Vector operators in curvilinear coordinates

Lesson 8 defined divergence and curl in rectangular coordinates and connected them to flux and circulation through the theorems of Gauss, Green and Stokes. Many engineering problems have round geometry: pipes and cables, pressure vessels and storage tanks, rotating machinery, the field around a wire or a charged sphere, flow toward a well. In these problems rectangular coordinates are clumsy, and the natural choice is cylindrical or spherical coordinates. This lesson develops the gradient, divergence, curl and Laplacian in those coordinates and shows where their extra factors come from. It also collects the vector identities that engineers use constantly, explains why fields with zero curl have scalar potentials and fields with zero divergence have vector potentials, and applies the results to heat loss from insulated pipes, the critical radius of insulation, spherical conduction and vortex flows.

The del operator and the Laplacian

The symbol \(\nabla = \mathbf{i}\,\partial/\partial x + \mathbf{j}\,\partial/\partial y + \mathbf{k}\,\partial/\partial z\) acts like a vector of derivatives. Applied to a scalar it gives the gradient \(\nabla f\); dotted with a field it gives the divergence \(\nabla \cdot \mathbf{F}\); crossed with a field it gives the curl \(\nabla \times \mathbf{F}\). The divergence of a gradient is the Laplacian:

The Laplacian compares the value of f at a point with its average over a small surrounding sphere: if \(\nabla^2 f > 0\), the point is lower than its surroundings. This is why it appears in the heat equation (a point warms when it is colder than its neighbourhood), in diffusion, in the wave equation and in the equations for electrostatic and gravitational potential. A function with \(\nabla^2 f = 0\) is harmonic: its value at every point equals the average around it, so it has no interior maxima or minima. A steady temperature distribution with no internal heat sources is harmonic, so the hottest and coldest points are always on the boundary.

Vector identities

A handful of identities, each provable by writing out components, simplify a great deal of engineering analysis:

The first two identities have converses that matter in practice. In a region without holes, a field with zero curl is the gradient of a scalar potential, \(\mathbf{F} = \nabla\phi\) (Lesson 7 found such potentials for conservative forces). Similarly, a field with zero divergence is the curl of a vector potential, \(\mathbf{F} = \nabla \times \mathbf{A}\). The Helmholtz theorem goes further: any reasonable field can be split into an irrotational part and a solenoidal part, \(\mathbf{F} = \nabla\phi + \nabla \times \mathbf{A}\). Fluid dynamicists use this to separate the compressing and swirling parts of a flow, and electrical engineers use it to describe electric and magnetic fields with potentials.

Cylindrical coordinates

In cylindrical coordinates \((r, \theta, z)\), the unit vectors \(\mathbf{e}_r\), \(\mathbf{e}_\theta\), \(\mathbf{e}_z\) point outward, around and along the axis. A small step dθ in angle moves a distance \(r\,d\theta\), not dθ, and this scale factor r is the source of every extra term below:

The factor r inside the radial derivative of the divergence has a direct physical meaning. Consider a thin cylindrical shell between r and r + dr. Its outer surface is larger than its inner surface, in proportion to r. A purely radial field with no sources must therefore weaken as 1/r so that the same total flux passes through each shell: the quantity \(rF_r\) must be constant. The z-component of the curl, needed for swirling flows, is \((\nabla \times \mathbf{F})_z = \frac{1}{r}\left[\frac{\partial (rF_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta}\right]\).

The critical radius of insulation

Adding insulation to a cylinder does two things: it adds conduction resistance \(\ln(r_2/r_1)/(2\pi k)\), which reduces heat loss, and it enlarges the outer surface, which increases convection to the air, with resistance \(1/(2\pi r_2h)\). Minimizing the total resistance with respect to \(r_2\) (a one-variable optimization) shows that the heat loss is greatest when \(r_2 = k/h\), the critical radius. For a large steam pipe the critical radius (a few millimetres) is far smaller than the pipe, and insulation always helps. For a thin electrical wire with PVC insulation (k ≈ 0.2 W/(m·K), h ≈ 10 W/(m²·K)), the critical radius is about 20 mm, so the insulation on a thin wire actually increases its heat loss, which helps keep cables cool.

Spherical coordinates

In spherical coordinates \((\rho, \phi, \theta)\), with ρ the distance from the origin, φ the angle from the positive z-axis and θ the angle around it, the scale factors are 1, ρ and \(\rho\sin\phi\). For a function that depends only on ρ, which covers spheres, point sources and spherical tanks, the Laplacian reduces to

and the divergence of a radial field \(F_\rho(\rho)\,\mathbf{e}_\rho\) is \(\frac{1}{\rho^2}\frac{d}{d\rho}(\rho^2F_\rho)\). The factor \(\rho^2\) is proportional to the area of a sphere of radius ρ, so a source-free radial field must fall off as \(1/\rho^2\), the inverse-square law.

Vortex flows

Swirling flows show clearly the difference between circulation around a path and rotation at a point. Compare two flows with circular streamlines:

• Rigid rotation, \(\mathbf{v} = \Omega r\,\mathbf{e}_\theta\), as in a tank of liquid spun steadily: the curl is \(\frac{1}{r}\frac{d}{dr}(\Omega r^2) = 2\Omega\), the same everywhere. Every fluid element spins.
• A free vortex, \(\mathbf{v} = \dfrac{\Gamma}{2\pi r}\mathbf{e}_\theta\), as in a draining bathtub or a tornado outside its core: the curl is \(\frac{1}{r}\frac{d}{dr}\left(\frac{\Gamma}{2\pi}\right) = 0\) for r > 0. The fluid circulates around the centre, yet small elements do not spin; a tiny paddle wheel carried by the flow keeps its orientation.

The free vortex still has circulation Γ around any loop enclosing the centre, which by Stokes' theorem means all its vorticity is concentrated at r = 0, just as the inverse-square field concentrates its divergence at the origin. Real vortices combine the two: a core in rigid rotation surrounded by a free vortex (the Rankine vortex), which is a useful model of tornadoes and of the tip vortices trailing behind aircraft wings.