Effects of fluid rheology and geometric disorder on the enhanced resistance of viscoelastic flows through porous media

Abstract

Recent works reveal the importance of chaotic flow fluctuations as a mechanism for the enhanced resistance observed in viscoelastic porous media flows, and also show how chaotic fluctuations are affected by the structural disorder of porous media. We seek further insight by performing pressure drop measurements and flow velocimetry on two viscoelastic fluids of contrasting rheology (one with constant viscosity, another strongly shear thinning) in flow through microfluidic post arrays. Ordered hexagonal arrays have posts either ``staggered'' or ``aligned'' along the mean flow direction and disorder is applied to each configuration by randomly displacing each post about its initial location. Both polymer solutions show the expected increase in flow resistance for Weissenberg numbers, Wi > 1. In both cases, the flow resistance enhancement increases with the geometric disorder in aligned arrays, but is independent of disorder in staggered arrays. At sufficient randomisation, aligned and staggered arrays become indistinguishable. Flow velocimetry performed over a range of Wi reveals no sign of chaotic fluctuations for the constant viscosity fluid. In this case, the observation of elastic wakes between the stagnation points of the posts evokes the coil-stretch transition and implicates the extensional viscosity as the cause of the enhanced flow resistance. For the shear thinning fluid chaotic fluctuations are observed for Wi > 1, which broadly correlate with the flow resistance in this case. We also show that the first normal stress is insufficient to account for the flow resistance observed for the constant viscosity fluid, but may account for the resistance observed in the shear thinning case. Our results suggest that the dominant mechanism for resistance enhancement in viscoelastic porous media flow may emerge depending on the specific combination of fluid rheology and geometric complexity.

Figures (16)

• Figure 1: Sketches of (a) a "staggered" hexagonal post array (lattice spacing $S$, post radius $R$), with the $\textbf{a}_1$ lattice vector at $30^{\circ}$ to the flow direction (i.e., along $x$), and (b) an identical "aligned" array with the $\textbf{a}_1$ lattice vector at $0^{\circ}$ to the flow direction. Part (b) illustrates the introduction of disorder to the arrays by random displacement of each post within a hexagon of circumradius $\beta S$ about its initial centrepoint, where $\beta$ is a variable parameter describing the degree of disorder (see Ref. Walkama2020). A hypothetical new location for the central post is shown by the transparent blue circle.
• Figure 2: (a) Experimental scheme (not to scale) illustrating an empty "blank" channel (top) and an identical channel containing an array of microposts (bottom). The coordinate system is indicated, with origin at the center of each channel. Channels have width $W=2.4~\text{mm}$, height $H=1~\text{mm}$ and length $L=25~\text{mm}$. The length of each post array is $L_a \approx 5~\text{mm}$, the radius of each post is $R=50~\upmu\text{m}$ and the lattice spacing (of ordered arrays) is $S=240~\upmu\text{m}$. Flow is driven through each channel along the $x$-direction at volumetric flow rate $Q$, while the pressure drop across each channel $\Delta P$ is measured. In post arrays, time-resolved optical interrogation of the flow field is performed using $\upmu$-PIV. Photographs of (b) "staggered" and (c) "aligned" arrays of posts with various degrees of disorder $\beta$ applied as per Ref. [Walkama2020]. Scale bars: $0.5~\text{mm}$.
• Figure 3: Rheological characterisation of the polymeric test solutions at $22^{\circ}$C. (a) Flow curves of viscosity $\eta$ as a function of the shear rate $\dot\gamma$, and (b) first normal stress difference $N_1$ as a function of $\dot\gamma$ measured under steady shear using a DHR3 stress-controlled rotational rheometer (TA Instruments) fitted with a stainless steel 40 mm diameter $1^{\circ}$ cone-and-plate geometry. In (a), the shear thinning flow curve for 200 ppm HPAA is fitted with a Carreau model (Eq. \ref{['Carreau']}, solid line), while respectively-coloured dashed and dotted lines indicate the solvent viscosity. Respectively-coloured dashed and dotted lines in (b) represent power-law fits to $N_1$. (c) Representative examples of the filament diameter vs. time ($D(t)$) measured during capillary driven thinning of the polymeric fluids, from which characteristic timescales can be estimated, see main text.
• Figure 4: Pressure drop measurements with the Newtonian reference fluid. (a) Example traces of the pressure drop versus time ($\Delta P(t)$) across the blank channel and the ordered staggered array for stepwise increments of the volumetric flow rate between $70 \leq Q\leq 205~\upmu\text{L~min}^{-1}$. (b) The steady plateau pressure drop $\Delta P(Q)$ shows the expected proportionality (dashed lines). Insert shows the apparent viscosity $\eta_{app}$ in the staggered array (derived from Darcy's law, Eq. \ref{['Darcy2']}) as a function of the superficial flow velocity $U$. The shaded region about the data points indicates the standard deviation over five repetitions. (c) Normalised apparent viscosity $\eta_{app}/\eta$versus the Reynolds number $\text{Re}$ showing a constant value $\approx 1$ for all fourteen post arrays. For better clarity, shaded error bounds are only shown for the two ordered arrays (i.e., $\beta=0$).
• Figure 5: Normalised time-averaged flow speed $\langle u \rangle_t/U$ for the Newtonian reference fluid flowing at $\text{Re} \approx 0.02$ in a few of the post arrays: (a) Staggered, $\beta = 0$; (b) Staggered, $\beta = 0.4$; (c) Aligned, $\beta = 0$. Scale bar represents $0.2~\text{mm}$; flow is from left to right.