 #1
aclaret
 24
 9
 Homework Statement:

Let ##u: R^3 \times R \rightarrow R^3## be flow velocity of incompressible fluid. Let fluid be subject to potential force ##\nabla \chi##. To prove
$$\frac{d}{dt} \int_{V} \frac{1}{2} \rho \langle u, u \rangle dV + \int_{\partial V} H \langle u, n \rangle dA = 0$$where ##H := \frac{1}{2}\rho \langle u, u \rangle + p + \chi##, and notation ##\langle x, y \rangle## denote standard inner product on ##R^3##.
 Relevant Equations:
 fluid dynamic, euler's equation of the motion
$$\frac{Du}{Dt} = \frac{\nabla p}{\rho}  \nabla \chi$$I rewrite the Euler equation for incompressible fluid using suffix notation
$$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$what theorems applies to the problem?
$$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$what theorems applies to the problem?