Syncopore Dynamics: Comprehensive Equation Matrix

Navier-Stokes Equations

∂v/∂t + (v ⋅ ∇)v = -(1/ρ)∇p + ν∇²v + f

Navier-Stokes Equation Diagram

Continuity Equation

∇ ⋅ v = 0

Continuity Equation Diagram

Vorticity Equation

∇ × v = ω

Vorticity Equation Diagram

Cylindrical Coordinates

v = (vr, vθ, vz), ∇ = (∂/∂r, 1/r ∂/∂θ, ∂/∂z)

Cylindrical Coordinates Diagram

Laser Energy Input

E = ∫ P(t) dt, P(t) = I ⋅ A

Laser Energy Input Diagram

Femtosecond Pulse Dynamics

Δt = λ/c, Epulse = P/f

Femtosecond Pulse Diagram

Vortex Strength

Γ = ∮C v ⋅ dl

Vortex Strength Diagram

Reynolds Number

Re = ρuL/μ

Reynolds Number Diagram

Photoacoustic Effect

p(t) = β(dT/dt), β = 1/(ρc²)

Photoacoustic Effect Diagram

Molecular Transport

J = -D∇c, D = kB T / (6πηr)

Molecular Transport Diagram

Cassie-Baxter Wettability

cos(θ_c) = φ_s cos(θ_y) + φ_l - 1

Cassie-Baxter Wettability Diagram

Diffusion Coefficient

D = kB T / (6πηr)

Diffusion Coefficient Diagram

Energy Conservation

ΔE = mc²

Energy Conservation Diagram

Entropy Equation

S = kB ln Ω

Entropy Equation Diagram

Thermal Expansion

ΔL = αL₀ΔT

Thermal Expansion Diagram

Ideal Gas Law

PV = nRT

Ideal Gas Law Diagram