The circulation component of the scalar Navier–Stokes equation on the meridional plane #
eq:aniso:circulation:pde is the swirl component of the momentum equation written in terms of
the circulation Γ = r u_θ. On the meridional plane {x₂ = 0} away from the axis, the
classical partials of Γ are expressed in terms of those of u_θ via the product rule; the
resulting equation follows directly from scalar_nse_swirl multiplied by r.
Away from the meridional plane Γ = x₁ u₂ - x₂ u₁ picks up an extra term from the second
coordinate, so the product-rule expansions of its partials are proved first at every point of
the cylinder and only specialised to the plane (where that extra term vanishes) afterward; the
second derivatives are obtained by differentiating the plane-independent first-derivative
identity on a full neighbourhood of the base point and setting the second coordinate to zero
only in the final step.
First radial derivative of the circulation on the meridional plane: ∂_r Γ = u_θ + r ∂_r u_θ.
First vertical derivative of the circulation on the meridional plane: ∂_z Γ = r ∂_z u_θ.
Second radial derivative of the circulation: ∂_r² Γ = 2 ∂_r u_θ + r ∂_r² u_θ.
Second vertical derivative of the circulation: ∂_z² Γ = r ∂_z² u_θ.
The circulation component eq:aniso:circulation:pde of the scalar Navier–Stokes equation:
for a classical solution with axisymmetric velocity and axisymmetric force, on the meridional
plane off the axis, ∂_t Γ + u_r ∂_r Γ + u_z ∂_z Γ = ∂_r² Γ - r⁻¹ ∂_r Γ + ∂_z² Γ + r f_θ.