Pressure drop reduction due to coupling between shear-thinning fluid flow and a weakly deformable channel wall: A reciprocal theorem approach

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 in terms of the shear rate and the viscosity function (and its derivative) of the underlying rigid-channel flow. 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. When the viscosity model allows for a closed-form solution for the axial velocity profile in a straight and rigid channel, the pressure drop reduction can be evaluated in closed form, which we demonstrate for the power-law and Ellis viscosity models as featured examples and to enable comparisons to previous works. Importantly, the pressure drop reduction under the Ellis model is valid for both small and large Carreau (or Ellis) numbers, and we show that it reduces to the analytical expression under the power-law model for large Carreau (small Ellis) numbers.