Question

A fluid has density 860 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 = 25, 0 ≤ z ≤ 1.

Answer 1

You can use the divergence theorem:

$\vec v=z\,\vec\imath+y^2\,\vec\jmath+x^2\,\vec k$

has divergence

$\mathrm{div}\vec v=\dfrac{\partial z}{\partial x}+\dfrac{\partial y^2}{\partial y}+\dfrac{\partial x^2}{\partial z}=2y$

Then the rate of flow out of the cylinder (call it R) is

$\displaystyle\iint_{\partial R}\vec v\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec v\,\mathrm dV$

(by divergence theorem)

$=\displaystyle2\int_0^{2\pi}\int_0^5\int_0^1r^2\sin\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$

(after converting to cylindrical coordinates)

whose value is 0.