EXISTENCE OF SIMILARITY SOLUTIONS FOR SURFACE-TENSION DRIVEN FLOWS IN FLOATING RECTANGULAR CAVITIES

Abstract

We consider the following problem: f''' + Q . [Af f'' - (f')2] = beta, (Q, beta is-an-element-of R, A greater-than-or-equal-to 0), f(0) = f(1) = f''(1) = f''(0) + 1 = 0, which arises from the steady surface-tension driver flows in floating rectangular cavities. In this paper, we first classify all possible solutions and obtain that the given problem can only possess, at most, three types of solutions for A greater-than-or-equal-to 1. By the classification, we further verify that for every Q greater-than-or-equal-to 0 or beta greater-than-or-equal-to 0, the problem has at least one solution if A greater-than-or-equal-to 1. Moreover, if 1 less-than-or-equal-to A less-than-or-equal-to 3/2, multiple solutions also exist for sufficiently large Q > 0.