Pressure drop reduction of shear-thinning fluids in weakly deformable channels via the Lorentz reciprocal theorem

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 reciprocity relation is obtained for a generalized Newtonian fluid whose effective viscosity is a function of shear stress. Importantly, our approach does not assume any restriction on the non-Newtonian correction being "small" or "weak'' (such as a low Carreau number). The only limitation of the approach is that the viscosity model must allow for a closed-form solution for the axial velocity profile in a straight and rigid channel. As a featured example, we consider the Ellis viscosity model, which captures the incipience of shear thinning. The analytical expression for the pressure drop reduction obtained via the reciprocal theorem under the Ellis model allows us to recover the Newtonian case (resp. the power-law regime) for small (resp. large) Carreau number or, equivalently, large (resp. small) Ellis number, as special cases. The approach based on the Lorentz reciprocal theorem offers the possibility of calculating the pressure drop of non-Newtonian fluid flows in deformable channels without solving a coupled elastohydrodynamics problem.