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egoroff_w [7]
3 years ago
8

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 co

mponents of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.
Mathematics
1 answer:
mixas84 [53]3 years ago
6 0

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}

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