The authors seek the solutions of the Navier Stokes equations in the form: \[ U_ x=xH'(z)-G(z)y+g(z);\quad \] \[ U_ y=yH'(z)+G(z)x- f(z);\quad U_ z=-H(z) \] and showed that H(z), G(z), g(z), f(z) satisfy a boundary value problem which constitutes the nonlinear von Kármán equations for axially symmetric swirling flow for functions H(z,\(\epsilon)\), G(z,\(\epsilon)\) and a linear set of equations for f(z,\(\epsilon)\), g(z,\(\epsilon)\) with coefficients depending on H(z,\(\epsilon)\), G(z,\(\epsilon)\). When the planes rotate with different angular velocities about a common axis or distinct axes it is shown that there is a one parameter family of solutions for large viscosities.