Question

A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.

Answer 1

Denote the cylindrical surface by $S$, and its interior by $R$. By the divergence theorem, the integral of $\vec v$ across $S$ (the outward flow of the fluid) is equal to the integral of the divergence of $\vec v$ over the space it contains, $R$:

$\displaystyle\iint_S\vec v\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec v)\,\mathrm dV$

The given velocity vector has divergence

$\vec v=z\,\vec\imath+y^2\,\vec\jmath+x^2\,\vec k\implies\nabla\cdot\vec v=\dfrac{\partial(z)}{\partial x}+\dfrac{\partial(y^2)}{\partial y}+\dfrac{\partial(x^2)}{\partial z}=2y$

Then the total outward flow is

$\displaystyle\iiint_R2y\,\mathrm dV$

Converting to cylindrical coordinates gives the integral

$\displaystyle2\int_0^{2\pi}\int_0^3\int_0^4r^2\sin\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{0}$