Concept

Helmholtz decomposition — where it appears

The splitting of a vector field into a part carrying all of its divergence and a part carrying all of its curl. The two are not unique: any field that is both divergence-free and curl-free can be moved between them, and only a boundary condition decides.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

One field, and the two parts the theorem splits it into. A velocity field made of a smooth source, a smooth vortex and a uniform stream, and the two fields the Helmholtz decomposition returns for it. The first carries the whole divergence and has no curl anywhere; the second carries the whole curl and has no divergence. They are computed by solving two Poisson problems on a grid, with the divergence and the vorticity differenced from the field rather than taken from the expressions that built it. Adding the two back together does not recover the field.

Every flow is two flows

Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.

kinematics · Helmholtz

Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

kinematics · Stream function

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary condition Divergence Model limit Streamfunction Velocity potential Axisymmetric Convergence Flow visualisation Irrotational Laplace's equation Mass conservation Null space

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