Weak-pressure Couette-Poiseuille expansion ($u=\frac{U}{h}y+\frac{G}{2{\eta }_d(U/h)}(h^2-y^2)+\cdots ,Q=\frac{2G{h}^3}{3{\eta }_d(U/h)}+\cdots$)

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 $nk(U/h{)}^{n-1}$.