Laminar equalizing flows in high Reynolds number Couette-type initial flows terminating in free-surface flows

This paper deals with the equalizing (secondary) flows in a two-dimensional channel in which the flows are limited by an infinite and a semi-infinite wall moving forward at different speeds. On this flow the initial Couette-type upstream flow in the closed part of the channel develops into a zero-shear flow far downstream of the edge (open to the atmosphere) of the semi-infinite wall. A method of matched asymptotic expansions is used to calculate the shape of the free surface and the boundary layer growth for several different initial velocity profiles. The description of this flow requires the use of higher order terms which the assumptions made in the traditional boundary layer theory do not satisfy. Therefore a theory of 'higher order' is necessary. A separate theory is used to study the flow equalization far downstream when the boundary layer fills the whole flow cross section.