SynopsisWe study Berman's problemsubjects to the conditions f(0) =f″(0) = f′(1) = f(1)− 1 = 0, which arises from the study of laminar flows in channels with porous walls. Positive (negative) Re denotes the case of suction (injection) flows. Preliminary numerical studies indicated the existence of three different types of solutions and a small portion of mathematical evidence was provided. In this paper, we are able to show that there exist connected sets in the Re-K plane on which different types of solutions occur. In particular, our result verifies that the problem possesses all three types of suction solutions for sufficiently large Re. Moreover, the limiting injection solution is also obtained.

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This study presents a nonautonomous boundary value problem, arising from the surface-tension driven flow in a cylindrical floating zone. It is shown that this problem possesses exactly eight different types of solutions. In addition, multiple solutions are also found when the Reynold's number Q is sufficiently large.