Infinitesimal consequences of rotation invariance #
An axisymmetric field on the unit cylinder is equivariant, u (Q_φ x, t) = Q_φ u (x, t)
(apply_rotZ). Differentiating this identity in the angle at φ = 0 gives the three
scalar identities of angularGenerator_zero, angularGenerator_one and
angularGenerator_two: with the generator J x = (-x₂, x₁, 0) of the rotations,
J u(x, t) = Du(x, t) (J x).
On the meridional plane {x₂ = 0} away from the axis these become the θ-derivative
relations ∂₂u₁ = -u₂/r, ∂₂u₂ = u₁/r, ∂₂u₃ = 0 (design note R4: the cylindrical
coefficients of an axisymmetric field are read on that plane, where u_r = u₁,
u_θ = u₂, u_z = u₃ and ∂_r = ∂₁, ∂_z = ∂₃). They are what turns the Cartesian
divergence, curl and Laplacian into their cylindrical forms
eq:aniso:scalar:nse:div, eq:aniso:vorticity and the right-hand sides of
eq:aniso:scalar:nse.
Differentiating the equivariance in the space variable instead gives
fderiv_comp_rotZ, the commutation Du(Q_φ x) Q_φ = Q_φ Du(x). At φ = π it yields
the structure on the axis quoted in the standing assumptions of Section
sec:aniso:equations: u_r, u_θ, ∂_z u_r, ∂_z u_θ and ∂_r u_z all vanish
there, and the components are odd, odd and even under the reflection
(x₁, 0, x₃) ↦ (-x₁, 0, x₃) of the meridional plane.
Spatial slices of a field on the cylinder #
A field of class C^n on the unit cylinder has C^n spatial slices on the unit ball.
The spatial slice of a C¹ field is differentiable at every point of the cylinder.
A component of a C^n field is C^n.
The spatial derivative of a component is the corresponding component of the spatial derivative.
Each classical spatial partial derivative of a component is read off the full spatial derivative.
The generator of the rotations #
The angle derivative at φ = 0 of a C¹ scalar transported along the rotations is the
derivative of the scalar along the generator.
Equivariance of an axisymmetric field #
An axisymmetric field on the unit cylinder is equivariant under the rotations:
u (Q_φ x, t) = Q_φ u (x, t) (Section sec:aniso:notation).
The three infinitesimal identities #
Master form of the infinitesimal identity: whatever the angle derivative at φ = 0 of
φ ↦ g (Q_φ x, t) is, it equals the derivative of g along the generator.
The infinitesimal form of rotation invariance, first component: -x₂ ∂₁u₁ + x₁ ∂₂u₁ = -u₂
(sec:aniso:equations).
The infinitesimal form of rotation invariance, second component:
-x₂ ∂₁u₂ + x₁ ∂₂u₂ = u₁.
The infinitesimal form of rotation invariance, third component:
-x₂ ∂₁u₃ + x₁ ∂₂u₃ = 0.
For a rotation-invariant scalar the derivative along the generator vanishes:
-x₂ ∂₁g + x₁ ∂₂g = 0.
The θ-derivative relations on the meridional plane #
On the meridional plane, off the axis, ∂_θ u_r = -u_θ: the second Cartesian partial
derivative of the first component is -u_θ / r (design note R4).
On the meridional plane, off the axis, ∂_θ u_θ = u_r: the second Cartesian partial
derivative of the second component is u_r / r (design note R4).
On the meridional plane, off the axis, ∂_θ u_z = 0: the second Cartesian partial
derivative of the third component vanishes (design note R4).
The derivative intertwines the rotations #
The spatial derivative of an axisymmetric field intertwines the rotations:
Du(Q_φ x, t) Q_φ = Q_φ Du(x, t).
The axis and the reflection of the meridional plane #
The rotation by π fixes the points of the vertical axis.
The rotation by π is the reflection (x₁, 0, x₃) ↦ (-x₁, 0, x₃) of the meridional
plane.
The radial component of an axisymmetric field vanishes on the axis (standing assumptions
of Section sec:aniso:equations).
The swirl of an axisymmetric field vanishes on the axis (standing assumptions of Section
sec:aniso:equations).
The rotation by π reverses the first basis vector.
The rotation by π reverses the second basis vector.
The rotation by π fixes the vertical basis vector.
At an axis point the spatial derivative anticommutes with the reflection Q_π.
On the axis the radial derivative of the vertical component vanishes: ∂_r u_z = 0
(standing assumptions of Section sec:aniso:equations).
On the axis the second Cartesian derivative of the vertical component vanishes
(standing assumptions of Section sec:aniso:equations).
On the axis the vertical derivative of the radial component vanishes: ∂_z u_r = 0
(the paragraph after eq:aniso:circulation:pde).
On the axis the vertical derivative of the swirl vanishes: ∂_z u_θ = 0.
Under the reflection of the meridional plane the radial component is odd.
Under the reflection of the meridional plane the swirl is odd.
Under the reflection of the meridional plane the vertical component is even.
On the meridional plane, off the axis, the second Cartesian partial derivative of a
rotation-invariant scalar vanishes: ∂_θ g = 0 (design note R4).