Impact of interfacial rheology on finger tip splitting

Fluid-fluid interfaces, laden with polymers, surfactants, lipid bilayers, proteins, solid particles, or other surface-active agents, often exhibit a rheologically complex response to deformations. Despite its academic and practical relevance to fluid dynamics and various other fields of research, the role of interfacial rheology in viscous fingering remains fairly underexplored. A noteworthy exception is the work by Li and Manikantan [Phys. Rev. Fluids 6, 074001 (2021)], who used linear stability analysis to show that surface rheological stresses act to stabilize the development of radial viscous fingering at the linear regime. In this paper, we perform a perturbative, second-order mode-coupling analysis of the system and investigate the influence of interfacial rheology on the morphology of the fingering structures at early nonlinear stages of the dynamics. In particular, we focus on understanding how interfacial rheology impacts the emblematic finger tip-widening and finger tip-splitting phenomena that take place in radial viscous fingering in Hele-Shaw cells. We describe the viscous Newtonian fluid-fluid interface by using a Boussinesq-Scriven model, and derive a generalized Young-Laplace pressure jump condition at the fluid-fluid interface. In this framing, we go beyond the purely linear description and use Darcy's law to obtain a perturbative mode-coupling differential equation which describes the time evolution of the perturbation amplitudes, accurate to second order. Our early nonlinear mode-coupling results indicate that regardless of their stabilizing action at the linear regime, interfacial rheology effects favor finger tip widening, leading to the occurrence of enhanced finger tip-splitting events.

Figure 1

A schematic of the radial viscous fingering problem in a Hele-Shaw cell of gap thickness $b$. The inner (outer) fluid has viscosity ${\eta }_1$ (${\eta }_2$). The fluids are separated by a structured, rheological interface. Fluid 1 is injected into the cell, previously filled with fluid 2, with constant injection rate $Q$. The time-dependent unperturbed, circular interface (dashed circle) has radius $R\left(t\right)$, and the interface perturbation is denoted by $\zeta (\theta ,t)$, where $\theta$ is the azimuthal angle. The deformed interface is represented as $\mathcal{R}(\theta ,t)=R\left(t\right)+\zeta (\theta ,t)$. The rheological interface has surface tension $\gamma$, surface shear viscosity ${\eta }_s$, and dilatational viscosity ${\kappa }_s$.

Figure 2

Time evolution of the weakly nonlinear interfacial patterns generated by solving Eq. (33) for $0\le t\le t_f$, and three increasingly larger values of the Boussinesq number (a) $\mathrm{Bq}=0$, (b) $\mathrm{Bq}=15$, and (c) $\mathrm{Bq}=30$. For a given $\mathrm{Bq}$, the final time $t_f$ is the maximum allowed time before unphysical interface crossings begin. These final times are (a) $t_f=2754$, (b) $t_f=5087$, and (c) $t_f=6499$. Interface profiles are plotted in time intervals of $t_f/10$, where the interface at $t=t_f$ is depicted by a thicker curve in black. The viscous outer fluid is represented by the shaded region, and the inner fluid of negligible viscosity is represented by the white region. Here we consider the second-order nonlinear coupling of cosine modes, where the fundamental mode $n=n_{\max }^{\zeta }=7$, with its harmonic $2n=14$. The other flow parameters are $A=1$, $\mathrm{Ca}=150$, and $R_0=3$. The initial perturbation amplitudes are $a_n\left(0\right)=R_0/2500$ and $a_{2n}\left(0\right)=0$

Figure 3

Time evolution of the rescaled cosine interfacial amplitudes for the fundamental mode $a_n\left(t\right)/R\left(t\right)$ (dashed curves) and for the first-harmonic mode $a_{2n}\left(t\right)/R\left(t\right)$ (solid curves) associated with the weakly nonlinear patterns illustrated Fig. 2, for increasing values of the Boussinesq number: $\mathrm{Bq}=0,15,30$.

Figure 4

Behavior of $a_{2n}\left(t\right)/R\left(t\right)$ with respect to $a_n\left(t\right)/R\left(t\right)$ for the pattern evolutions depicted in Fig. 2, for different values of the Boussinesq number: $\mathrm{Bq}=0,15,30$. The vertical dashed line is drawn for $a_n\left(t\right)/R\left(t\right)=0.05$.

Figure 5

Time evolution of the weakly nonlinear interfacial patterns generated by solving Eq. (33) for $1\le n\le 25$, $0\le t\le t_f$, and equal time intervals $\mathrm{\Delta }t=t_f/10$. The patterns are obtained for $\mathrm{Bq}=0$ [(a), (d), (g)], $\mathrm{Bq}=15$ [(b), (e), (h)], and $\mathrm{Bq}=30$ [(c), (f), (i)]. In addition, the values of $t_f$ are (a) 1076, (b) 1867, (c) 2423, (d) 983, (e) 1726, (f) 2210, (g) 1144, (h) 1958, and (i) 2461. Three different sets of random phases are used, set I for [(a)–(c)], set II for [(d)–(f)], and set III for [(g)–(i)]. The most prominent occurrences of finger tip-widening and finger tip-splitting events in (c), (f), and (i) for the largest value of the Boussinesq number considered ($\mathrm{Bq}=30$) are indicated by small arrows. Here $A=1$, $\mathrm{Ca}=150$, and $R_0=3$.

Figure 6

Time evolution of the linear [(a)–(c)] and weakly nonlinear [(d)–(f)] interfacial patterns for $1\le n\le 25$, $0\le t\le t_f$, and equal time intervals $\mathrm{\Delta }t=t_f/10$. The patterns are obtained for $\mathrm{Bq}=30$. Moreover, the values of $t_f$ are (a) 2462, (b) 2199, (c) 2435, (d) 2423, (e) 2210, and (f) 2461. The sets of random phases utilized here are equal to those used in Fig. 5: Set I for (a) and (d), set II for (b) and (e), and set III for (c) and (f). The rest of the physical parameters are the same as the ones employed in Fig. 5. By comparing the linear [(a)–(c)] with the corresponding weakly nonlinear [(d)–(f)] structures, it is apparent that the nonlinear fingers are typically wider, having a greater tendency to split at their tips due to the action of surface rheological stresses.