Use cylindrical coordinates $(R, \varphi, z)$ $$ \mathbf{u} = (0, 0, u_z(R)) $$

$\varphi, R, z$ momentum equations $$ 0 = \frac{\partial p}{\partial \varphi}, \quad 0 = \frac{\partial p}{\partial R}, \quad 0 = -\frac{dp}{dz} + \frac{\mu}{R} \frac{d}{dR} \left( R \frac{du_z}{dR} \right) $$ $$ \Rightarrow u_z(R) = \frac{R^2}{4\mu} \frac{dp}{dz} + A \ln R + B $$

Boundary conditions

• @ $R = a:\; u_z = 0$
• @ $R = 0:\; u_z$ is bounded

$$ \Rightarrow u_z(R) = \frac{R^2 - a^2}{4\mu} \frac{dp}{dz} $$

Volume flow rate $$ Q = \int_0^a u(R)\, 2\pi R \, dR = -\frac{\pi a^4}{8\mu} \frac{dp}{dz} $$

Average velocity $$ V = \frac{Q}{\pi a^2} = -\frac{a^2}{8\mu} \frac{dp}{dz} $$