  We employ the Lorentz reciprocal theorem to derive a closed-form expression
for the pressure drop reduction due to the coupling between shear-thinning
fluid flow and a weakly deformable channel wall. The methodology is applied in
parallel to fluids for which the generalized Newtonian viscosity depends on
either the shear rate or the shear stress magnitude. The only limitation on the
viscosity model is that it allows for a closed-form solution for the axial
velocity profile in a straight and rigid channel. The power-law and Ellis
viscosity models are considered as featured examples and to enable comparisons
to previous works. The analytical expression obtained via the reciprocal
theorem shows that the pressure drop reduction under the power-law model can
also be obtained from the pressure drop reduction under the Ellis model in the
regime of large Carreau number (small Ellis number).
