Simulating fluid-fluid displacement in a soft capillary tube: How compliance delays interfacial instability and bubble pinch-off

TL;DR

This work develops a monolithic fluid–structure interaction framework that couples Navier–Stokes–Cahn–Hilliard immiscible flow with a nonlinear neo‑Hookean solid to simulate fluid displacement in soft capillary tubes. Using isogeometric analysis on a boundary‑fitted mesh and a variational multiscale–stabilized formulation, it resolves topological changes, dynamic wetting, and large solid deformations in axisymmetry. The key findings show that tube compliance delays or suppresses interfacial instability and subsequent bubble pinch‑off for both imbibition and drainage, with distinct stabilization mechanisms in wetting and non‑wetting cases and a strong dependence on the elastocapillary and inlet capillary numbers. The results have implications for flow in soft porous media, bio‑microfluidics, and manufacturing processes, offering a pathway to control interfacial dynamics in deformable confinements.

Abstract

The displacement of a more viscous fluid by a less viscous immiscible fluid in confined geometries is a fundamental problem in multiphase flows. Recent experiments have shown that such fluid-fluid displacement in micro-capillary tubes can lead to interfacial instabilities and, eventually, bubble pinch-off. A critical yet often overlooked aspect of this system is the effect of the tube's deformability on the onset of interfacial instability and bubble pinch-off. Here, we present a computational fluid-structure interaction model and an algorithm to simulate this fluid-fluid displacement problem in a soft capillary tube. We use a phase-field model for the fluids and a nonlinear hyperelastic model for the solid. Our fluid-structure interaction formulation uses a boundary-fitted approach and we use isogeometric analysis for the spatial discretization. Using this computational framework, we study the effects of inlet capillary number and tube stiffness on the control of interfacial instabilities in a soft capillary tube for both imbibition and drainage. We find that tube compliance delays or even suppresses interfacial instability and bubble pinch-off, a finding that has important implications for flow in soft porous media, bio-microfluidics, and manufacturing processes.

Figures (11)

• Figure 1: (a) Schematic of the computational domain, initial conditions, domain boundaries (see Eq. \ref{['eq:strongform_bcs']}) and geometrical parameters. Our 2D axisymmetric domain is obtained by slicing the original 3D tube along the center line axis. We only illustrate a portion of the tube in the 3D schematic to show the presence of air and glycerol. (b-c) Air-glycerol interface at two times, $t_1$ and $t_2$. In (b), $R_c^{\prime}$ denotes the radius of the deformed tube and $R_s$ is the radius of a spherical cap fitted to the interface. The zoomed inset in (b) illustrates our definition of the angles $\beta$ and the apparent contact angle $\theta_\text{a}$. The air-glycerol interface is stable in (b) and unstable in (c). The blue solid line representing the air-glycerol interface at $t_1$ in (b-c) is the level set of $c = 0$.
• Figure 2: Comparison of our simulations with experiments zhao_etal_prl_2018 and previous numerical simulations esmaeilzadeh_etal_pre_2020 in a rigid capillary tube. (a) Steady state air-glycerol interface in the wetting capillary tube ($\theta = 68^\circ$) for different $\mathrm{Ca}$. (b) Steady state air-glycerol interface in the wetting capillary tube ($\theta = 68^\circ$) for two different values of $D_w$ when $\mathrm{Ca} = 0.012$. (c) Contact line capillary number $\mathrm{Ca_{cl}}$ as a function of $\mathrm{Ca}$ in wetting ($\theta = 68^\circ$) and non-wetting ($\theta = 115^\circ$) capillary tubes. The green and blue vertical dotted lines represent the critical inlet capillary numbers $\mathrm{Ca_{cr}}$. The insets in the left subfigure show sketches of the stable glycerol-air interface regime when $\mathrm{Ca} < \mathrm{Ca_{cr}}$ and unstable air-glycerol interface regime when $\mathrm{Ca} > \mathrm{Ca_{cr}}$. The zoomed insets on the right subfigure show a close up of the comparison between our simulation results and the results from the literature zhao_etal_prl_2018esmaeilzadeh_etal_pre_2020.
• Figure 3: Variation of the thickness of entrained film $h_f$ with the interface tip capillary number $\mathrm{Ca_t}$ (top) and dependence of $\mathrm{Ca_t}$ on the inlet capillary number $\mathrm{Ca}$ (bottom) for a rigid tube. Panel (a) shows the wetting case ($\theta = 68^\circ$) and panel (b) shows the non-wetting case ($\theta = 115^\circ$).
• Figure 4: Time evolution of air-glycerol interface profiles and radial solid displacements at the fluid-solid interface in a soft wetting capillary tube ($\theta = 68^\circ$) for different values of inlet capillary number ($\mathrm{Ca}$). Air is gray, glycerol is blue and the solid is brown. The inset in (b) shows the time variation of the pressure $p_m$ along the symmetry axis. In (d), the radius of the finger $R_f$ at its midpoint is specified at two different times.
• Figure 5: Air-glycerol interfacial instability metrics in a wetting capillary tube ($\theta = 68^\circ$) for three different values of inlet capillary number ($\mathrm{Ca}$). (a) Temporal variation of interface arc length $L_\text{arc}$. The inset shows also the time evolution of $\mathrm{Ca_{cl}}$ for $\mathrm{Ca} = 0.04$. (b) Time evolution of apparent contact angle $\theta_\text{a}$. The insets show the snapshots of the solution in both rigid and soft tubes for $\mathrm{Ca} = 0.04$ near the onset of instability. In the insets, $R_s$ denotes the radius of the spherical cap, which at that specific time instant, is nearly perpendicular to the fluid-solid interface profile at the triple contact point. In (a-b), the solid lines denote the results in a soft tube and the dashed lines denote the results in a rigid tube. The air-glycerol interface is unstable for $\mathrm{Ca} = 0.015$ and $0.040$, but stable for $\mathrm{Ca} = 0.012$.

Theorems & Definitions (3)

• Remark 3.1
• Remark 3.2
• Remark 3.3