In fluid mechanics (specifically lubrication theory), the Reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films. It was first derived by Osborne Reynolds in 1886. The classical Reynolds Equation can be used to describe the pressure distribution in nearly any type of fluid film bearing; a bearing type in which the bounding bodies are fully separated by a thin layer of liquid or gas.
General usage
The general Reynolds equation is:
∂ ∂ x ( ρ h 3 12 μ ∂ p ∂ x ) + ∂ ∂ y ( ρ h 3 12 μ ∂ p ∂ y ) = ∂ ∂ x ( ρ h ( u a + u b ) 2 ) + ∂ ∂ y ( ρ h ( v a + v b ) 2 ) + ρ ( w a − w b ) − ρ u a ∂ h ∂ x − ρ v a ∂ h ∂ y + h ∂ ρ ∂ t {\displaystyle {\frac {\partial }{\partial x}}\left({\frac {\rho h^{3}}{12\mu }}{\frac {\partial p}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\rho h^{3}}{12\mu }}{\frac {\partial p}{\partial y}}\right)={\frac {\partial }{\partial x}}\left({\frac {\rho h\left(u_{a}+u_{b}\right)}{2}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\rho h\left(v_{a}+v_{b}\right)}{2}}\right)+\rho \left(w_{a}-w_{b}\right)-\rho u_{a}{\frac {\partial h}{\partial x}}-\rho v_{a}{\frac {\partial h}{\partial y}}+h{\frac {\partial \rho }{\partial t}}}
Where:
p {\displaystyle p} is fluid film pressure.
x {\displaystyle x} and y {\displaystyle y} are the bearing width and length coordinates.
z {\displaystyle z} is fluid film thickness coordinate.
h {\displaystyle h} is fluid film thickness.
μ {\displaystyle \mu } is fluid viscosity.
ρ {\displaystyle \rho } is fluid density.
u , v , w {\displaystyle u,v,w} are the bounding body velocities in x , y , z {\displaystyle x,y,z} respectively.
a , b {\displaystyle a,b} are subscripts denoting the top and bottom bounding bodies respectively. The equation can either be used with consistent units or nondimensionalized. The Reynolds Equation assumes:
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