The skin friction of flat plates to Oseen's approximation

Abstract

Recent observation that flow past tangential flat plates may remain steady up to Reynolds’ numbers so great as 5 x 10 5 has renewed interest in the problem of calculating the motion. For large motions, such as are characterized by Prandtl’s thin boundary layer of viscous effects, there has long existed the well-known theory of Blasius which recent experiments by Hansen tend to confirm. Approaching the problem from the opposite extreme, Bairstow and Misses Cave and Lang have obtained a solution according to Oseen’s approximation to the equations of viscous flow. Their result is given in the form of an integral equation for the distribution of doublets along the plate which will exactly satisfy Oseen’s suggestion and the boundary conditions for an infinite fluid. But the solution of the equation has depended so far upon constructing a group of simultaneous equations with numerical coefficients determined by graphical means. The process is cumbersome and only two evaluations have been attempted, viz., at Reynolds’ numbers 4 and 4 x 10 4 . Exact treatment of Bairstow and Misses Cave and Lang’s integral equation presents difficulties, but it is possible to find an analytical solution of the equation whose errors throughout the experimental range are probably less than those involved in graphical manipulation. This enquiry is the subject of Section I of the present paper. Section II gives the streamlines and other details of the flow.