The vorticity and the circulation of an axisymmetric field #
The Cartesian curl curlComp u i of a field is written with CKN's classical spatial
partial derivatives. For a smooth axisymmetric field, on the meridional plane {x₂ = 0}
away from the axis, its components are the cylindrical ones of eq:aniso:vorticity,
ω_r = -∂_z u_θ, ω_θ = ∂_z u_r - ∂_r u_z, ω_z = ∂_r u_θ + u_θ/r; the middle one is
the definition itself, the outer two use the θ-derivative relations of design note R4.
The circulation Γ = r u_θ of eq:aniso:vorticity is x₁ u₂ on that plane.
The i-th Cartesian component of the curl, (∇ × u)_i = ∂_{i+1} u_{i+2} - ∂_{i+2} u_{i+1}
with the indices read cyclically.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The first Cartesian component of the curl is ∂₂u₃ - ∂₃u₂.
The second Cartesian component of the curl is ∂₃u₁ - ∂₁u₃; on the meridional plane this
is the azimuthal vorticity ω_θ = ∂_z u_r - ∂_r u_z of eq:aniso:vorticity.
The third Cartesian component of the curl is ∂₁u₂ - ∂₂u₁.
On the meridional plane, off the axis, the first component of the curl of a smooth
axisymmetric field is the radial vorticity ω_r = -∂_z u_θ of eq:aniso:vorticity.
On the meridional plane, off the axis, the third component of the curl of a smooth
axisymmetric field is the vertical vorticity ω_z = ∂_r u_θ + u_θ/r of
eq:aniso:vorticity.
On the meridional plane the circulation Γ = r u_θ of eq:aniso:vorticity is x₁ u₂.