Viscous fingering in yield stress fluids: a numerical study

Abstract

The effect of yield stress is numerically investigated on the viscous fingering phenomenon in a rectangular Hele–Shaw cell. It is assumed that the displacing fluid is Newtonian, while the displaced fluid is assumed to obey the bi-viscous Bingham model. The lubrication approximation together with the creeping-flow assumption is used to simplify the governing equations. The equations so obtained are made two-dimensional using the gap-averaged variables. The initially flat interface between the two (immiscible) fluids is perturbed by a waveform perturbation of arbitrary amplitude/wavelength to see how it grows in the course of time. Having treated the interfacial tension like a body force, the governing equations are solved using the finite-volume method to obtain the pressure and velocity fields. The volume-of-fluid method is then used for interface tracking. Separate effects of the Bingham number, the aspect ratio, the perturbation parameters (amplitude/wavelength), and the inlet velocity are examined on the steady finger width and the morphology of the fingers (i.e., tip-splitting and/or side-branching). It is shown that the shape of the fingers is dramatically affected by the fluid’s yield stress. It is also shown that a partial slip has a stabilizing effect on the viscous fingering phenomenon for yield-stress fluids.

Appendix

1.1 The effect of wall slip

In the experimental work performed by Lindner et al. [21], they have noticed that their viscoplastic foam has a tendency to slip. And, the slip was found to delay the tip-splitting in Hele–Shaw cell [21]. To see if our code is capable of corroborating this finding, we can enter the slip velocity at the stage of integrating Eqs. (3) and (4) in the z-direction; that is,

$$\begin{aligned} {\mathbf{v}}(x,y,z,t)=-{\mathbf{P}}(x,y,t)\int \limits _z^{h/2} {\frac{\xi \,{\mathrm {d}}\xi }{\eta (\xi )} + } {\mathbf{v}}_\mathrm{{s}} \left( {x,y,t} \right) , \end{aligned}$$

(25)

where ${\mathbf{v}}_\mathrm{{s}}$ is the slip velocity vector. In the present work, we use the Navier’s slip boundary condition [36] to include slip effects into our analysis. To that end, we write

$$\begin{aligned} \theta u_\mathrm{{s}} \left( {x,y,t} \right) +\left( {1-\theta } \right) {\tau _{xz} } \left. \right| _{z=h/2}= & {} 0, \end{aligned}$$

(26)

$$\begin{aligned} \theta v_\mathrm{{s}} \left( {x,y,t} \right) +\left( {1-\theta } \right) {\tau _{yz} } \left. \right| _{z=h/2}= & {} 0, \end{aligned}$$

(27)

where $0\le \theta \le 1$ is the so-called slip factor [36]. It is to be noted that for $\theta =0$, Eqs. (26) and (27) reduce to the so-called free-slip boundary condition. On the other hand, for $\theta =1$, these equations reduce to the no-slip boundary condition. As regards the shear stress terms appearing in Eqs. (26) and (27), based on the lubrication approximation, they can be simplified as

$$\begin{aligned} {\tau _{xz}}\left. \right| _{z=h/2}= & {} 2\eta {D_{xz} }\left. \right| _{z=h/2} =\eta \left. {\frac{\partial u}{\partial z}} \right| _{z=h/2}, \end{aligned}$$

(28)

$$\begin{aligned} {\tau _{yz}}\left. \right| _{z=h/2}= & {} 2\eta {D_{yz}}\left. \right| _{z=h/2} =\eta \left. {\frac{\partial v}{\partial z}} \right| _{z=h/2}. \end{aligned}$$

(29)

Thus, from Eqs. (3) and (4), we obtain

$$\begin{aligned} {\tau _{xz} } \left. \right| _{z=h/2}= & {} \frac{h}{2}P_x, \end{aligned}$$

(30)

$$\begin{aligned} {\tau _{yz} } \left. \right| _{z=h/2}= & {} \frac{h}{2}P_y. \end{aligned}$$

(31)

Therefore, Eq. (25) is transformed to

$$\begin{aligned} {\mathbf{v}}(x,y,z,t)=-{\mathbf{P}}(x,y,t)\left[ {\int \limits _z^{h/2} {\frac{\xi {\mathrm {d}}\xi }{\eta (\xi )} + } \left( {\frac{1-\theta }{\theta }} \right) \frac{h}{2}} \right] . \end{aligned}$$

(32)

Also, the gap-averaged velocity can be written as

$$\begin{aligned} {\bar{{\mathbf{v}}}}(x,y,t)=-\frac{h^{2}}{12}{\mathbf{P}}\left[ {24h^{-3}\int \limits _0^{h/2} {\frac{z^{2}}{\eta }{\mathrm {d}}z} +\left( {\frac{1-\theta }{\theta }} \right) \frac{6}{h}} \right] . \end{aligned}$$

(33)

Thus, the effective viscosity for the fluid 2 becomes

$$\begin{aligned} \eta _{2,\mathrm{{eff}}} =\left[ {24h^{-3}\int \limits _0^{h/2} {\frac{z^{2}{\mathrm {d}}z}{\eta _2 \left( {\dot{\gamma }} \right) }+\left( {\frac{1-\theta }{\theta }} \right) \frac{6}{h}} } \right] ^{-1}. \end{aligned}$$

(34)

Figure 14 shows the effect of the wall slip on the results for the case in which the Bingham fluid is allowed to slip (i.e., $\theta \le 1$). As can be seen in this figure, the slip of fluid 2 has a stabilizing effect on the time evolution of the finger. As a matter of fact, for $\theta =0.7$, the perturbation is seen to have been stabilized.

Fig. 14

The effect of the wall slip, ${{\varvec{\theta }} }$, on the shape of the finger plotted at equal time steps ($S=0.05,\,\varepsilon =0.01,\,\Gamma =0.002,\,$Bn = 200 and $t_f =0.5$)