Reciprocal theorem for calculating the flow rate–pressure drop relation for complex fluids in narrow geometries

We study the mechanically driven flows of non-Newtonian fluids in narrow and confined configurations. Using the Lorentz reciprocal theorem, we derive a closed-form expression for the flow rate–pressure drop relation of complex fluids in such geometries, which holds for a wide class of non-Newtonian constitutive models. For the weakly non-Newtonian limit, our theory provides the first-order non-Newtonian correction for the flow rate–pressure drop relation solely using the corresponding Newtonian solution, eliminating the need to solve the non-Newtonian flow problem. In particular, for the flow-rate-controlled situation, we find that the first-order non-Newtonian pressure drop correction may increase, decrease, or not change the total pressure drop for a viscoelastic second-order fluid, depending on the geometry, but always decreases it for a shear-thinning Carreau fluid.

Figure 1

Schematic illustration of the geometry consisting of a two-dimensional spatially varying shallow channel of height $h\left(x\right)$ and length $\ell$. The channel contains a non-Newtonian fluid steadily driven by an imposed flow rate $q$ resulting in the pressure drop $\mathrm{\Delta }p$.