Navier-Stokes Equations
∂v/∂t + (v ⋅ ∇)v = -(1/ρ)∇p + ν∇²v + f
Continuity Equation
∇ ⋅ v = 0
Vorticity Equation
∇ × v = ω
Cylindrical Coordinates
v = (vr, vθ, vz), ∇ = (∂/∂r, 1/r ∂/∂θ, ∂/∂z)
Laser Energy Input
E = ∫ P(t) dt, P(t) = I ⋅ A
Femtosecond Pulse Dynamics
Δt = λ/c, Epulse = P/f
Vortex Strength
Γ = ∮C v ⋅ dl
Reynolds Number
Re = ρuL/μ
Photoacoustic Effect
p(t) = β(dT/dt), β = 1/(ρc²)
Molecular Transport
J = -D∇c, D = kB T / (6πηr)
Cassie-Baxter Wettability
cos(θ_c) = φ_s cos(θ_y) + φ_l - 1
Diffusion Coefficient
D = kB T / (6πηr)
Energy Conservation
ΔE = mc²
Entropy Equation
S = kB ln Ω
Thermal Expansion
ΔL = αL₀ΔT
Ideal Gas Law
PV = nRT