The existence of axially symmetric flow above a rotating disk

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 ''' + ff ''+½( g 2 – f ' 2 ) = ½ Ω 2 ∞ , g "+ fg ' = f ' g , with the boundary conditions f (0) = a , f '(0) = 0, g (0) = Ω 0 ); f '(∞) = 0, g (∞) = Ω ∞ , where a is a constant measuring possible suction at the disk, Ω 0 is the angular velocity of the disk, and Ω ∞ is an angular velocity to which the fluid is subjected at infinity. When Ω ∞ = 0, existence of solutions has previously been proved by the ‘shooting technique’. This method breaks down when Ω 0 ǂ 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 Ω 0 is close to Ω ∞ and then extended for all Ω 0 , with the important restriction that Ω 0 and Ω ∞ be of the same sign.