Water flows steadily through a pipe of length L and radius R=75 mm. Calculate the uniform inlet

step 1 In this question we have to find the u which is the uniform inlet velocity. So first we can use

Water flows steadily through a pipe of length L and radius...

Water flows steadily through a pipe of length $L$ and radius $R=75 \mathrm{mm}$. Calculate the uniform inlet velocity, $U$, if the velocity distribution across the outlet is given by
\[
u=u_{\max }\left[1-\frac{r^{2}}{R^{2}}\right]
\]
and $u_{\max }=3 \mathrm{m} / \mathrm{s}$

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Key Concepts

Continuity Equation

The continuity equation states that mass is conserved in a fluid flow, meaning that the mass flow rate at any cross-section of a steady flow remains constant. This concept is crucial in determining relationships between inlet and outlet velocity profiles, as the product of area and velocity must remain the same along the length of the pipe for incompressible flows.

Velocity Profile Integration

When a velocity distribution is non-uniform across a cross-section, the average or uniform velocity is found by integrating the local velocity over the entire area and then dividing by the cross-sectional area. This process converts a known distribution, such as a parabolic profile common in laminar flow, into an effective or average velocity that represents the flow rate across the section.

Integration in Cylindrical Coordinates

Evaluating flow in cylindrical geometries like pipes requires integration in polar coordinates, where the differential area element is expressed as r dr d?. This approach simplifies the integration of radially dependent functions, allowing the calculation of quantities such as the average velocity by summing contributions from all radial elements in the circular cross-section.

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