AtlasCalcsUtility Calculation Suite
2026-07-30 10 min

Reynolds Lubrication Calculator

Apply the Reynolds lubrication equation to analyze pressure distribution in lubricated contacts. Calculate load capacity and friction for slider bearings and journal bearings.

Calculator

Unit System
Bearing Length L (mm)
Bearing Width B (mm)
Inlet Film h_in (um)
Viscosity mu (Pa-s)
Sliding Speed U (m/s)
Minimum Film h_min (um)

Results

Normal

Load Capacity W

2070784.3N

Maximum load the lubricant film can support

Max Pressure p_max

83.333MPa

Peak pressure in the lubricant film

Friction Force F_f

304.7N

Shear force resisting sliding

Coefficient of Friction

0.0001-

Hydrodynamic friction coefficient

Convergence Ratio K

5.00-

Ratio of inlet to minimum film thickness

Visualizations

Results Overview

2,070,784NLoad Capacity W
83.3NMax Pressure p_max
304.7NFriction Force F_f
0NCoefficient of Friction
5NConvergence Ratio K

Formula

The Reynolds equation describes pressure distribution in a thin lubricant film between surfaces in relative motion.

d/dx(h^3/mu x dp/dx) + d/dz(h^3/mu x dp/dz) = 6U x dh/dx

Alternative Form

W = 6 mu U B L^2 / h_min^2 x [ln(K) - 2(K-1)/(K+1)]

Variables

SymbolNameUnitDescription
hFilm ThicknessmLocal lubricant film thickness
muDynamic ViscosityPa-sLubricant dynamic viscosity
pPressurePaHydrodynamic pressure in the film
USliding Velocitym/sRelative velocity of the sliding surface

Explanation

Derived from Navier-Stokes with lubrication approximations: thin film, negligible inertia, no body forces, no slip. Left side = Poiseuille flow, right side = Couette flow.

Derivation

Start from Navier-Stokes: (1) h << L,B, (2) Re << 1, (3) p constant across film, (4) Newtonian fluid, (5) no-slip BCs. Integrate twice for velocity, then integrate with continuity.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Define bearing geometry and convergence ratio.

h_in:50.0 um
h_min:10.0 um
Result: K = 5.00
Step 2

Solve Reynolds equation for pressure distribution.

dp/dx = 6 mu U (h-h*)/h^3
mu:0.050 Pa-s
U:5.00 m/s
Result: p_max = 83.333 MPa
Step 3

Integrate pressure over area for load capacity.

W = integral(p dA)
Result: W = 2070784.3 N
Step 4

Calculate friction force from shear stress.

F_f = integral(tau dA)
Result: mu = 0.0001
Final Answer
2070784.3

Worked Examples

See how the calculator works with different scenarios.

Scenario

L=100mm, B=50mm, h_in=50um, h_min=10um, mu=0.05 Pa-s, U=5 m/s.

Inputs

L:100mm
B:50mm
h_in:50um
h_min:10um
mu:0.05 Pa-s
U:5m/s

Calculation

K=5. W = 6*0.05*5*0.05*0.1^2/(10^-5)^2 * [ln(5)-2(4/6)]

Result

W=22,050N, p_max=13.2MPa, friction coefficient=0.0045

Interpretation

22kN load with extremely low friction, demonstrating full-film hydrodynamic lubrication advantage.

Understanding Your Result

Your Result: Load Capacity W: 2070784.3 N

Typical range: p_max=1-20MPa, mu=0.001-0.1, load=1-100kN. Journal bearings: Sommerfeld No=0.01-1.0.

What This Means

The Reynolds equation mathematically describes how a thin fluid film supports load through hydrodynamic pressure generation via the wedge effect.

Why It Matters

It is the foundation of bearing design, predicting load capacity, friction, flow, and heat generation.

Professional Interpretation

Assumes laminar Newtonian flow with no slip. For high-speed use turbulent corrections. For EHL couple with pressure-viscosity and elasticity.

Recommendations

  • •Use Sommerfeld number for journal bearing design
  • •Design convergence ratio K between 2 and 3 for optimal load
  • •Verify laminar: Taylor number < 41.3 for journal bearings
  • •Account for thermal viscosity reduction at high speeds

Educational Content

Introduction

The Reynolds lubrication equation, derived by Osborne Reynolds in 1886, governs fluid film lubrication pressure distribution.

Concept Explanation

Combines Poiseuille (pressure-driven) and Couette (drag-driven) flow. The converging gap wedge generates hydrodynamic pressure that supports load.

Applications

▸Hydrodynamic journal bearings
▸Thrust bearings
▸Slider bearings in machine tools
▸Squeeze film dampers
▸Air bearings
▸Biomedical joint lubrication

Benefits

  • •Predicts load capacity without empiricism
  • •Optimizes bearing geometry
  • •Enables friction/power loss estimation
  • •Provides film thickness for reliability

Limitations

  • •Assumes laminar Newtonian flow
  • •Neglects thermal effects
  • •Constant viscosity across film
  • •No surface roughness effects

Best Practices

  • •Use for preliminary design
  • •Validate with CFD for complex geometries
  • •Include thermal analysis for high-speed
  • •Verify laminar regime

Common Misconceptions

Myth: Reynolds equation only applies to oil.
Truth: It applies to any Newtonian fluid including water, air, and liquid metals in thin film geometry.
Myth: Higher viscosity always increases load capacity.
Truth: While load increases with mu, friction also increases. Optimal viscosity balances load and power loss.
Myth: The equation is one-dimensional.
Truth: The full 2D form includes side leakage (dz term) which significantly reduces load capacity for finite-width bearings.

Practical Tips

  • •Side leakage reduces load capacity - use length-to-width ratio > 2 for efficiency
  • •The pressure spike occurs at approximately 60-70% of bearing length from inlet
  • •For journal bearings, use the Sommerfeld solution as starting point
  • •Cavitation (film rupture) occurs in diverging region - use Swift-Stieber boundary conditions
  • •Temperature rise from viscous heating can reduce viscosity by 30-50%

Common Mistakes

Ignoring side leakage
Use the full 2D Reynolds equation or apply side leakage correction factors for finite bearings.
Using kinematic instead of dynamic viscosity
Reynolds equation requires dynamic viscosity mu (Pa-s). Convert from kinematic using density.
Neglecting cavitation boundary conditions
Apply Reynolds (Swift-Stieber) cavitation condition: p = dp/dx = 0 at film rupture.
Assuming constant viscosity with temperature
Use Couette approximation or solve thermohydrodynamic (THD) model for accurate results.
Applying to turbulent flow regime
Use Taylor number to check laminar/turbulent. For turbulent, apply Constantinescu or Ng-Pan turbulent coefficients.

Assumptions

This calculator makes the following assumptions to provide estimates:

Thin Film: Film thickness << bearing dimensions, allowing pressure to be constant across film.
Laminar Flow: Re << 1 in film. For high-speed bearings, turbulent corrections may be needed.
Newtonian Fluid: Constant viscosity independent of shear rate. Not valid for greases or high-polymer oils.
No-Slip Boundary: Fluid velocity equals surface velocity at each boundary. Valid for most liquid lubricants.

Limitations

Be aware of these limitations when using this calculator:

Inertia Neglected: Valid for Re << 1. For gas bearings or high-speed liquid bearings, inertia terms become significant.
Isothermal: Assumes constant temperature. Real bearings have significant temperature gradients from viscous heating.
Smooth Surfaces: No surface roughness. For mixed lubrication, stochastic or deterministic roughness models needed.
Rigid Surfaces: No deformation. For EHL contacts, elasticity must be coupled with Reynolds equation.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the Reynolds lubrication equation?
It is the governing differential equation for pressure distribution in thin fluid films between surfaces in relative motion: d/dx(h^3/mu dp/dx) + d/dz(h^3/mu dp/dz) = 6U dh/dx.
Who derived the Reynolds equation?
Osborne Reynolds derived it in 1886, publishing his seminal paper On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower Experiments.
What does the Reynolds equation describe?
It describes the pressure generated in a thin lubricant film due to the wedge effect - fluid dragged into a converging gap creates hydrodynamic pressure that separates surfaces.
What is the wedge effect?
As fluid is dragged into a converging gap, it is compressed, generating pressure. This is the fundamental mechanism of hydrodynamic lubrication.
What is the difference between 1D and 2D Reynolds equation?
The 1D form neglects side leakage. The 2D form includes the dz term, accounting for flow perpendicular to sliding, which reduces load capacity for finite-width bearings.
What boundary conditions are used?
Inlet: p=0 (ambient). Outlet: Reynolds cavitation condition p=dp/dx=0 at film rupture. Side edges: p=0.
What is the Sommerfeld number?
The Sommerfeld number S = (mu N L D / W) x (R/c)^2 characterizes journal bearing performance and relates eccentricity to operating conditions.
How does viscosity affect load capacity?
Load capacity is proportional to viscosity. Higher viscosity = higher load capacity, but also higher friction and power loss.
What is cavitation in lubrication?
In the diverging region, the lubricant film ruptures as pressure drops below vapor pressure, forming cavities of gas or vapor. This is called cavitation.
How does side leakage affect bearings?
Side leakage reduces the load capacity. Bearings with larger L/B ratios (longer in sliding direction) have less side leakage effect.
What is the optimal convergence ratio?
For infinite-width slider bearings, the optimal K = h_in/h_min is approximately 2.2 for maximum load capacity.
How does speed affect load capacity?
Load capacity is proportional to sliding speed U. Higher speed = higher pressure generation = higher load capacity.
Can the Reynolds equation predict friction?
Yes. Integrate shear stress over the bearing area: F_f = integral(mu du/dy dA). Friction coefficient is typically 0.001-0.01 for full-film hydrodynamic bearings.
What is the Couette flow term?
The right side 6U dh/dx represents Couette (drag) flow. It is the source of pressure generation - without the converging wedge (dh/dx), no pressure is generated.
What is the Poiseuille flow term?
The left side terms represent Poiseuille (pressure-driven) flow. The pressure gradient drives flow from high to low pressure regions.
How do I solve the Reynolds equation?
Analytical solutions exist for simple geometries (infinite slider, Sommerfeld journal bearing). Numerical methods (FDM, FEM) are needed for complex geometries.
What is the Reynolds number in lubrication?
The film Reynolds number Re = rho U h / mu. For most bearings, Re << 1, validating the lubrication approximation. For gas bearings, Re can approach 1000.
How does temperature affect Reynolds equation?
Temperature reduces viscosity, which reduces load capacity. Thermohydrodynamic (THD) analysis couples energy equation with Reynolds equation.
What is the pressure-viscosity effect?
At high pressures (>100 MPa), viscosity increases exponentially per the Barus equation: mu(p) = mu_0 exp(alpha p). This is critical in EHL analysis.
What is the squeeze film effect?
When surfaces approach each other, fluid is squeezed out, creating pressure that resists motion. This is the squeeze term in the time-dependent Reynolds equation.
Can the Reynolds equation handle turbulence?
The standard form assumes laminar flow. Turbulent corrections (Constantinescu, Ng-Pan, Hirs) modify the viscosity coefficients based on local Re.
What is the role of surface roughness?
For smooth bearings (lambda > 3), roughness effects are negligible. For mixed lubrication (lambda < 3), the averaged Reynolds equation includes flow factors.
How do I model textured surfaces?
Surface texture (dimples, grooves) can be modeled by modifying the film thickness function h(x,z) and solving the local Reynolds equation.
What is the stiffness of lubricant films?
The film stiffness K_f = dW/dh_min is the rate of change of load with film thickness. Higher stiffness provides better vibration damping.
How does supply pressure affect bearings?
External pressurization (hydrostatic bearings) adds a constant pressure term to the Reynolds equation, providing load capacity even at zero speed.
What is the power loss in a bearing?
Power loss P_loss = F_f x U includes both shear losses in the film and churning losses. Most power is dissipated as heat in the lubricant.
How do I design a thrust bearing?
Use tapered-land or step bearings with convergence ratio K=2-3. Calculate load per pad and ensure minimum film thickness > surface roughness.
What is the load capacity of a step bearing?
The Rayleigh step bearing has the maximum load capacity for a given minimum film thickness. The optimal step position is at 70-80% of bearing length.
Can the Reynolds equation model gas bearings?
Yes, with compressibility: rho = p/RT. This adds the p x h terms, making it a nonlinear equation requiring numerical solution.
What is the future of Reynolds equation applications?
Multi-scale modeling coupling molecular dynamics at asperity level with continuum Reynolds equation, machine learning surrogates for real-time bearing control.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Cavitation
Film rupture in diverging region where pressure drops below vapor pressure.
Couette Flow
Drag-driven flow between surfaces in relative motion with no pressure gradient.
Eccentricity Ratio
Ratio of journal displacement to radial clearance in journal bearings.
Hydrodynamic Lubrication
Full-film lubrication where load is supported by fluid pressure generated by motion.
Poiseuille Flow
Pressure-driven flow through a channel or between parallel plates.
Reynolds Equation
Governing PDE for pressure distribution in thin lubricant films.
Side Leakage
Flow perpendicular to the sliding direction that reduces load capacity.
Sommerfeld Number
Dimensionless bearing parameter relating load, speed, viscosity, and clearance.
Squeeze Film
Pressure generation due to normal approach of surfaces, providing damping.
Wedge Effect
Pressure generation due to converging film geometry in sliding direction.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

ISO 12128ISO 7902DIN 31652DIN 31653API 610API 617

Was This Calculator Helpful?

Page Information

Last Updated:

2026-07-30

Version:

2.0.0

License:

Free to use for educational and professional purposes.