Model-free analysis of physical equations by tracking the units of measurement (M, L, T, Θ, I, N, J) of each variable. Core theorem: Buckingham 1914 π-theorem — a physical relation F(q_1,…,q_n)=0 among n dimensioned quantities with…
dimensional-analysis
Buckingham 1914 π-theorem: n−r independent dimensionless groups
Buckingham 1914 (Phys. Rev. 4:345) synthesis and generalisation of Rayleigh's 1877 earlier work: any physical equation F(q_1,…,q_n) = 0…
Reynolds-number similarity: laminar ↔ turbulent transition at Re ~ 2300
Reynolds 1883 (Phil. Trans. Roy. Soc. 174:935) injected dye into pipe flow and identified a threshold flow speed above which laminar…
Rayleigh-Bénard convection onset at dimensionless threshold Ra_c
Rayleigh 1916 (Phil. Mag. 32:529) linear-stability analysis of a horizontal fluid layer heated from below identifies a dimensionless…
Pipe-flow π-count: 5 variables, rank 3 → 2 π-groups
Textbook application of the Buckingham π-theorem to pipe-flow pressure drop. Variables and their dimensions: Δp [ML⁻¹T⁻²], ρ [ML⁻³], V…
Reynolds-number value at reference state and its laminar/turbulent ratio
Elementary dimensional check of the Reynolds-number definition at a chosen reference state. With V = L = 1 (consistent units) and ν =…
Rayleigh-Bénard free-free critical Rayleigh number Ra_c = 27π⁴/4
Closed-form result for the free-free (stress-free) Rayleigh-Bénard problem. Rayleigh 1916's linear-stability analysis of the Boussinesq…