Past exam of the fluids and waves course of the University of Oxford / 2015 / 3 / a

Solution

Let

V (t)

be any material volume, moving with the velocity field

u

. The Reynolds transport theorem in the required form is

\frac{d}{d t} \int_{V (t)} f d V = \int_{V (t)} [\partial_{t} f + \nabla \cdot (f u)] d V,

applied componentwise when

f

0 = \frac{d}{d t} \int_{V (t)} ρ d V = \int_{V (t)} [ρ_{t} + \nabla \cdot (ρ u)] d V .

Since the material volume is arbitrary, the local continuity equation follows:

ρ_{t} + \nabla \cdot (ρ u) = 0.

For an inviscid fluid with no body force, the balance of linear momentum equates the rate of change of total momentum to the boundary pressure force:

\frac{d}{d t} \int_{V (t)} ρ u d V = - \int_{\partial V (t)} p n d S = - \int_{V (t)} \nabla p d V,

using the divergence theorem. The integrand on the left expands as

\partial_{t} (ρ u) + \nabla \cdot (ρ u \otimes u) = ρ \frac{D u}{D t} + u [ρ_{t} + \nabla \cdot (ρ u)] .

ρ \frac{D u}{D t} = - \nabla p, \frac{D}{D t} = \partial_{t} + u \cdot \nabla.

Unlike the incompressible flow case, the mass density need not be constant along a fluid element.

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