An uncountable number of solutions for a BVP governing Marangoni convection

This paper investigates a boundary value problem (BVP) governing Marangoni convection over a flat surface. The BVP involves a temperature gradient parameter k > − 1 . Previous numerical and analytical studies report the existence of one solution for each value of k . Here we show that the nature of the solutions varies greatly depending on the value of k . For each − 1 < k < − 1 / 2 we prove that an uncountable number of solutions to the BVP exist. For − 1 / 2 ≤ k < 0 a unique solution exists. For k ≥ 0 we prove the existence of a solution but, based on numerical integration of the differential equation, conjecture the existence of precisely two solutions when k > 0 .