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 Incoming links: Harmonic pressure and vorticity in Stokes flow

Past exam of the fluids course of the University of Oxford / 2022 / 4 / a / Solution 2026-10-25

A small Reynolds number,

Re = ρ UL / η ≪ 1

, means that inertia is small compared with viscous forces. It can arise from small speed, small length or large dynamic viscosity. Examples are the locomotion of a microswimmer and the slow sedimentation of a small particle in a viscous liquid.

To neglect local acceleration as well, an externally imposed time scale

T

must satisfy

ρ L^{2} / (η T) = L^{2} / (ν T) ≪ 1

. This holds for a steady flow or a sufficiently slowly varying quasi-steady Stokes approximation. Small Reynolds number alone is insufficient for that extra step. For an incompressible flow with constant material coefficients and uniform gravity, the Navier-Stokes equation then reduces to the Stokes equation:

0 = - \nabla p - ρ g \hat{z} + η \nabla^{2} u, \nabla \cdot u = 0.

Taking the divergence and using commuting spatial derivatives gives

0 = - \nabla^{2} p + η \nabla^{2} (\nabla \cdot u), \nabla^{2} p ≃ 0.

Taking the curl, which annihilates the pressure gradient and uniform gravitational force, gives

0 = η \nabla^{2} (\nabla \times u), \nabla^{2} ω ≃ 0.

These are the Harmonic pressure and vorticity in Stokes flow results.

A concrete counterexample to inferring both solely from small Reynolds number is Stokes' second problem: an oscillating plate with arbitrarily small velocity amplitude has negligible inertia, but

\partial_{t} ω = ν \nabla^{2} ω

is nonzero. Its vorticity is not a harmonic function. The quasi-steady qualification is therefore necessary.

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 39A / b / i / Solution Created 2026-09-24 Updated 2026-09-25

The geometry and boundary velocities are axisymmetric and linear in the axial pseudovector

Ω

. The pressure is a true scalar. No nonzero parity-even scalar can be formed linearly from

Ω

and the radial vector

x

:

Ω \cdot x

is a pseudoscalar. Hence the flow-induced pressure is spatially constant.

Equivalently, taking the divergence of the Stokes momentum equation gives the Harmonic pressure and vorticity in Stokes flow result

\nabla^{2} p = 0,

and symmetry plus the homogeneous boundary data excludes every nonconstant harmonic pressure mode. Since pressure is defined up to an additive constant, we choose

p = 0 .

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / a / Solution Created 2026-09-24 Updated 2026-09-25

The steady incompressible Stokes flow equations are

- \nabla p + μ \nabla^{2} u = 0, \nabla \cdot u = 0.

Taking the curl eliminates the pressure gradient and commutes with the Laplacian, so for the vorticity

ω = \nabla \times u

,

\nabla^{2} ω = 0 .

This is the vorticity part of Harmonic pressure and vorticity in Stokes flow.

For a two-dimensional incompressible flow, choose the stream function convention

u_{x} = ψ_{y}, u_{y} = - ψ_{x} .

Then

ω_{z} = \partial_{x} u_{y} - \partial_{y} u_{x} = - \nabla^{2} ψ .

Applying the harmonic-vorticity equation gives

\nabla^{4} ψ = 0,

the Biharmonic stream function for planar Stokes flow equation.

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