## Abstract

The paper studies the boundary-value problem arising from the behaviour of a fluid occupying the half space x > 0 above a rotating disk which is coincident with the plane x = 0 and rotates about its axis which remains fixed. The equations which describe axially symmetric solutions of this problem are $f^{\prime \prime \prime}+ff^{\prime \prime}+\frac{1}{2}(g^{2}-f^{\prime 2})=\frac{1}{2}\Omega _{\infty}^{2}$, $g^{\prime \prime}+fg^{\prime}=f^{\prime}g$, with the boundary conditions f(0) = a, $f^{\prime (0)=0}$, $g(0)=\Omega _{0}$; $f^{\prime}(\infty)=0$, $g(\infty)=\Omega _{\infty}$, where a is a constant measuring possible suction at the disk, $\Omega _{0}$ is the angular velocity of the disk, and $\Omega _{\infty}$ is an angular velocity to which the fluid is subjected at infinity. When $\Omega _{\infty}=0$, existence of solutions has previously been proved by the 'shooting technique'. This method breaks down when $\Omega _{0}\neq 0$ because of oscillations in the functions f and g, but in the present paper existence is first proved by a fixed point method when $\Omega _{0}$ is close to $\Omega _{\infty}$ and then extended for all $\Omega _{0}$, with the important restriction that $\Omega _{0}$ and $\Omega _{\infty}$ be of the same sign.