Viscoelastic instability in an asymmetric geometry

Abstract

Viscoelastic flow through porous media is important in industrial applications such as enhanced oil recovery (EOR), microbial mining, and groundwater remediation. It is also relevant in biological processes such as drug delivery, infectious biofilm formation, and transport during respiration and fertilization. The porous medium is highly disordered and viscoelastic instability-induced flow states at the pore-scale regulate the transport in porous media. In the present study, we systematically explore the effect of geometrical asymmetry on pore-scale viscoelastic instability. The asymmetric geometry used in the present study consists of two cylinders confined inside a channel, where the front cylinder is located on the centerline of the channel and the rear cylinder is situated off-center of the channel. The geometrical asymmetry facilitates asymmetric flow around both cylinders. An eddy also appears in the region between the cylinders at intermediate Weissenberg numbers, where the Weissenberg number characterizes the relative importance of elastic and viscous forces in viscoelastic flows. We further explore the effect of the strength of geometrical asymmetry and fluid rheological properties on flow asymmetry and eddy formation.

Appendix

1.1 A. Entrance and exit effects

Fig. 12

The velocity profile at different locations along the length of the channel at $\mathrm {Wi=0.62}$. The entrance and the exit of the channel are at $x=-9.4d$ and $x=15.6d$, respectively. The front cylinder is located at $x=0$

The hydrodynamic entrance length for a Newtonian channel flow can be estimated as [58]:

$$\begin{aligned} L_{\text {entrance}}=0.05Re_DD, \end{aligned}$$

(7)

where D is the hydraulic diameter of the channel and Re$_D=\rho U_{in} D/\mu$ is the Reynolds number based on the hydraulic diameter. In the present study, the value of hydrodynamic entrance length for Newtonian flow varies from $L_{\text {entrance}}=10^{-4}d$ to $L_{\text {entrance}}=6 \times 10^{-4} d$, where d is the cylinder diameter. The locations of the entrance, exit and front cylinder are $x/d=-9.4$, $x/d=15.6$, and $x/d=0$, respectively. Thus, the length of the channel in the present study is 25d and the front cylinder is located 9.4d downstream from the inlet, which is much larger than the hydrodynamic entrance length. We have also plotted the velocity profile at different locations along the length of the channel for viscoelastic flow, which clearly shows that the velocity profile sufficiently upstream of the front cylinder becomes fully developed (Fig. 12). The flow also becomes fully developed downstream of the rear cylinder much before the exit (Fig. 12).

1.2 B. Mesh and time-step dependence tests

Fig. 13

Normalized pressure drop across the channel at a $\mathrm {Wi=0.62}$ and b $\mathrm {Wi=3.75}$. Other parameters are $\beta =0.05$ and $L^2=1000$

We use pressure drop ($\Delta p$) across the channel as a metric for mesh and time-independent studies [10]. The pressure drop across the channel for different numerical meshes and the different values of the Courant numbers have been shown in Fig. 13a for a small Wi ($\mathrm {Wi=0.62}$). At a small Wi, the simulation achieves a steady-state for $t>5$ and $\Delta p$ becomes constant. The simulation becomes mesh independent for $n_x \times n_y>2000 \times 200$, where $n_x$ and $n_y$ are the numbers of grid points along the length and the width of the channel (Fig. 13a). The simulations in the present study have been performed using $n_x \times n_y=2560 \times 256$. The Courant number ($\mathrm {Co}$) controls the time-step size in the present study and it has been defined as:

$$\begin{aligned} {\mathrm{{Co}}}=\tau \Delta t, \end{aligned}$$

(8)

where $\Delta t$ is the simulation time-step. $\tau$ is a characteristic time scale based on the local cell flow scales and defined as:

$$\begin{aligned} \tau =\frac{1}{2V}\sum _{faces_i}|\phi _i|, \end{aligned}$$

(9)

where V and $\phi$ are cell volume and the cell-face volumetric flux. $\sum _{faces_i}$ shows the summation over all cell faces. The simulation becomes time-step independent for $\mathrm {Co_{\max }<0.035}$ (Fig. 13a). We use $\mathrm {Co_{\max }=0.025}$ in the present study. We also check the convergence at the maximum Wi ($\mathrm {Wi=3.75}$) used in the present study and ensure that the results are mesh independent even at the maximum Wi (Fig. 13b). The instability becomes fully developed for $t>5$ and $\Delta p$ fluctuates around a well-defined mean (Fig. 13b). However, the standard deviation of the fluctuation is very small ($<1 \%$ of the mean value). Therefore, the fluctuation is very weak and the flow remains almost steady even at the maximum Wi in the present study.

1.3 C. Time dependent flow asymmetry around a cylinder

Fig. 14

Flow asymmetry around the front cylinder in the asymmetric geometry ($\Delta y=d/16$) at $\mathrm {Wi=3.75}$ for $\beta =0.05$ and $L^2=1000$. The standard deviation of the fluctuation of $I_1$ in the fully-developed regime is $0.24 \%$ of the mean value

The value of flow asymmetry fluctuates around a well-defined mean once the instability becomes fully developed (Fig. 14). The standard deviation of the fluctuation is $0.24 \%$ of the mean value at $\mathrm {Wi=3.75}$.

1.4 D. Elastic instability criteria

Fig. 15

The Pakdel–McKinley (M) parameter in the asymmetric geometry ($\Delta y=d/16$) at a $\mathrm {Wi=0.62}$ and b $\mathrm {Wi=1.25}$. Other parameters are $\beta =0.05$, $L^2=1000$, and $t=17.5$ (steady state). White circles indicate the regions of $\mathrm {M_{\max }}$. The values of $\mathrm {M_{\max }}$ represent the mean and the standard deviation obtained over $3 \times 3$ pixel area centered at the point of maximum value of M

The Pakdel–McKinley parameter (M) is widely used to characterize the criteria for elastic instability in curved geometry [1, 2]. The Pakdel-McKinley parameter is defined as:

$$\begin{aligned} M=\left[ \frac{\tau _{11}}{\eta _0 \dot{\gamma }} \lambda U \kappa \right] ^{1/2} , \end{aligned}$$

(10)

where $\tau_{11}$, $\dot{\gamma}$, and $\kappa$ are the local tensile stress along the streamline direction, the magnitude of the shear rate, and streamline curvature, respectively. The details to calculate these variables can be found in the literature [10, 38]. The elastic instability occurs when $\mathrm {M \ge M_{\text {crit}}}$. The spatial profiles of the M parameter in the asymmetric geometry at different Wi ($<\mathrm {Wi_{cr1}}$) have been shown in Fig. 15. The location where the value of M is maximum is the most sensitive region to the instability [10]. Similar to the symmetric geometry [10], the location of $\mathrm {M_{\max }}$ shifts from the side of the rear cylinder to the region in between the cylinders as $\mathrm {Wi \rightarrow Wi_{cr1}}$ (Fig. 15) and hence the formation of new flow state at $\mathrm {Wi_{cr1}<Wi<Wi_{cr2}}$ occurs due to instability in the region between the cylinders.