solutions are not necessarily the proper solutions for the second-order fluid in either three-dimensional or plane creeping flow. Bourgin and Tichy [8] obtained a similarity solution for the plane creeping flow of a second-order fluid with non-parallel porous walls by prescribing an additional boundary condition in addition to the usual no slip conditions.