Flow rate–pressure drop relations for shear-thinning fluids in deformable configurations: Theory and experiments

We provide an experimental framework to measure the flow rate–pressure drop relation for Newtonian and shear-thinning fluids in two common deformable configurations: (i) a rectangular channel and (ii) an axisymmetric tube. Using the Carreau model to describe the shear-dependent viscosity, we identify the key dimensionless rheological number $\mathrm{Cu}$, which characterizes shear thinning, and we show that our experiments lie within the power-law regime of shear rates. To rationalize the experimental data, we derive the flow rate–pressure drop relation taking into account the two-way-coupled fluid-structure interaction between the flow and its compliant confining boundaries. We thus identify the second key dimensionless number $\alpha$, which characterizes the compliance of the conduit. We then compare the theoretical flow rate–pressure drop relation to our experimental measurements, finding excellent agreement between the two. We further contrast our results for shear-thinning and Newtonian fluids to highlight the influence of $\mathrm{Cu}$ on the flow rate–pressure drop relation. Finally, we delineate four distinct physical regimes of flow and deformation by mapping our experimental flow rate–pressure drop data for Newtonian and shear-thinning fluids into a $\mathrm{Cu}-\alpha$ plane.

Figure 1

Illustration of the configurations used for experiments and modeling. (a),(c) Image of the (a) experimental device and (c) schematic of a 3D channel of an initially rectangular cross section with a top deformable wall. (b),(d) Image of the (b) experimental device and (d) schematic of a 3D tube extruded from a large block of elastic material, exhibiting radial deformation. Both configurations have a deformable section of length $\ell$ and contain either a Newtonian or shear-thinning fluid steadily driven by the imposed flow rate $q$. Our interest is to determine the pressure drop $\mathrm{\Delta }p$ over a streamwise distance ${\ell }_{\mathrm{\Delta }p}$ between two pressure ports. For the channel, we have $\ell \approx {\ell }_{\mathrm{\Delta }p}$, while for the tube, $\ell >{\ell }_{\mathrm{\Delta }p}$.

Figure 2

Experimental data for viscosity as a function of shear rate for the xanthan gum ($\times$) and the glycerin ($\circ$) solutions. The solid black curve represents the fit of the xanthan gum solution's rheological data to the Carreau model (1). The rheological parameters obtained from the fitting are summarized in the figure. Error bars are smaller than the symbols.

Figure 3

Comparison between our “soft” theory (solid curves), based on (7) and (10), and the experimental data (symbols) for the flow rate–pressure drop relation in (a) the channel and (b) the tube. The shaded regions indicate the combined uncertainty in (a) $h_0$ or (b) $a_0$ and the viscosity of the glycerin solution. Dashed curves denote the respective rigid-conduit relations (6) and (9). Error bars represent the standard deviation based on more than three individual experiments.

Figure 4

The experimental data from Fig. 3 shown on a $\mathrm{Cu}-\alpha$ qualitative diagram, in which four regimes of low-Reynolds-number fluid-structure interaction of Newtonian and shear-thinning fluids can be identified (shaded regions). Note the truncated horizontal axis.