 Reynolds equation

Incompressible mass conservation in a moving gap givesCombined with the local Couette-Poiseuille flux, this is the Reynolds lubrication equation.

For a thin gap $h$ with tangential wall velocities ${\mathbf{v}}_0,{\mathbf{v}}_h$, lubrication theory and the no-slip boundary condition give the integrated volume fluxConservation of mass gives $h_t+{\nabla }_{\parallel }\cdot \mathbf{q}=0$. This is the Reynolds equation; it determines the lubrication pressure from wall motion and gap geometry.

Reynolds equation is a fundamental equation in hydrodynamics that describes the flow of a viscous fluid between two surfaces. It is particularly important in the study of lubrication theory, which deals with the behavior of lubricants in mechanical systems like bearings and seals. The equation is derived from the Navier-Stokes equations by making certain simplifying assumptions, including the assumption of a thin film of fluid.