Classical fluid mechanics: continuity equation, Euler equations, Navier-Stokes, Bernoulli's principle, Reynolds number, vorticity. The substrate for reactor coolant flow, aerodynamics, oceanography, and atmospheric science.
fluid-dynamics
Continuity equation
Local conservation of mass: ∂ρ/∂t + ∇·(ρv) = 0. Any increase in density at a point equals the net inflow.
Euler equations (inviscid flow)
Momentum equation for an inviscid fluid: ρ(∂v/∂t + v·∇v) = −∇p + ρg. Newton's second law written for a fluid element.
Navier-Stokes equations
Euler equations + viscous stress: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + ρg. Existence and smoothness of 3D solutions is a Millennium Problem.
Bernoulli's principle
Along a streamline in steady, incompressible, inviscid flow: ½ρv² + ρgh + p = const. Energy conservation applied to fluid flow.
Reynolds number
Re = ρvL/μ = inertial/viscous force ratio. Low Re: laminar. High Re: turbulent. Re is the single most useful dimensionless number in fluid…
Turbulence (open problem)
Turbulent flow — despite being governed by the deterministic Navier-Stokes equations — has no closed-form predictive theory. Kolmogorov's…
Vorticity ω = ∇×v
Local rotation rate of a fluid element; ω = ∇×v. Evolves via the vorticity equation; vortex lines are advected with the flow in barotropic…
Kelvin's circulation theorem
In an inviscid, barotropic fluid under conservative forces, the circulation Γ = ∮_C v·dl around any material loop C is constant in time.
Kolmogorov K41 theory
Statistical theory of homogeneous isotropic turbulence: in the inertial range, the energy spectrum E(k) ∝ ε^{2/3} k^{−5/3}; structure…
Compressible flow
Flow regime where ρ varies with p; Mach number M = v/c_s distinguishes subsonic (M<1) from supersonic (M>1); shocks form when M crosses 1.
Boundary layer
Thin viscous region adjacent to a solid surface where velocity transitions from no-slip to free-stream; governed by Prandtl's equations;…
Mach number
Dimensionless ratio of flow speed u to local sound speed c: Ma = u/c. Demarcates flow regimes: subsonic (Ma < 1), transonic (≈ 1),…
Reynolds transport theorem
d/dt ∫_V ρφ dV = ∫_V ∂(ρφ)/∂t dV + ∫_∂V ρφ(v·n) dA; converts Lagrangian to Eulerian descriptions.
Vorticity equation
Dω/Dt = (ω·∇)v + ν∇²ω (incompressible); vortex stretching in 3D key to turbulence; 2D conservation of ω.
Kelvin–Helmholtz instability
Shear between parallel streams of different density unstable at all k; σ = k U₁U₂ρ₁ρ₂/(ρ₁+ρ₂)²; clouds, Jupiter's bands.
Rayleigh–Taylor instability
Heavy fluid over light in gravity: σ = √(kg(ρ₂-ρ₁)/(ρ₂+ρ₁)); mushroom plumes; SN ejecta, ICF capsules.
Taylor–Couette flow & instability
Flow between rotating cylinders; Taylor number T > T_c causes toroidal vortices; route to turbulence; analog to MRI.
Rayleigh–Bénard convection
Convection onset at Ra_c ≈ 1708 for rigid-rigid; hexagonal/roll patterns; paradigm for pattern formation and chaos.
Kolmogorov K41 theory
Inertial range E(k) ~ ε^(2/3) k^(-5/3); universal statistics; intermittency corrections (multifractal She–Leveque).
Richardson energy cascade
'Big whorls have little whorls...'; energy transfers from large eddies to dissipative scale η ~ (ν³/ε)^(1/4); Kolmogorov scale.
2D turbulence inverse cascade
2D flows conserve enstrophy; energy cascades to large scales (inverse), enstrophy to small; relevant for atmosphere, oceans.
Prandtl boundary layer
Thin layer δ ~ √(νx/U) where viscous effects concentrate at high Re; Blasius similarity solution; separation under adverse ∂P/∂x.
Drag crisis
Sudden drop in C_D at Re ~ 3×10⁵ for sphere as boundary layer transitions to turbulent, delaying separation.
Potential flow and complex analysis
Inviscid, irrotational: v=∇φ, ∇²φ=0; conformal maps (Joukowski airfoil); D'Alembert paradox: no drag in pure potential.
Kutta–Joukowski theorem
Lift per unit span L' = ρ U Γ; circulation Γ fixed by Kutta condition (smooth trailing-edge flow).
Bernoulli's equation
½v²+P/ρ+gz = const along streamline (steady, inviscid, incompressible); basis of Venturi, Pitot, airfoil lift intuition.
Compressible flow & normal shocks
Isentropic relations A*/A = M[(2+(γ-1)M²)/(γ+1)]^... ; normal shock jump: entropy increases, M>1→M<1; oblique shocks (wedge).
Stokes flow (low Re)
∇²u = ∇P/μ; linear, reversible; scallop theorem — no net motion from reciprocal stroke; swimming microorganisms.
Sedov–Taylor blast wave
Self-similar expansion R(t) = (E/ρ)^(1/5) t^(2/5); applies to SN remnants, atomic bomb test (Taylor 1950).
RANS & closure models
Decompose flow into mean+fluctuation; Reynolds stress -ρ⟨u'u'⟩ requires closure (k-ε, k-ω, RSM); engineering CFD workhorse.
DNS, LES, and turbulence modeling
DNS resolves all scales (grid N³ ~ Re^(9/4)); LES resolves large, models subgrid (Smagorinsky); hybrid RANS-LES for aerodynamics.
Non-Newtonian fluids & rheology
Shear-thinning (power-law), Bingham plastics, viscoelastic (Maxwell, Oldroyd-B); Deborah number De=τ/t_flow; polymer, blood, magma.