= Reynolds lubrication equation
{c}
{wiki=Reynolds_equation}
Incompressible mass conservation in a moving gap gives
$$
\partial_t h+\partial_xQ=0.
$$
Combined 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 flux>
$$
\mathbf q=-\frac{h^3}{12\mu}\nabla_\parallel p+\frac h2(\mathbf v_0+\mathbf v_h).
$$
<Conservation 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.
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