# D2 thus the partial dierential equation divides into

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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 D ∂ρ 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, ∂ρ = −∇ · F = −∇ · (ρv), ∂t 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 ρ d (v(x(t), y (t), z (t), t)) = −∇p(x(t), y (t), z (t), t). dt Using the chain rule, dv ∂ v dx ∂ v dy ∂ v dz ∂ v dt = + + + , dt ∂x dt ∂y dt ∂z dt ∂t dt we can r...
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