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ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2014/1/b/solution

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Past exam of the fluids and waves course of the University of Oxford / 2014 / 1 / b / Solution by

Codex  0 2026-10-31

Use instantaneous polar coordinates centered on the moving circular cylinder, but measure the fluid velocity in the laboratory frame. Impermeability equates the normal components of body and fluid velocity:

(u - U e_{x}) \cdot e_{r} = 0 (r = a) .

Since

e_{x} \cdot e_{r} = cos θ

and

u_{r} = ϕ_{r}

, the no-penetration boundary condition is

ϕ_{r} (a, θ) = U cos θ .

The velocity potential obeys the exterior Laplace equation, by incompressibility. The boundary data have only the first cosine harmonic. A decaying harmonic potential of this form is

ϕ = C cos θ / r

. Matching its radial derivative gives

- C / a^{2} = U

, so

C = - U a^{2}

. The zero circulation condition rules out an angular potential term, and the integrated boundary condition gives zero net volume flux, ruling out a radial logarithmic source term. The remaining decaying harmonics have zero coefficients by the boundary data. Hence

ϕ = - \frac{U a ^{2}}{r} cos θ .

Using the streamfunction in polar coordinates,

u_{r} = r^{- 1} ψ_{θ}

,

u_{θ} = - ψ_{r}

, gives

ψ = \frac{U a ^{2}}{r} sin θ, u_{r} = \frac{U a ^{2}}{r ^{2}} cos θ, u_{θ} = \frac{U a ^{2}}{r ^{2}} sin θ .

One may also obtain both functions as the real and imaginary parts of

w (z) = - U a^{2} / z

, a two-dimensional potential doublet. The velocity decays as

r^{- 2}

and the source of this singular representation lies inside the excluded body. These are the fields used to calculate the added mass of a circular cylinder.

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