Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / b / i

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- Solution i

Solution

In the exterior domain

r > a

, the streamfunction obeys

\nabla^{4} ψ = 0 .

Using the Streamfunction in polar coordinates convention

u_{r} = \frac{1}{r} ψ_{θ}, u_{θ} = - ψ_{r},

the no-slip condition on the stationary disk is

u_{r} (a, θ) = u_{θ} (a, θ) = 0.

Thus

ψ

is constant on the disk and

ψ_{r} = 0

there. Choose the arbitrary constant to be zero:

ψ (a, θ) = 0, ψ_{r} (a, θ) = 0 .

The imposed shear

u_{\infty} = γ y e_{x}

has streamfunction

ψ_{\infty} = \frac{γ}{2} y^{2} = \frac{γ r ^{2}}{4} (1 - cos 2 θ) .

The far-field condition is therefore

\nabla ψ - \nabla ψ_{\infty} ⟶ 0 (r \to \infty) .

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