lubrication-theory

name	lubrication-theory
description	Lubrication theory and tribology — Reynolds equation derivation and solutions (journal bearing, slider bearing, squeeze film), Sommerfeld number, minimum film thickness (Ocvirk, Sommerfeld solutions), elasto-hydrodynamic lubrication (EHL) film thickness (Hamrock-Dowson), Stribeck curve (boundary/mixed/hydrodynamic regimes), viscosity and viscosity-pressure (Barus equation), journal bearing design, thrust bearing, gear tooth lubrication (flash temperature), ISO VG viscosity grades, oil film parameter Λ, surface roughness effects, ASTM D445 (kinematic viscosity), ISO 15312 (bearing speed rating).
metadata	{"priority":7,"promptSignals":{"phrases":["lubrication","Reynolds equation","EHL","Sommerfeld number","hydrodynamic bearing"],"minScore":3}}

Lubrication Theory — Complete Skill

Reynolds Equation

Derivation and Form

Reynolds equation (full form for Newtonian fluid, incompressible, thin film): ∂/∂x[ρh³/(12μ) × ∂p/∂x] + ∂/∂z[ρh³/(12μ) × ∂p/∂z] = (U₁+U₂)/2 × ∂(ρh)/∂x + ∂(ρh)/∂t

Simplified (incompressible, steady, 2D): ∂/∂x[h³/μ × ∂p/∂x] + ∂/∂z[h³/μ × ∂p/∂z] = 6 × (U₁ + U₂) × ∂h/∂x + 12 × ∂h/∂t

Where: h = film thickness [m]; p = pressure [Pa]; μ = dynamic viscosity [Pa·s] U₁, U₂ = surface velocities in x-direction [m/s]; z = axial coordinate ∂h/∂t = squeeze film velocity [m/s] (approach speed)

Physical interpretation: Left side: pressure-driven (Poiseuille) flow in x and z directions Right side (first term): wedge action — converging film geometry generates pressure Right side (second term): squeeze film — surfaces approaching each other build pressure

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Slider Bearing (Inclined Plane)

Pressure Distribution

Inclined plane slider (1D, infinite width): h(x) = h₁ − (h₁ − h₂) × x/L [linearly converging; h₁ = inlet; h₂ = outlet; L = bearing length] Taper ratio n_t = h₁/h₂

Reynolds equation (1D, steady): d/dx[h³ × dp/dx] = 6μU × dh/dx Solution: p(x) = 6μUL × (h − h_m) / (h_m × h²) [h_m = film thickness where dp/dx = 0]

Load capacity per unit width: W/b = 6μUL²/(h₂²) × [ln(n_t)/(n_t−1) − 2(n_t−1)/(n_t+1)²] [W in N/m; b = width] Optimal taper for max load: n_t_optimal ≈ 2.2

Friction coefficient: f = F_friction/W [F_friction from viscous shear; f ≈ 0.001–0.01 for hydrodynamic regime]

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Journal Bearing

Sommerfeld Number and Minimum Film Thickness

Sommerfeld number: S = (μ × N × L × D) / W × (R/c)² [dimensionless] Or: S = (η × N × (R/c)²) / (P) [η = viscosity; N = rev/s; P = load per unit projected area = W/(L×D); c = radial clearance; R = journal radius] Alternative: S = μ × n × (R/c)² / P [n = rev/s; μ in Pa·s; P in Pa; R, c in same units]

Minimum film thickness (h_min): From Sommerfeld solution (long bearing, L/D >> 1) or Ocvirk (short bearing, L/D << 1): h_min = c × (1 − ε) [ε = eccentricity ratio = e/c; e = journal center displacement] S relates to ε by Sommerfeld charts; for S = 0.1: ε ≈ 0.85; S = 1.0: ε ≈ 0.2

Ocvirk short-bearing approximation (L/D ≤ 0.5): p_max ≈ (3μ × U × L²)/(2h_min³) × ε/(1−ε²)^(3/2) W = π × μ × ω × R × L³ × ε / [4c² × (1−ε²)^(1/2)] [ω = angular velocity [rad/s]]

Design criterion: h_min ≥ R_a1 + R_a2 (surface roughness sum) × safety factor (3–5) Typically h_min ≥ 2–5 μm for precision; 10–25 μm for industrial

Film parameter Λ (Lambda): Λ = h_min / √(R_q1² + R_q2²) [R_q = RMS roughness of bearing surfaces] Λ > 3: full hydrodynamic (no asperity contact) 1 < Λ < 3: mixed lubrication (partial asperity contact; wear occurs) Λ < 1: boundary lubrication (full asperity contact; additives required)

Bearing stability — Whirl: Half-frequency whirl instability if S > S_critical (depends on L/D ratio and bearing clearance) Fixed pad bearings: prone to whirl at high speed; tilting pad bearings: stable at all speeds (zero cross-coupling)

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Elastohydrodynamic Lubrication (EHL)

EHL Film Thickness (Hamrock-Dowson)

EHL applies when: contact pressure high enough to elastically deform surfaces (gears, rolling bearings, cams) Contact pressures: 1–4 GPa → significant viscosity increase and elastic deformation of surfaces

Dimensionless groups: U* = μ₀ × u_s / (E' × R') [speed parameter; u_s = sum velocity; E' = reduced modulus; R' = reduced radius] G* = α × E' [material parameter; α = pressure-viscosity coefficient [Pa⁻¹]] W* = W / (E' × R'²) [load parameter; W = normal load per unit width for line contact]

Minimum film thickness (Hamrock-Dowson, point contact): H_min = 3.63 × U*^0.68 × G*^0.49 × W*^(−0.073) × (1 − e^(−0.68k)) h_min = H_min × R' [k = ellipticity parameter]

Hamrock-Dowson (line contact, infinite width): H_min = 2.65 × U*^0.70 × G*^0.54 × W*^(−0.13) [Dowson-Higginson formula] h_min = H_min × R'

Barus viscosity-pressure equation: μ(p) = μ₀ × exp(α × p) [μ₀ = atmospheric viscosity; α = pressure-viscosity coeff [Pa⁻¹]] Typical: α = 10–30 GPa⁻¹ for mineral oils At p = 1 GPa: μ/μ₀ = exp(10×10⁻⁹ × 10⁹) = exp(10) ≈ 22,000 [dramatic viscosity increase → enables EHL film]

Reduced modulus E': 1/E' = (1−ν₁²)/E₁ + (1−ν₂²)/E₂ [plane strain combined modulus] Steel on steel: E₁ = E₂ = 207 GPa; ν = 0.3; E' = 226 GPa

Equivalent radius R': R' = R₁ × R₂ / (R₁ + R₂) [parallel cylinders or equivalent sphere for Hertz contact]

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Stribeck Curve

Lubrication Regimes

Stribeck number: Sf = μ × N / P [μ = viscosity; N = speed; P = contact pressure] or Gumbel number: G = η × u / (P × h)

Friction vs. Stribeck number: Boundary lubrication (Sf < 10⁻⁸): f ≈ 0.1–0.4 (metal contact); additives dominate Mixed lubrication (10⁻⁸ < Sf < 10⁻⁶): f = 0.01–0.1 (asperity + film) Hydrodynamic (Sf > 10⁻⁶): f ≈ 0.001–0.01 (full film); friction increases with speed (viscous drag) Transition visible as characteristic U-shape (Stribeck curve minimum at mixed-to-hydrodynamic transition)

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Viscosity and ISO VG Grades

Kinematic Viscosity

Kinematic viscosity: ν = μ / ρ [m²/s; measured in cSt = mm²/s by ASTM D445 glass capillary viscometer] Viscosity index (VI): measure of viscosity change with temperature; high VI = small change; mineral oil VI ≈ 90–120; synthetic PAO VI ≈ 140–160

ISO VG (viscosity grade) system (ISO 3448):

ISO VG	Kinematic viscosity at 40°C [cSt]	Range
32	32	28.8–35.2
46	46	41.4–50.6
68	68	61.2–74.8
100	100	90–110
150	150	135–165
220	220	198–242
320	320	288–352

Selection rule: higher speed → lower viscosity; higher load → higher viscosity; check h_min ≥ Λ_min × √(R_q1² + R_q2²)

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Standards

Standard	Scope
ASTM D445	Kinematic viscosity of transparent/opaque liquids
ISO 3448	Lubricant viscosity classification (ISO VG)
ASTM D2270	Viscosity index calculation
DIN 51517	CLP gear oil specification
ISO 15312	Rolling bearing speed rating (lubrication-dependent)
ASTM D2783	Load-carrying capacity of lubricating fluids (4-ball EP test)
ISO 281	Rolling bearing life calculation (lubrication factor a₂₃)
AGMA 9005-F16	Industrial gear drive lubrication

Output

Provide: contact geometry (bearing type: journal/slider/thrust/gear; surfaces: material, R₁ [mm], R₂ [mm]; R' = R₁R₂/(R₁+R₂) [mm]; E' [GPa] from reduced modulus formula), operating conditions (W [N]; N [rpm]; u_s [m/s]; contact pressure P = W/(L×D) [MPa] for journal), oil selection (ISO VG grade; μ₀ at operating T [Pa·s] from viscosity-temperature; α [GPa⁻¹]), lubrication regime (journal bearing: S = μnR²/(Pc²) [Sommerfeld]; h_min = c(1−ε) [μm]; or EHL: U*, G*, W*; h_min from Hamrock-Dowson [μm]), film parameter (Λ = h_min/√(Rq1²+Rq2²); regime: full-film/mixed/boundary; risk level), temperature rise (bulk oil temperature rise from viscous dissipation; flash temperature from Blok equation if gear tooth), design check (h_min ≥ 3×Ra for full-film?; bearing stability: not in whirl zone?; journal bearing L/D range; operating clearance c/R [ppm]), recommendations (ISO VG upgrade if Λ < 3; EP additive if boundary regime; speed reduction or surface finish improvement), and applicable standard (AGMA 9005 for gears; ISO 281 for rolling bearings; DIN 51517 for industrial gearboxes).