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Biharmonic stream function for a fixed disk in planar shear

For a stationary disk of radius

a

in the far-field shear

u_{\infty} = γ y e_{x}

, using

u_{r} = ψ_{θ} / r

and

u_{θ} = - ψ_{r}

, the exterior solution is

ψ = \frac{γ}{4} [r^{2} - a^{2} - 2 a^{2} lo g \frac{r}{a} - (r^{2} - 2 a^{2} + \frac{a ^{4}}{r ^{2}}) cos 2 θ] .

It satisfies no slip at

r = a

and approaches the imposed shear in velocity.

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- Hydrodynamic torque on a fixed disk in planar shear Biharmonic stream function for a fixed disk in planar shear 27

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