 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2023/1/b/i/solution

Use the spherical symmetry of the expansion, with fluid at rest at infinity and no superposed imposed flow. The radial Laplace equation gives

\partial_{r} (r^{2} ϕ_{r}) = 0, ϕ_{r} = \frac{A ( t )}{r ^{2}}, ϕ = G (t) - \frac{A ( t )}{r} .

The moving sphere is a material boundary: its kinematic boundary condition is

u_{r} (R, t) = \dot{R}

. Consequently

A (t) = R^{2} \dot{R}

ϕ (r, t) = G (t) - \frac{R ^{2} R ˙}{r}, u = \frac{R ^{2} R ˙}{r ^{2}} r .

Equivalently, its volumetric flow rate through any concentric sphere is

4 π r^{2} u_{r} = 4 π R^{2} \dot{R}

, the rate of change of the enclosed volume.

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