Question

In fully developed laminar flow in a circular pipe, the velocity at \(R / 2\) (midway between the wall surface and the centerline) is measured to be $6 \mathrm{~m} / \mathrm{s}$. Determine the velocity at the center of the pipe. Answer: \(8 \mathrm{~m} / \mathrm{s}\)

Short answer

Answer: The fluid velocity at the center of the pipe is 8 m/s.

Step-by-step solution

Identify the velocity profile for laminar flow in a pipe

For laminar flow in a circular pipe, the velocity profile is parabolic and can be expressed as: \(u(r) = U_{max}\left(1 - \frac{r^2}{R^2}\right)\) where \(u(r)\) is the fluid velocity at a radial distance \(r\) from the center of the pipe, \(U_{max}\) is the maximum velocity at the center of the pipe (i.e., when \(r = 0\)), and \(R\) is the pipe radius.

Set up the equation with known values

We know that the velocity at \(R/2\) is 6 m/s. Let's plug in the known values: \(6 = U_{max}\left(1 - \frac{(R/2)^2}{R^2}\right)\)

Simplify the equation and solve for \(U_{max}\)

Simplifying the fraction in the equation, we get: \(6 = U_{max}\left(1 - \frac{1}{4}\right) \Rightarrow 6 = U_{max}\left(\frac{3}{4}\right)\) Now, solving for \(U_{max}\) (centerline velocity): \(U_{max} = \frac{6}{(3/4)} = 8~\mathrm{m} / \mathrm{s}\) Hence, the velocity at the center of the pipe is \(8 \mathrm{~m} / \mathrm{s}\).