 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/k/k1/c/solution

p_{s}

, solve the two pressure expressions from part (b) for

z

. With

C = (B - p_{s}) / (ρ g)

, this gives

z (r) = ⎩ ⎨ ⎧ C - \frac{Γ ^{2} ( 2 a ^{2} - r ^{2} )}{2 g a ^{4}}, C - \frac{Γ ^{2}}{2 g r ^{2}}, r \leq a, r > a .

Substituting

r^{2} = x^{2} + y^{2}

recovers the required Cartesian form. The two values coincide at

r = a

, and both radial slopes there equal

Γ^{2} / (g a^{3})

. The surface is parabolic in the core and approaches height

C

r \to \infty

; at the centre its depression below

C

Γ^{2} / (g a^{2})

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