Capillary imbibition of shear-thinning fluids: From Lucas-Washburn to oscillatory regimes

The study of capillary imbibition has ramifications in many fields, such as energy, biology, process industry, and subsurface flows. Although the capillary rise of Newtonian liquids has been the subject of several studies since the seminal works of Lucas (1918) and Washburn (1921), its generalization to the case of non-Newtonian fluids is still an open question. To fill this gap, starting from first principles, we derive a transient one-dimensional model describing the rising dynamics of shear-thinning fluid, whose viscosity is described by the Ellis viscosity model. Our model identifies the scaling for the different imbibition regimes accounting for the interplay of inertial, gravity, and viscous non-Newtonian effects (i.e., the zero-shear-rate and the shear-thinning behavior). Specifically, the rising dynamics is described by the interplay of three dimensionless parameters: the Richardson number (i.e., the ratio between potential and kinetic energies), the Ellis number (i.e., the ratio between the characteristic shear-stress of the fluid and gravity), and the shear-thinning index which quantify the degree of shear-thinning of the fluid. At early times the system follows a universal inertial regime, followed by two possible limiting regimes, i.e., the classical Lucas-Washburn and the oscillatory regimes. The competition between the governing dimensionless numbers dictates the transition between the two. We show that when the viscous effect dominates over inertia, the identification of a (time-dependent) scaling law for the effective viscosity leads to a generalization of the Lucas-Washburn theory and the rescaled trajectories toward equilibrium collapse over the classical 1/2 scaling law. On the contrary, when inertia dominates the later stage of the imbibition, the filling length oscillates around the equilibrium. By means of linear control theory, we discuss the physical mechanisms that lead to such oscillating behavior and map the different regimes in terms of the governing dimensionless parameters.

Figure 1

Sketch of the considered problem: the capillary rise of a shear-thinning fluid in a cylindrical tube of radius $R$. Note that air viscosity and density are negligible if compared to the liquid ones.

Figure 2

(a) Time evolution of the column height as a function of the Ellis number for $\alpha =2$: the inset show the slope of the $h-t$ curve in logarithmic scale; (b) time evolution of the effective viscosity computed at the tube wall as a function of the Ellis number for $\alpha =2$.

Figure 3

(a) Evolution of the pressure gradient, $P$, as a function of the speed of the column, $\dot{h}$, for different El and $\alpha =2$; (b) evolution of the effective viscosity computed at the tube wall as a function of the speed of the column, $\dot{h}$ for different El and $\alpha =2$.

Figure 4

(a) Rescaled time evolution of the column height for $\alpha =2$: the inset shows the slope of the $h-t^{★}$ curve in logarithmic scale; (b) master curve for the effective viscosity for $\alpha =2$ as a function of $1/\mathrm{\Gamma }$ with $\mathrm{\Gamma }$ defined in Eq. (33) where $\mathrm{\Gamma }$ is the effective viscosity in the $\text{El}\to 0$ limit.

Figure 5

(a) Time evolution of the column height as a function of the Ellis number. (b) Time evolution of the column height as a function of the Richardson number.

Figure 6

(a) Map of the system behavior as a function of El and Ri for $\alpha =2$ and $\text{Bo}=0.01.$ (b) Map of the system behavior as a function of El and Ri for different $\alpha$ and $\text{Bo}=0.01$.

Figure 7

(a) Overshoot (i.e., the maximal liquid height reached by the liquid column exceeding the equilibrium configuration) as a function of Ri and El for $\alpha =2$ and $\text{Bo}=0.01$; (b) decay ratio (i.e., the ratio between the height of first oscillation right after the overshoot and the overshoot) as a function of Ri and El for $\alpha =2$ and $\text{Bo}=0.01$.

Figure 8

(a) Evolution of the column speed (solid lines), $\dot{h}$, as a function of time for different Ri when $\text{El}\to \infty$ and $\alpha =2$; Estimate of the entrance length (dashed lines), $\ell$, as a function of time for different Ri when $\text{El}\to \infty$ and $\alpha =2$; (b) Evolution of the column speed (solid lines), $\dot{h}$, as a function of time for different Ri when $\text{El}=0.1$ and $\alpha =2$; Estimate of the entrance length (dashed lines), $\ell$, as a function of time for different Ri when $\text{El}=0.1$ and $\alpha =2$.

Figure 9

(a) Evolution of the dimensionless velocity $f^{\prime }(\eta ,\alpha )$ as function of the similarity variable $\eta$. (b) Values of the similarity values where the dimensionless velocity equal to 99% of the velocity outside the boundary layer.

Figure 10

(a) Evolution of the shape factor, $\gamma$, as a function of the column speed for different El and $\alpha =2$. (b) Evolution of the product $\dot{h}\gamma$ as a function of the column speed for different El and $\alpha =2$.