Expected Flow Velocity

The first step to solve this is to compute the volumetric flow rate through the cross-section. Consider an elementary ring of radius $r$ and thickness $dr$ . The elementary volumetric flow rate through this area is:

$dQ = (2 \pi r \ dr)\mathrm{v}(r)$ $\implies dQ = 2 \pi k\left(r - \frac{r^7}{R^6}\right) \ dr$

The total flow rate through this cross-section is: $Q = 2 \pi k\int_{0}^{R} \left(r - \frac{r^7}{R^6}\right) \ dr$ $Q = \frac{3 \pi R^2k}{4}$

The expected, or average velocity can be found by dividing the total velocity flux, or volumetric flow rate by the cross-section area. This can also be thought of in terms of the continuity equation as follows:

$\pi R^2\mathrm{v}_{\mathrm{exp}} = \frac{3 \pi R^2k}{4}$ $\implies \boxed{\mathrm{v}_{\mathrm{exp}}= \frac{3 k}{4}}$

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The volumetric flow rate can be defined as:

$Q = \int_{A} \vec{\mathrm{v}} \cdot d\vec{A}$