Stokes stream function
In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function encloses a streamtube, everywhere tangential to the flow velocity vectors. Further, the volume flux within this streamtube is constant, and all the streamlines of the flow are located on this surface. The velocity field associated with the Stokes stream function is solenoidal—it has zero divergence. This stream function is named in honor of George Gabriel Stokes. Cylindrical coordinates. Consider a cylindrical  coordinate system ( "ρ" , "φ" , "z" ), with the "z"–axis the line around which the incompressible flow is axisymmetrical, "φ" the azimuthal angle and "ρ" the distance to the "z"–axis. Then the flow velocity components "u" and "u" can be expressed in terms of the Stokes stream function formula_1 by: The azimuthal velocity component "u" does not depend on the stream function. Due to the axisymmetry, all three velocity components ( "u" , "u" , "u" ) only depend on "ρ" and "z" and not on the azimuth "φ". The volume flux, through the surface bounded by a constant value "ψ" of the Stokes stream function , is equal to "2π ψ". Spherical coordinates. In spherical coordinates ( "r" , "θ" , "φ" ), "r" is the radial distance from the origin, "θ" is the zenith angle and "φ" is the azimuthal angle. In axisymmetric flow, with "θ" = 0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth "φ". The flow velocity components "u" and "u" are related to the Stokes stream function formula_1 through: Again, the azimuthal velocity component "u" is not a function of the Stokes stream function "ψ". The volume flux through a stream tube, bounded by a surface of constant  ψ, equals "2π ψ", as before. Vorticity. The vorticity is defined as: with formula_7 the unit vector in the formula_8–direction. As a result, from the calculation the vorticity vector is found to be equal to: Comparison with cylindrical. The cylindrical and spherical coordinate systems are related through Alternative definition with opposite sign. As explained in the general stream function article, definitions using an opposite sign convention – for the relationship between the Stokes stream function and flow velocity – are also in use. Zero divergence. In cylindrical coordinates, the divergence of the velocity field u becomes: as expected for an incompressible  flow. And in spherical coordinates: Streamlines as curves of constant stream function. From calculus it is known that the gradient vector formula_14 is normal to the curve formula_15 (see e.g. Level set#Level sets versus the gradient). If it is shown that everywhere formula_16 using the formula for formula_17 in terms of formula_18 then this proves that level curves of formula_1 are streamlines. In cylindrical coordinates, and So that And in spherical coordinates and So that