lubricationekvation

Lubricationekvation, commonly referred to in English as the Reynolds equation, is a fundamental partial differential equation used to predict the pressure distribution within a thin viscous lubricant film between closely spaced, moving surfaces. It is central to hydrodynamic lubrication theory and is applied in the design and analysis of bearings, gears, seals, and other tribological contacts.

The equation is derived from the Navier–Stokes equations under lubrication approximations: the lubricant film thickness h(x,y) is small compared with other dimensions; flow is laminar and inertial effects are negligible; the fluid is Newtonian and incompressible; and thermal effects are often neglected for isothermal cases. In two dimensions, the steady-state Reynolds equation can be written as:

∂/∂x[(h^3/12μ) ∂p/∂x] + ∂/∂y[(h^3/12μ) ∂p/∂y] = ∂/∂x(hU/2),

where p(x,y) is the pressure, μ is the dynamic viscosity, h(x,y) is the film thickness, and U is the relative sliding velocity in the x-direction. When h varies mainly in x, a one-dimensional form d/dx[(h^3/12μ) dp/dx] = d/dx(hU/2) is often used.

Key variables and boundary conditions are chosen to reflect device geometry and operating conditions, with ambient pressure at outlets or specified pressure boundaries commonly imposed.

Applications include predicting load-bearing capacity, friction, and film stability in bearings and seals. Extensions of the basic equation account for surface deformation (elastohydrodynamic lubrication) and non-Newtonian or thermal effects, which broaden its applicability but also introduce additional complexity.

Limitations include neglect of inertial effects at higher speeds, into highly compressible or non-Newtonian fluids, and the omission of thermal coupling unless explicitly incorporated.