Numerical investigation of multistability in the unstable flow of a polymer solution through porous media

The flow of viscoelastic polymeric fluids through porous media is common in industrial applications such as oil recovery and groundwater remediation. Polymeric stresses can lead to an elastic-induced instability of the flow. Here, we numerically study the flow of a polymeric fluid in a channel consisting of multiple diverging and converging physical constraints, mimicking the pore bodies and throats of an ordered porous medium. Inertial stresses here are negligible, and instead the flow is dominated by elasticity and viscosity; their relative effects are characterized by the Weissenberg number. There is a critical Weissenberg number below which eddies appear on the top and the bottom of each pore. Above the critical Weissenberg number, eddies form in different regions of the pores and multiple distinct unstable flow structures occur. The stretched polymeric chains inside the pore facilitate eddy formation, whereas relaxed chains lead to eddy-free regions. We quantify the eddy area and correlations between the flow patterns of different pairs of pores, as well as polymeric stress and pressure drop across the tortuous channel to better understand the mechanism behind the observed flow patterns.

Figure 1

The geometry used for numerical simulations. $D=1.6$ mm is the diameter of the pillar and $W=2$ mm is the width of the channel. The region enclosed with the square box depicts a sample pore.

Figure 2

(a) The streamlines depicting the eddies formation in the upstream of a single throated channel at $\mathrm{Wi}=26$. The contour represents the trace of polymeric stress tensor in the channel. (b) x-component of velocity along the length of channel close to the wall (at a distance $\mathrm{\Delta }y/W=0.015$ from the wall, where $\mathrm{\Delta }y$ is the height of first grid element next to the wall). (c) The instantaneous and time-averaged length of eddies at different Wi in the upstream of a single-pore channel. The time is nondimensionalized with timescale $t_{\text{pv}}$. (d) Power spectral density (PSD) of the normalized eddies' length ($L_{\text{eddy}}/W$). Frequency is normalized with $1/t_{\text{pv}}$.

Figure 3

(a) Probability density function of dimensionless eddy length ($L_{\text{eddy}}/W$) in a single throated channel. (b) Mean ($\mu$) and standard deviation ($\sigma$) of normalized eddies' length.

Figure 4

The instantaneous and time-averaged length of eddies in the upstream of throat-1 and throat-2 at different $\mathrm{Wi}$ of a double throated channel. Time is normalized with volumetric flow timescale $t_{\text{pv}}$.

Figure 5

(a) Streamlines and trace of polymeric stress tensor in a channel with two throats at $\mathrm{Wi}=34$. (b) The correlation between instantaneous eddy lengths of throat-1 and throat-2 at $\mathrm{Wi}=34$ in a channel of two pore throats. (c) Instantaneous $f_{1,2}$ and time-average $\langle f_{1,2}\rangle$ value of the correlation function between the eddy of throat 1 and throat 2 at $\mathrm{Wi}=34$.

Figure 6

Power spectral density of normalized eddies' length ($L_{\text{eddy}}/W$) at (a) $\mathrm{Wi}=26$, (b) $\mathrm{Wi}=34,$ and (c) $\mathrm{Wi}=42$.

Figure 7

(a) Probability density function of dimensionless eddy length in two widely separated throats channel. (b) Mean ($\mu$) and standard deviation ($\sigma$) of normalized eddies' length.

Figure 8

(a) Instantaneous streamlines in a channel of 10 closely interconnected pores at $\mathrm{Wi}=18$ and dimensionless time, $t\approx 6$. (b) Instantaneous streamlines in a channel of 10 closely interconnected pores at $\mathrm{Wi}=34$ and dimensionless time, $t\approx 10$.

Figure 9

The snapshot of the trace of polymeric stress tensor at (a) $\text{Wi}=18,t\approx 6$ and (b) $\text{Wi}=34,t\approx 10$. These plots of trace of polymeric stress correspond to the streamlines shown in Figs. 8 and 8, respectively.

Figure 10

(a) The ratio of eddy to pore area for “pore 2” as a function of time at $\mathrm{Wi}=18$. $A_{\text{eddy}}$ is the area of eddies in a particular half region of the pore, while $A_{\text{pore}}$ is the total area of the pore. The streamlines inside the pore represent the flow pattern at $\mathrm{Wi}=18$. (b) The contour of the trace of polymeric stress tensor across the channel at the center of the pore [i.e., along the red line shown in the inset of Fig. 10]. (c) The ratio of eddy to pore area for “pore 2” as a function of time at $\mathrm{Wi}=34$. The snapshots of streamlines inside the pore represent the flow pattern at specific time indicated by red solid circles. (d) The contour of the trace of the polymer stress tensor across the channel at the center of the pore. The upper limit of time in these plots ($t=9.5$) corresponds to $t^{*}/\lambda =13.4$ for $\mathrm{Wi}=18$ and $t^{*}/\lambda =3.3$ for $\mathrm{Wi}=34$.

Figure 11

(a) Instantaneous $f_{5,6}$ and time-average $\langle f_{5,6}\rangle$ values of the correlation function between pore 5 and pore 6 at $\mathrm{Wi}=34$. (b) The time average value of correlation function for different pairs of the pores at $\mathrm{Wi}=34$. (c) The value of $\langle f_{i,j}\rangle$, further averaged across the pores of the same separation, as a function of pore separation at different $\mathrm{Wi}$.

Figure 12

(a) PDF of the ratio of eddies to pore area ($A_{\text{eddy}}/A_{\text{pore}}$) at different Wi for a channel of 10 closely located pores. $A_{\text{eddy}}$ represents total area occupied by eddies in an individual pore and $A_{\text{pore}}$ is the total area of the pore. Above a threshold Wi, there appears to be multistability, and the eddy areas take on a broad range of values. (b) Mean ($\mu$) and standard deviation ($\sigma$) of normalized eddies' area.

Figure 13

Eddies at the center of the pore at $\mathrm{Wi}=34$. (a) Eddy at the center as well as the top and bottom regions of the pore. Here, the eddy at the center persists in two pores. (b) Eddies at the center and the top region of the pore, while bottom region is eddy free. (c) Eddies only at the center of the pore. Top and bottom regions are eddy free.

Figure 14

Dimensionless pressure for the flow field shown in Fig. 8 (i.e., $\text{Wi}=34$, $t\approx 10$): (a) The contours of pressure field inside the channel. (b) Pressure along the centerline of the channel [i.e., red solid line in Fig. 14]. (c) Pressure across the channel at the center of the pore [i.e., dashed yellow line in Fig. 14].

Figure 15

(a) Instantaneous and time averaged pressure drop across the channel of 10 pores at $\mathrm{Wi}=34$. (b) Averaged pressure drop ($\langle \mathrm{\Delta }p\rangle$) across the channels at different Wi.

Figure 16

(a) Dimensionless pressure drop across the channel along the centerline at $\mathrm{Wi}=0.3$ (almost Newtonian fluid). Flow converges to steady state for $t>0.2$. (b) Dimensionless pressure drop across the channel along the centerline at $\mathrm{Wi}=18,34$. Fully developed instability occurs for $t>1$.

Figure 17

The area of eddies on the top and bottom of “pore-2” at $\mathrm{Wi}=0.3$.

Figure 18

Instantaneous streamlines in a channel of 10 closely interconnected pores at $\mathrm{Wi}=16.4$ for FENE-CR constitutive model.