Pressure shifts in pulsatile shear: A microfluidic method to probe the normal stress response of complex fluids

TL;DR

The paper addresses measuring the first normal-stress difference $N_1$ in viscoelastic fluids under high-shear, pulsatile microfluidic flows by introducing a pressure-shift metric in large-amplitude pulsatile shear (laps). It calibrates a transfer-function model to correct pressure signals and defines a Chronos number $Ch = \sqrt{\tau_e \tau_v^3}/\tau_c^2$ that collapses $\,\Delta\mathcal{P}_S$ data across fluids with similar viscosity but different elasticity, linking to $N_1$ through a scale $\Delta\mathcal{P}_S \propto {\mathcal{N}_1}^{2/m}$ with $m \approx 0.9$. The study demonstrates that $\Delta\mathcal{P}_S$ scales with $Ch$ and correlates with $N_1$ measured by traditional rheometry, providing a compact, low-volume method to quantify nonlinear elastic effects in microflows. This approach is relevant for biological fluids and industrial processes where high shear and time-dependent flows are common, offering a new tool for microfluidic rheometry of nonlinear viscoelasticity.

Abstract

A microfluidic approach to probing the first normal stress difference from single-point pressure measurements in transient shear flows is presented. Using an original experimental design, we examine the near-zero-mean pulsatile flow of polymeric solutions in a straight microchannel at low Reynolds and Womersley numbers. An important aspect of this work is that the enhanced fluid elastic stresses can be efficiently determined via the pressure shift measured from pressure-controlled pulsatile shear experiments. We find a scaling law that collapses pressure-shift data from viscoelastic fluids of different molecular weights onto a single master curve that can then be used to predict this phenomenology. Taken together, these results could help shed light on our understanding of the non-linear normal stress responses in time-dependent flows.

Figures (7)

• Figure 1: Schematic of the $x$--$y$ plane of the microchannel and pressure tap ($p_\text{out}$) placed perpendicular to the flow direction ($r=750$ µ m, $l_2=850$ µ m). Net volumetric flow is left to right, driven by a sinusoidal pressure gradient. Flat, plug-like velocity profile $v_x$ typical of shear-thinning polymer solution flows.
• Figure 2: Fluid rheological characterisation. Shear viscosity $\eta$ and first normal stress coefficient $\Psi_1$ (determined where possible, inset plot) as a function of applied shear rate $\dot\gamma$, with fits of c--y and power-law models, respectively. Minimum torque and secondary flow limits (slopes $-1$ and 1, respectively) shown in light and dark grey (pp and cc geometries, respectively).ewoldt2015 Inset: Resolution limit (slope $-2$) shown in light grey.walters1975_book The shear rate value reaches as high as $O(10^3~\text{s}^{-1})$ in the laps experiments.
• Figure 3: Frequency response of the pressure measuring system. Bode diagrams of experimental data for the transfer functions of the (a) Newtonian ($\omega_\text{n}=0.29$ rad/s, $\zeta=2.63$) and (b) non-Newtonian ($\omega_\text{n}=0.07$ rad/s, $\zeta=11.77$) second-order systems, with overdamped dynamics ($\zeta>1$) reminiscent of that of soft materials. $\omega_\text{n}$ and $\zeta$ are the natural frequency and damping ratio of the system, respectively, which are typically present in the transfer function of a second-order system. The non-Newtonian data correspond to $10\agt\text{Wi}_\text{max}\agt 7$ and $8.2\times 10^{-5}\alt\text{De}\alt 1.3\times 10^{-3}$ (hollow squares); $\text{Wi}_\text{max}\simeq\{26,7\}$ and $\text{De}\simeq\{2.6,5.2\}\times 10^{-3}$ (filled squares). Different symbols represent different input mean-amplitude pairs $(\langle p_\text{in}\rangle,|p_\text{in}|)$ (see also Fig. S4 in the supplementary material). The inset in (a) shows $p_\text{out}$versus$p_\text{in}$, used to determine $g=0.45$. Error bars (based on pressure sensor resolution) are less than marker size and are not shown here for clarity. Colour code as in Fig. \ref{['fig:rheol']}.
• Figure 4: Strain amplitude sweep for xg and paa at a fixed frequency $\omega=0.79$ rad/s. Beyond the linear regime, both $G'$ and $G"$ monotonically decrease. The non-linear laps region is marked as the pink area. The minimum torque and instrument inertia limits are shown in light and dark grey, respectively.ewoldt2015 Error bars are less than marker size and are not shown here for clarity. Colour code as in Fig. \ref{['fig:rheol']}.
• Figure 5: Pressure shift evolution. Top: Frequency sweep responses at $p_0=200$ Pa (left column for water, middle column for xg, and right column for paa). The non-Newtonian data show a vertical shift, indicating the presence of viscoelasticity. This effect becomes more pronounced for increasing $\omega$, and the corresponding pressure shifts are labelled as dash-dotted lines. This phenomenology was subsequently reproduced using a blood-mimicking fluid and a physiological waveform (this will be discussed in a future paper), suggesting that our results should apply more widely. Bottom: Magnitude of pressure shift $\Delta\mathcal{P}_\text{S}$ as a function of imposed frequency $\omega$ and pressure amplitude $p_0$. In the white region, there are no data.