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Unformatted text preview: (x, y, z, t)dxdydz
∂t F · NdA, − and S which must be equal. It follows from the divergence theorem that the second of
these expressions is
− ∇ · F(x, y, z, t)dxdydz,
D and hence
dxdydz = −
∂t ∇ · Fdxdydz.
D Since this equation must hold for every region D in (x, y, z )-space, we conclude
that the integrands must be equal,
= −∇ · F = −∇ · (ρv),
which is just the equation of continuity.
The second of the Euler equations is simply Newton’s second law of motion,
(mass density)(acceleration) = (force density).
We make an assumption that the only force acting on a ﬂuid element is due to
the pressure, an assumption which is not unreasonable in the case of a perfect
gas. In this case, it turns out that the pressure is deﬁned in such a way that
the force acting on a ﬂuid element is minus the gradient of pressure:
Force = −∇p(x(t), y (t), z (t), t). 131 (5.17) The familiar formula Force = Mass × Acceleration then yields
(v(x(t), y (t), z (t), t)) = −∇p(x(t), y (t), z (t), t).
dt Using the chain rule,
∂ v dx ∂ v dy ∂ v dz
∂ v dt
we can r...
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- Winter '14