= Solution
Let $H$ be a representative positive gap thickness and set
$$
W=|U|+\frac{VL}{H},
$$
the larger-order horizontal speed after including the squeeze-induced flux. The <lubrication-limit scaling for a moving thin gap> requires
$$
\boxed{
\frac HL\ll1,\qquad
\max_{0\leq x\leq L}|h'(x)|\ll1,\qquad
\frac{\rho W H^2}{\mu L}\ll1.}
$$
Equivalently, the inertial condition contains
$$
\frac{\rho|U|H^2}{\mu L}\ll1,
\qquad
\frac{\rho VH}{\mu}\ll1.
$$
The gap must also remain open, $h(x)>0$. These conditions make streamwise derivatives small relative to transverse derivatives, keep the pressure uniform across the gap to leading order, and reduce the <Navier-Stokes equation> to <Stokes flow> locally.
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