D2 thus the partial dierential equation divides into

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online