Functions§
- apparent_
weight - Apparent weight in fluid: W_app = W - F_b = mg - ρ_fluid * V * g
- archimedes_
number - Archimedes number: Ar = g × d³ × ρf × (ρp - ρf) / μ²
- bernoulli_
pressure - Bernoulli’s equation: P1 + 0.5ρv1^2 + ρgh1 = P2 + 0.5ρv2^2 + ρgh2 Returns P2 given all other quantities.
- bond_
number - Bond number: Bo = Δρ × g × L² / σ (gravity vs surface tension)
- buoyancy_
velocity - Characteristic buoyancy velocity: v = √(gβΔTL)
- buoyant_
force - Buoyant force (Archimedes’ principle): F_b = ρ_fluid * V_displaced * g
- capillary_
rise - Capillary rise: h = 2 * γ * cos(θ) / (ρ * g * r)
- circulation
- Circulation (uniform vorticity): Γ = ω × A
- continuity_
velocity - Continuity equation: A1 * v1 = A2 * v2 → v2 = A1 * v1 / A2
- darcy_
friction_ factor_ laminar - Darcy friction factor for laminar pipe flow: f = 64/Re
- darcy_
weisbach_ head_ loss - Darcy-Weisbach head loss: h_L = f × (L/D) × v²/(2g)
- drag_
force - Drag force: F_d = 0.5 * C_d * ρ * A * v^2
- dynamic_
pressure - Dynamic pressure: q = ½ρv²
- flow_
rate - Volume flow rate: Q = A * v
- fraction_
submerged - Fraction of object submerged (floating): f = ρ_object / ρ_fluid
- froude_
number - Froude number: Fr = v / √(gL) (inertia vs gravity in free surface flow)
- hydraulic_
diameter - Hydraulic diameter: D_h = 4A/P
- hydrostatic_
pressure - Hydrostatic pressure: P = ρ * g * h
- isentropic_
pressure_ ratio - Isentropic pressure ratio: P/P₀ = (1 + (γ-1)/2 × M²)^(-γ/(γ-1))
- isentropic_
temperature_ ratio - Isentropic temperature ratio: T/T₀ = (1 + (γ-1)/2 × M²)^(-1)
- kinematic_
viscosity - Kinematic viscosity: ν = μ/ρ
- kutta_
joukowski_ lift - Kutta-Joukowski lift per unit span: L = ρ × V × Γ
- mach_
number - Mach number: M = v / a
- marangoni_
number - Marangoni number: Ma = -(dσ/dT) × L × ΔT / (μ × α)
- mass_
flow_ rate - Mass flow rate: ṁ = ρ * A * v
- natural_
convection_ nu_ horizontal_ hot - Natural convection Nusselt number for hot horizontal plate facing up: Nu = 0.54 × Ra^(1/4), valid for 10⁴ ≤ Ra ≤ 10⁷
- natural_
convection_ nu_ vertical - Churchill-Chu correlation for natural convection on a vertical plate (air, Pr≈0.71): Nu = (0.825 + 0.387 × Ra^(1/6) / 1.1936)²
- pascal_
force - Pascal’s principle: F2 = F1 * (A2 / A1)
- peclet_
number - Peclet number: Pe = vL/α (advection vs diffusion)
- poiseuille_
flow_ rate - Poiseuille’s law (volume flow rate through a pipe): Q = π * r^4 * ΔP / (8 * μ * L)
- pressure
- Pressure: P = F / A
- pressure_
gradient_ pipe - Pressure gradient in a pipe (Poiseuille inverse): dP = 8μLQ/(πr⁴)
- reynolds_
number - Reynolds number: Re = ρ * v * L / μ
- stagnation_
pressure - Stagnation pressure: P₀ = P + q
- stokes_
drag - Stokes’ drag (low Reynolds number): F = 6π * μ * r * v
- surface_
tension_ force - Surface tension force along a line: F = γ * L
- terminal_
velocity - Terminal velocity: v_t = sqrt(2 * m * g / (ρ * A * C_d))
- thermal_
conductivity_ gas - Sutherland’s law for thermal conductivity (same form as viscosity)
- thermal_
expansion_ coefficient_ ideal_ gas - Thermal expansion coefficient for ideal gas: β = 1/T
- torricelli_
velocity - Torricelli’s theorem: v = sqrt(2 * g * h)
- total_
pressure - Total pressure at depth: P = P_atm + ρ * g * h
- venturi_
velocity - Venturi effect velocity from pressure difference: v2 = sqrt(2 * (P1 - P2) / (ρ * (1 - (A2/A1)^2)))
- viscosity_
sutherland - Sutherland’s law for viscosity: μ = μ₀ × (T/T₀)^(3/2) × (T₀ + S)/(T + S)
- vorticity_
2d - Vorticity in 2D: ω = ∂v_y/∂x - ∂v_x/∂y
- weber_
number - Weber number: We = ρv²L / σ (inertia vs surface tension)