 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2008/k/1/iv/solution

(u \cdot \nabla) u = (sin x cos x, - sin y cos y) .

The steady Euler equations therefore require

p_{x} = - ρ sin x cos x

and

p_{y} = ρ sin y cos y

. Integrating both derivatives consistently gives

p = C + \frac{ρ}{4} (cos 2 x - cos 2 y) .

The additive constant is arbitrary. With a corresponding redefinition of this constant, the Bernoulli function is

p / ρ + ∣ u ∣^{2} /2 = C_{0} - ψ^{2}

, making its constancy along streamlines explicit.

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