 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2008/k/k1/iv/solution

u u_{x} + v u_{y} = sin x cos x, u v_{x} + v v_{y} = - sin y cos y .

Steady Euler equations therefore give

p_{x} = - ρ sin x cos x

and

p_{y} = ρ sin y cos y

. Integrating these compatible equations yields

p (x, y) = p_{0} - \frac{ρ}{2} sin^{2} x + \frac{ρ}{2} sin^{2} y,

p_{0}

. As a check,

p / ρ + (u^{2} + v^{2}) /2 = p_{0} / ρ + 1/2 - ψ^{2}

 New to topics? Read the docs here!