Derivation from Navier-Stokes Equations
Lubrication theory begins with the incompressible Navier-Stokes equations for a Newtonian fluid, which describe the conservation of momentum and mass:
where u=(u,v,w)\mathbf{u} = (u, v, w)u=(u,v,w) is the velocity vector, ppp is pressure, ρ\rhoρ is density, μ\muμ is dynamic viscosity, and g\mathbf{g}g is the body force per unit mass (often gravity). These equations are simplified under the thin-film limit characteristic of lubrication flows, where the film thickness HHH is much smaller than the lateral dimensions LLL, defining the small aspect ratio ϵ=H/L≪1\epsilon = H/L \ll 1ϵ=H/L≪1.
To derive the lubrication approximations, the equations are non-dimensionalized using characteristic scales appropriate to the geometry: x,y∼Lx, y \sim Lx,y∼L, z∼Hz \sim Hz∼H, u,v∼Uu, v \sim Uu,v∼U (lateral velocity scale), w∼ϵUw \sim \epsilon Uw∼ϵU (vertical velocity scale), p∼μUL/H2p \sim \mu U L / H^2p∼μUL/H2 (viscous pressure scale), and t∼L/Ut \sim L/Ut∼L/U (convective time scale). Substituting these into the Navier-Stokes equations reveals the relative magnitudes of terms. The inertial terms scale as Reϵ2\mathrm{Re} \epsilon^2Reϵ2, where Re=ρUL/μ\mathrm{Re} = \rho U L / \muRe=ρUL/μ is the Reynolds number; for typical lubrication regimes, Reϵ2≪1\mathrm{Re} \epsilon^2 \ll 1Reϵ2≪1, allowing neglect of inertia. The pressure gradient in the zzz-direction scales as ϵ2\epsilon^2ϵ2 times the lateral gradient, so ∂p/∂z∼ϵ2∂p/∂x≪∂p/∂x\partial p / \partial z \sim \epsilon^2 \partial p / \partial x \ll \partial p / \partial x∂p/∂z∼ϵ2∂p/∂x≪∂p/∂x. Body forces are often negligible unless buoyancy or other effects dominate.
The simplified momentum equations follow from this scaling. The zzz-momentum equation reduces to ∂p/∂z=0\partial p / \partial z = 0∂p/∂z=0, implying pressure is constant across the film thickness: p=p(x,y)p = p(x, y)p=p(x,y). The xxx-momentum equation balances pressure gradient and viscous diffusion, yielding 0=−∂p/∂x+μ∂2u/∂z20 = -\partial p / \partial x + \mu \partial^2 u / \partial z^20=−∂p/∂x+μ∂2u/∂z2 (Poiseuille-Couette flow). The yyy-momentum equation is analogous: 0=−∂p/∂y+μ∂2v/∂z20 = -\partial p / \partial y + \mu \partial^2 v / \partial z^20=−∂p/∂y+μ∂2v/∂z2. Integrating the xxx-momentum equation twice with no-slip boundary conditions—at z=0z = 0z=0, u=0u = 0u=0 (stationary lower surface), and at z=h(x,y)z = h(x, y)z=h(x,y), u=Uu = Uu=U (upper surface moving in the xxx-direction, as in a simple slider bearing)—gives the velocity profile:
A similar expression holds for v(z)v(z)v(z). These profiles capture the parabolic pressure-driven (Poiseuille) component superimposed on the linear shear-driven (Couette) component.
Integrating the continuity equation across the film thickness from z=0z = 0z=0 to z=hz = hz=h enforces mass conservation. For steady, two-dimensional flow (neglecting yyy-variation for simplicity), this yields ∂/∂x∫0hu dz=0\partial / \partial x \int_0^h u , dz = 0∂/∂x∫0hudz=0, or more generally, the divergence of the depth-averaged flux q=∫0hu dz=0\mathbf{q} = \int_0^h \mathbf{u} , dz = 0q=∫0hudz=0. The xxx-component of the flux is
with an analogous qyq_yqy. The first term represents diffusive transport due to pressure gradients, while the second is advective transport due to surface motion. For unsteady or three-dimensional cases, additional terms arise from film thickness changes, but the steady form ∇⋅q=0\nabla \cdot \mathbf{q} = 0∇⋅q=0 provides the foundation for pressure determination in lubrication problems.
Reynolds Equation and Boundary Conditions
The Reynolds equation governs the pressure distribution in thin lubricant films under the assumptions of lubrication theory, integrating the simplified Navier-Stokes equations across the film thickness. For steady, incompressible, internal flows between two surfaces moving with velocities UUU in the xxx-direction and VVV in the yyy-direction, the two-dimensional form is
where h(x,y)h(x,y)h(x,y) is the film thickness, μ\muμ is the constant dynamic viscosity, and p(x,y)p(x,y)p(x,y) is the pressure. This equation, first derived by Osborne Reynolds, balances the Poiseuille pressure-driven flow with the Couette shear-driven flow. For unsteady flows, a time-dependent term is added to the right-hand side: ∂h∂t\frac{\partial h}{\partial t}∂t∂h, yielding the general three-dimensional form when extended to the zzz-direction if needed.
In free-surface lubrication, such as coating flows or meniscus-driven films, the Reynolds equation couples with the kinematic boundary condition at the liquid-air interface to enforce mass conservation:
where the depth-averaged flux q=−h312μ∇p+Uh2\mathbf{q} = -\frac{h^3}{12\mu} \nabla p + \frac{U h}{2}q=−12μh3∇p+2Uh incorporates both pressure gradients and mean surface velocity UUU. This formulation applies to thin films where surface tension effects dominate at the edges.
Boundary conditions are essential for solving the Reynolds equation and reflect physical constraints at the film domain. For internal flows, ambient pressure p=pap = p_ap=pa (typically pa=0p_a = 0pa=0 gauge) is imposed at the inlet and outlet edges where the film opens to the surroundings. In regions prone to cavitation, such as journal bearings under load, the pressure cannot drop below the vapor pressure (often approximated as p≥0p \geq 0p≥0); the Jakobsson-Floberg-Olsson (JFO) model handles this by switching off the diffusive terms in cavitated zones, enforcing p=0p = 0p=0 and ∂p∂n=0\frac{\partial p}{\partial n} = 0∂n∂p=0 (normal derivative zero) across the film-rupture interface while conserving mass flux. For free surfaces, the dynamic boundary condition arises from the stress balance σ⋅n=−pn+∇⋅(σ∇h)\boldsymbol{\sigma} \cdot \mathbf{n} = -p \mathbf{n} + \nabla \cdot (\sigma \nabla h)σ⋅n=−pn+∇⋅(σ∇h), where σ\boldsymbol{\sigma}σ is the stress tensor, n\mathbf{n}n the normal, and σ\sigmaσ the surface tension; this simplifies to the Young-Laplace equation p=σκp = \sigma \kappap=σκ at the interface, with κ\kappaκ the mean curvature.
Solutions to the Reynolds equation range from analytical for idealized cases to numerical for practical geometries. In simple one-dimensional slider bearings with constant viscosity and no side leakage, an analytical pressure profile is p(x)=6μUL2h2(xL)(1−xL)p(x) = \frac{6\mu U L^2}{h^2} \left( \frac{x}{L} \right) \left(1 - \frac{x}{L} \right)p(x)=h26μUL2(Lx)(1−Lx), illustrating the parabolic pressure build-up that supports load via the wedge effect. Complex configurations, including variable film thickness, thermal effects, or irregular boundaries, require numerical methods such as finite differences, finite elements, or finite volumes to discretize and solve the elliptic-parabolic PDE iteratively.