The convective term of an axisymmetric field in cylindrical form

On the meridional plane {x₂ = 0} away from the axis, the Cartesian convective term (u · ∇) u_i of a smooth axisymmetric field is the cylindrical transport term of eq:aniso:scalar:nse: the radial component picks up the centrifugal term -u_θ²/r, the swirl picks up u_r u_θ / r, and the vertical component is transported by the meridional velocity alone. The three identities are the θ-derivative relations of design note R4 inserted into the middle summand; they are pointwise and do not use the equation.

theorem CIV.sum_convective_zero_meridional {u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3} (h : IsAxisymmetricOn u unitCylinder) (hu : ContDiffOn ℝ 1 (fun (z : CKN.Foundation.Parabolic.Vec3 × ℝ) => u z) unitCylinder) {z : CKN.Foundation.Parabolic.ParabolicPoint} (hz : z ∈ unitCylinder) (hplane : z.1 1 = 0) (hr : z.1 0 ≠ 0) : ∑ j : Fin 3, u z j * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 0) j z = u z 0 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 0) 0 z + u z 2 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 0) 2 z - u z 1 ^ 2 / z.1 0

The convective term of the radial component on the meridional plane, off the axis: u_r ∂_r u_r + u_z ∂_z u_r - u_θ²/r, the left-hand side of eq:aniso:scalar:nse:r without the time derivative.

theorem CIV.sum_convective_one_meridional {u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3} (h : IsAxisymmetricOn u unitCylinder) (hu : ContDiffOn ℝ 1 (fun (z : CKN.Foundation.Parabolic.Vec3 × ℝ) => u z) unitCylinder) {z : CKN.Foundation.Parabolic.ParabolicPoint} (hz : z ∈ unitCylinder) (hplane : z.1 1 = 0) (hr : z.1 0 ≠ 0) : ∑ j : Fin 3, u z j * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 1) j z = u z 0 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 1) 0 z + u z 2 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 1) 2 z + u z 0 * u z 1 / z.1 0

The convective term of the swirl on the meridional plane, off the axis: u_r ∂_r u_θ + u_z ∂_z u_θ + u_r u_θ / r, the left-hand side of eq:aniso:scalar:nse:swirl without the time derivative.

theorem CIV.sum_convective_two_meridional {u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3} (h : IsAxisymmetricOn u unitCylinder) (hu : ContDiffOn ℝ 1 (fun (z : CKN.Foundation.Parabolic.Vec3 × ℝ) => u z) unitCylinder) {z : CKN.Foundation.Parabolic.ParabolicPoint} (hz : z ∈ unitCylinder) (hplane : z.1 1 = 0) (hr : z.1 0 ≠ 0) : ∑ j : Fin 3, u z j * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 2) j z = u z 0 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 2) 0 z + u z 2 * CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 2) 2 z

The convective term of the vertical component on the meridional plane, off the axis: u_r ∂_r u_z + u_z ∂_z u_z, the left-hand side of eq:aniso:scalar:nse:z without the time derivative.