PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity
Fluids and Continuum Physics, Ch Dimensional analysis, dynamical similarity and the dimensionless groups
Problem
Four flows are described below. For each, decide which single dimensionless group governs the behaviour named, say which two influences that group compares, and compute its value. Use these data: water ρ=1000 kg m−3, surface tension with air γ=0.072 N m−1; air ρ=1.2 kg m−3, μ=1.8×10−5 Pa s, with a small squeeze measured to travel through it at 343 m s−1; g=9.8 m s−2. Take the length named in each description as the length scale, and treat each flow as incompressible except where the group you choose says otherwise.
((a)) A fire hose sends a smooth water jet of diameter 20 mm into the air at 25 m s−1; the question is whether surface tension keeps its surface smooth until it pinches off gently into drops about its own size, or inertia tears it apart into fine spray. Use the density of the water in the jet.
((b)) A spherical weather balloon of diameter 0.90 m rises through still air at 4.0 m s−1; the question is what kind of drag acts on it.
((c)) A tidal bore advances up an estuary of uniform depth 2.0 m at 5.0 m s−1; the question is whether the front can outrun the long surface waves of that depth, whose speed is gh.
((d)) The tip of a turbine blade moves through air at 260 m s−1; the question is whether the air must be treated as squeezable.