ASYMPTOTIC-BEHAVIOR OF SOLUTIONS OF A SIMILARITY EQUATION FOR LAMINAR FLOWS IN CHANNELS WITH POROUS WALLS

The boundary value problem f(iv) = R(ff'" - f'f") f(0) = f"(0) = f'(1) = 0, f(1) = 1, introduced by Berman (1953), possesses multiple solutions which display four distinct types of asymptotic behaviour as Absolute value of R --> infinity, one for negative R and the remaining three for positive R. In this paper, three of the four types are considered. First, a proof of the leading behaviour up to and including the third derivative of f is given for the case R --> -infinity. This enables the confirmation of certain numerical conjectures. For two of the remaining three cases (R --> +infinity), the authors prove that the graph of the solution is uniformly approximated by the straight line segment joining the two end-points, which implies there must be a boundary layer at the right end-point for each of these solutions. A more complete description of these two solutions is then studied using the method of matched asymptotic expansions. This results in a description of the asymptotic behaviour of the two solutions up to and including their seventh-order derivatives, uniformly valid on the closed interval [0, 1]. These results are closely related to previous work, but there are technical differences in how the method of matched asymptotic expansions is applied; these differences are explained in the paper.