Fluid Mechanics

Pressure grows linearly with depth (p = ρgh), the force on a submerged surface is ρg·h_c·A acting at the centre of pressure below the centroid, and buoyancy equals the weight of displaced fluid.

$\frac{P}{\rho g} + \frac{V^{2}}{2g} + z = constant$ (Modi & Seth)

Apply the impulse-momentum principle ΣF = ρQ(V_out − V_in) to a control volume, being careful with the vector directions of inlet and outlet velocities, to find forces on bends, nozzles and vanes.

Compute major friction loss from Darcy-Weisbach h_f = f(L/D)(V²/2g) with f from the Moody chart, add minor losses KV²/2g, and remember series pipes share discharge while parallel pipes share head loss.

Use the Buckingham π theorem to reduce the variables to (n − k) dimensionless groups, then achieve dynamic similarity between model and prototype by matching the governing group — Reynolds number for closed-conduit flow, Froude number for free-surface flow.