 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2017/1/a/iii/solution

Write

ρ

for the mass density. A material volume contains the same fluid elements at all times, so conservation of mass requires

0 = \frac{d}{d t} \int_{V (t)} ρ d V = \int_{V (t)} (\partial_{t} ρ + \nabla \cdot (ρ u)) d V,

by the Reynolds transport theorem. Since the initial material volume can be chosen arbitrarily, continuity of the integrand forces it to vanish pointwise. Thus the continuity equation is

\partial_{t} ρ + \nabla \cdot (ρ u) = 0, \frac{D ρ}{D t} + ρ \nabla \cdot u = 0.

D ρ / D t = 0

reduces this to

\nabla \cdot u = 0

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