Existence of secondary solutions to a generalized taylor problem

Abstract

In this paper the existence of secondary solutions to a generalization of the classical Taylor problem is considered. A viscous liquid is assumed to occupy the region interior to a right circular cylinder and exterior to a. surface formed by rotating a smooth, positive, periodic function about the axis of the cylinder. The cylinder is fixed while the inner surface rotates with a constant angular velocity. The existence of axisymmetric ceiiuiar solutions is estab¬lished by a generalization of the method of Lyapunov and Schmidt. By treating the branching equation as a function of three complex variables it is shown that a critical Reynolds number λ₁ ^* exists and that for λ-λ₁ ^* < 0 the problem has a unique solution while for λλ_:1 ^* positive and small there are three solutions