Skin friction on a strip of finite width moving parallel to its length

Abstract

A flat strip, of infinitesimal thickness and infinite lenght, is located in a viscous incompressible fluid, and both the strip and fluid are at rest initially. The strip is abruptly set into steady motion parallel to its length. An unsteady uni-directional flow of the fluid results, and there is a variable skin friction on the strip which must be overcome to maintain its velocity. In the early stages of motion, the skin friction is large, with a local behaviour which resembles that for a flat strip of infinite width. The skin friction near each edge of the strip can be more accurately represented by referring to the semi-infinite configuration that is realized on displacement of the other edge to infinity. At this stage, the results depend on separate consideration of the two limiting configurations, and the way to further improvements is not clearly delineated. The object of this paper is to provide a formulation which contains the preceding information and allows a systematic evaluation of all additional refinements. Thus, it is shown that the total skin friction on a strip of width 2a, moving with velocity V in a fluid whose coefficients of viscosity and kinematic viscosity are μ, ν, takes the form $D = 2\pi V \left[\frac {2a}{\surd (\pi vt)} + 1 - \frac {2}{\pi}\int ^\infty _1 \frac{1}{u^2} \surd \left(\frac {u-1}{u+1}\right) \rm {erfc} \left(\frac {au}{\surd (vt)} \right)du \right].$ Here the first two terms stem from the infinite and semi-infinite strip distributions, and the integral, containing a complementary error function, furnishes all corrections of the lowest exponential order, in the initial stages of motion, when $a| \surd (vt) \gg 1$ 1. The calculation is based on a new integral equation which makes explicit the effects of an isolated edge, and by iteration provides the interaction effects between edges at a finite separation.