= Weak-pressure Couette-Poiseuille expansion
{title2=$u=\frac Uh y+\frac G{2\eta_d(U/h)}(h^2-y^2)+\cdots,\quad Q=\frac{2Gh^3}{3\eta_d(U/h)}+\cdots$}
Perturb the <Couette flow> between walls moving at +-U by a small pressure gradient -G. The mean shear rate remains U/h, and the stress variation across the gap is -Gy. Linearizing the monotone <generalized Newtonian fluid> law divides this variation by differential viscosity, giving the displayed velocity and flux per unit width. Require $Gh\ll(U/h)\eta_d(U/h)$ and a smooth constitutive law on the perturbed rates. For a <power-law fluid>, the denominator is $nk(U/h)^{n-1}$.
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