A fluid flows in a horizontal pipe of diameter d (Fig. P6.74). The flow is suddenly increased to an average velocity V. Show that the appropriate Navier–Stokes equation, using the coordinates shown, simplifies to
\rho \frac{\partial u}{\partial t} =-\frac{\partial p}{\partial x} +\mu \left\lgroup\frac{\partial ^2u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r} \right\rgroupNormalize this equation using characteristic velocity V, length d, and time (a) d/V and (b) d^2/ν. Identify any dimensionless parameters that result.