A contribution to the problem of vortex breakdown

Associated with the breakdown process is the formation of a stagnation point on the axis of the vortex. This requires the deceleration of the axial velocity component, which must be enforced by a positive axial pressure gradient. The analysis presented here shows, how the pressure gradient along the axis of the vortex is influenced by the radial and azimuthal velocity components. An explicit expression for $\text{?p/?x (x, 0)}$ can be obtained by integration of the momentum equation for the radial velocity component with respect to the radial and subsequent differentiation of the integral with respect to the axial direction. In an order of magnitude analysis it is then demonstrated that for large Reynolds numbers one component of the frictional force in the azimuthal direction cannot be neglected. In order to obtain an estimate for the pressure gradient rigid body rotation is assumed for the vortex core, and a distribution similar to that of a potential vortex $\text{w = kr}{}^{\mathrm{?n}}$, for the outer portion. The estimate shows that a positive axial pressure gradient can exist only, if the radial velocity component is positive and if the exponent $\text{n}$ is less than unity. It is also verified that a potential vortex cannot support an axial pressure gradient, that the pressure gradient in magnitude is directly proportional to the square of the maximum of the azimuthal velocity, referenced to the freestream velocity.