Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C (Fluid Dynamics II)

The steady incompressible Stokes flow equations areTaking the curl eliminates the pressure gradient and commutes with the Laplacian, so for the vorticity $\mathbf{\omega }=\nabla \times \mathbf{u}$,This is the vorticity part of Harmonic pressure and vorticity in Stokes flow.

For a two-dimensional incompressible flow, choose the stream function conventionThenApplying the harmonic-vorticity equation givesthe Biharmonic stream function for planar Stokes flow equation.

In the exterior domain $r>a$, the streamfunction obeysUsing the Streamfunction in polar coordinates conventionthe no-slip condition on the stationary disk isThus $\psi$ is constant on the disk and ${\psi }_r=0$ there. Choose the arbitrary constant to be zero:

The imposed shear ${\mathbf{u}}_{\infty }=\gamma y{\mathbf{e}}_x$ has streamfunctionThe far-field condition is therefore

Only the axisymmetric and $\cos 2\theta$ modes are needed. The exterior biharmonic forms compatible with the far-field velocity areTerms $r^2\log r$ and $r^4\cos 2\theta$ are excluded because they would dominate the imposed shear.

The two conditions at $r=a$ applied separately to each Fourier mode giveHenceDirect differentiation gives $\psi ={\psi }_r=0$ at $r=a$. At infinity the remaining disturbance velocity is $O(r^{-1})$, so the imposed shear is recovered. This is the Biharmonic stream function for a fixed disk in planar shear.

The polar shear stress in a Newtonian fluid isOn $r=a$, no slip holds for every $\theta$, so $u_{\theta }=0$ and ${\partial }_{\theta }{u}_r=0$. ThereforeDifferentiating the solution from part (ii) givesand hence

The moment arm and line element are both $a$, so the torque exerted by the fluid on the disk per unit axial length isFor $\gamma >0$ the negative sign denotes the clockwise sense of the ambient shear rotation. This agrees with the hydrodynamic torque on a fixed disk in planar shear.