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=(\partial z)/(\partial x)+(\partial y^2)/(\partial y)+(\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.