Stability of viscous fingering in lifted Hele-Shaw cells with a hole

Lifted Hele-Shaw cells typically display viscous fingering of liquids, which in turn leads to branched fractal patterns in the absence of any anisotropies. Recently, experiments involving parallely lifted Hele-Shaw cells with holes in the cell plates, also termed as “multiport lifted Hele-Shaw cells,” have been used to generate more regular meshlike patterns in the liquid film. Although such patterns promise usefulness in several applications, their spatiotemporal evolution needs to be theoretically understood for better synthesis. As a first step, therefore, we examine the stability of fingers evolving from a single hole by focusing on flow of an annular film of liquid placed in a lifted Hele-Shaw cell. We use linear stability analysis to find the growth rate of azimuthally periodic perturbations of the inner and outer interfaces around the evolving base state of the liquid film. To validate the results of our stability analysis, we also perform resolved numerical simulations of the setup via an in-house solver based on lubrication theory, which uses front-tracking method to evolve the interface in time and space. For a wide range of parameters and wave numbers, we find excellent agreement in the growth rates predicted by the linear stability analysis with the numerical simulation. The results of the stability analysis are expressed in terms of the capillary number, initial nondimensional plate separation, and initial ratio of the interface radii. Furthermore, using the results from our linear stability analysis, we generate a phase map to demarcate the flow regimes corresponding to unstable and stable states for the interfaces. Numerical simulations of the interface evolution over finite times are consistent with the results predicted by the proposed analysis. These finite-time simulations successfully capture the presence of shielding of the fingers at both the inner and outer interface. The proposed theoretical analysis and insights obtained through numerical simulations thus provide a framework for accurately predicting and experimentally realizing stable fluid patterns in a multiport Hele-Shaw cell.

Figure 1

Schematic of different stages of lifted Hele-Shaw cell (LHSC) experiments. (a) Without and (b) with hole in the top plate. We place the liquid drop on plate (stage I), then squeeze the the fluid between plates (stage II), and separate the plates from each other (stage III) until the film on the two plates are completely detached (stage IV). For the LHSC setup with hole [panel (b)], the hole is kept sealed during stages I and II, and opened at the beginning of stage III. The image at the bottom shows fully separated plates with the fluid pattern formed on it. The vertical dash-dot line in the side view schematics (stages I–IV) denote the axis of revolution of the initial liquid drop placed between the plates.

Figure 2

Final top view of liquid film formed on a plate for different values of Ca and $h_0/R_0$. Physical radius of source hole is $R_i=0.5$ mm and initial outer radius of liquid film is $R_0=20$ mm. Further details of the experiments can be found in Ref. [16].

Figure 3

Plan view of LHSC without hole, with perturbation at outer interface; $R_{\text{out}}^0$ is the base radius of the outer interface, $R_{\text{out}}^{\prime }$ describes the perturbation to the outer interface.

Figure 4

Plan view of LHSC with hole; $R_{\alpha }^0{R}_{\alpha }^{\prime }$ are the base radius and interface perturbation, respectively, where $\alpha \in \{\text{in},\text{out}\}$.

Figure 5

Schematic showing fixed Cartesian grid for representing pressure (black lines) and Lagrangian nodes ($\square$) to discretely represent the liquid-gas interface shape. The unit tangent ($\mathbf{t}$) and normal (n) vector has also been shown at an interface node here. The shaded region represents the liquid film.

Figure 6

Schematic showing assignment of phases to pressure nodes. Yellow dots denote “Gas” nodes, green dots denote “Boundary” nodes, blue dots denote “Liquid” nodes. Inset shows stencil for calculating Laplacian operator at a “Boundary” point. Shaded region represents liquid film.

Figure 7

Normal pressure gradient at interface Lagrangian point $P0$ ($\square$), $▵$ nodes are in negative normal direction from Lagrangian nodes. Shaded region represents liquid film.

Figure 8

Logarithmic of Fourier coefficient ${\widehat{R}}_{\alpha ,n}$ vs time $t$ for different values of $n$. Red symbols are results from simulation and solid black lines are linear fits. Panels (a) and (b) are for outer interface of LHSC without hole, with parameters $dh/dt=0.7\mu \mathrm{m}/\mathrm{s}$, $h_0=0.3\mathrm{mm}$, $\mu =93\mathrm{Pa}\mathrm{s}$, $\sigma =20\mathrm{mN}/\mathrm{m}$, $R_{\text{out}}^0=20\mathrm{mm}$, $\mathrm{Ca}/{\left(h^{*}\right)}^3=964$, and panels (c) and (d) are for inner interface of LHSC with hole, with parameters $dh/dt=1.4\mu \mathrm{m}/\mathrm{s}$, $h_0=0.3\mathrm{mm}$, $\mu =93\mathrm{Pa}\mathrm{s}$, $\sigma =20\mathrm{mN}/\mathrm{m}$. Initially, $R_{\text{in}}^0=20\mathrm{mm}$, $R_{\text{in}}^0=10\mathrm{mm}$. The nondimensional parameters are $\mathrm{Ca}/{\left(h^{*}\right)}^3=1928$ and $\gamma =2$. The grid resolution is given by $N\times N=1200\times 1200$, $M_{\text{in}}=10000$, $M_{\text{out}}=10000$.

Figure 9

Plot of nondimensional growth rate ${\omega }_{\alpha ,i}^{*}$ vs wave number $n$ for LHSC (a) without hole ($\alpha =\text{out}$) and (b) with hole ($\alpha =\text{in}$). Theoretical expression for ${\omega }_{\alpha ,i}^{*}$ has been compared with measurements from the numerical simulation, for different grid sizes $N\times N=600\times 600$, $840\times 840$, $1200\times 1200$, with lifting velocity (a) $dh/dt=0.7\mu \mathrm{m}/\mathrm{s}$ and $\mathrm{Ca}/{\left(h^{*}\right)}^3=964$ (b) $dh/dt=1.4\mu \mathrm{m}/\mathrm{s}$, $\mathrm{Ca}/{\left(h^{*}\right)}^3=1928,\gamma =2$. Other parameters are $h_0=0.3\text{mm}$, $\mu =93\frac{\text{Ns}}{{\text{m}}^2}$, $\sigma =20\mathrm{mN}/\mathrm{m}$; initially $R_{\text{out}}^0=20\mathrm{mm}$. Initial interface resolution is $M_{\text{out}}=10000$ for panel (a) and $M_{\text{in}}=M_{\text{out}}=10000$ for panel (b).

Figure 10

Pattern formed on plate in LHSC without hole, Contour shows the pressure distribution and black line shows interface $\mathcal{C}$ from our simulation. Red circles show points from interface from Boundary Integral simulations by Ref. [3]. Initial film thickness for (b) $h_0=0.507\text{mm}$ and (c) $h_0=0.804\text{mm}$. Other parameters are $\frac{dh}{dt}=0.73\mu \mathrm{m}/\mathrm{s}$, $\mu =93\frac{\text{Ns}}{{\text{m}}^2}$, $\sigma =20\mathrm{mN}/\mathrm{m}$, $R_{\text{out}}^0=20\text{mm}$ initially. Here, $\tilde{\tau }=\frac{1}{96}\left[\frac{{\left(h^{*}\right)}^3}{\mathrm{Ca}}\right]$ and $t^{*}=t\dot{h}/h_0$. Grid size for our simulation is $N\times N=675\times 675$, $M_{\text{out}}=3000$.

Figure 11

(a) Shape of interface ${\mathcal{C}}_{\text{out}}$ for LHSC without hole for Ca/${\left(h^{*}\right)}^3=964$, $M_{\text{out}}=6000$, for different resolutions of FD grid $N\in \{300,450,675,1000\}$. Order of accuracy of FD (finite difference) method for outer interface is 1.30. (b) Shape of interface ${\mathcal{C}}_{\mathrm{in}}$ for LHSC with hole for Ca/${\left(h^{*}\right)}^3={10}^4$, $\gamma =8$, $M_{\text{in}}=1500$, $M_{\text{out}}=1500$, $L=0.045$ m for different FD grid resolutions $N\in \{300,450,675,1000\}$. Order of accuracy of inner interface for LHSC with the hole is 1.45.

Figure 12

Dimensionless growth rate as function of wave number for different lifting velocities $dh/dt$ with $R_{\text{out}}^0\left(0\right)=20\text{mm},R_{\mathrm{in}}^0\left(0\right)=10\text{mm},h_0=0.3\text{mm}$, $\mu =93\mathrm{Pa}\mathrm{s}$, $\sigma =20\mathrm{mN}/\mathrm{m}$. Nondimensional parameter values: $\mathrm{Ca}/{\left(h^{*}\right)}^3$ for $\square$, $◯$, and $▵$ is 2893, 1928, and 964, respectively, and $\gamma =2$. Filled symbols represent values from LSA theory, while unfilled symbols represent values from simulations. Grid size for our simulation is $N=1200\times 1200$, $M_{\text{in}}=10000$, $M_{\text{out}}=10000$.

Figure 13

Dimensionless growth rate as a function of wave number for different initial thickness $h_0$, and $R_{\text{out}}^0\left(0\right)=20\text{mm},R_{\mathrm{in}}^0\left(0\right)=10\text{mm},dh/dt=2.1\mu \text{m/s}$, $\mu =93\mathrm{Pa}\mathrm{s}$, $\sigma =20\mathrm{mN}/\mathrm{m}$. Nondimensional parameters: $\mathrm{Ca}/{\left(h^{*}\right)}^3$ for $\square$ and $◯$ is 2893 and 1220, respectively, and $\gamma =2$. Filled symbols represent values from LSA theory, while unfilled symbols represent values from simulations. Grid size is $N\times N=1200\times 1200$, $M_{\text{in}}=10000$, $M_{\text{out}}=10000$.

Figure 14

Dimensionless growth rate of different wave numbers for inner and outer interface with parameters $R_{\text{out}}^0\left(0\right)=20\text{mm},R_{\mathrm{in}}^0\left(0\right)=10\text{mm},dh/dt=1.4\mu \text{m/s}$, $\mu =93\mathrm{Pa}\mathrm{s}$, $\sigma =20\mathrm{mN}/\mathrm{m}$. Nondimensional parameters: $\mathrm{Ca}/{\left(h^{*}\right)}^3=1928,\gamma =2$ for both $◯$ and $\diamond$. Filled symbols represent values from LSA theory, while unfilled symbols represent values from simulations. Grid size is $N\times N=1200\times 1200$, $M_{\text{in}}=10000$, $M_{\text{out}}=10000$.

Figure 15

Neutral stability curves (solid, blue curves) for different values of radius ratio $\gamma$ for inner (a), (c), (e) and outer (b), (d), (f) interface, for lifted Hele-Shaw cell with hole. The shading (cyan) denotes the unstable region. The dashed (red) line is the approximation for $n_{+}$ versus $\mathrm{Ca}/{\left(h^{*}\right)}^3$ for inner and outer interface, given by Eqs. (41) and (44), respectively. The expression for $n_{+}$ for LHSC without hole [Eq. (29)] has also been denoted via (black) dash-dot line for reference in panels (b), (d), and (f).

Figure 16

Plots of $n_{max}$ vs $\mathrm{Ca}/{\left(h^{*}\right)}^3$ at (a) inner interface and (b) outer interface for different representative values of radius ratio $\gamma$ for LHSC with hole.

Figure 17

Phase map showing stability regime for LHSC with hole for (a) inner and (b) outer interface over the $\gamma$ and $\mathrm{Ca}/{\left(h^{*}\right)}^3$. The red curve separates the unstable (shaded) region from the stable (white) region. Isocontours of ${\left({\omega }_{i,max}\right)}^{0.1}$ have been shown for the unstable region of the phase map. Here ${\omega }_{i,max}$ denotes the maximum growth rate over a range of perturbation modes for a particular value of $\mathrm{Ca}/{\left(h^{*}\right)}^3$ and $\gamma$. The symbols indicate parameters for which simulations have been performed over finite time (Sec. 3c). $◯$, $\square$, and $◊$ are used to mark $\mathrm{Ca}/{\left(h^{*}\right)}^3=1$, ${10}^2$ and ${10}^4$, respectively, for $\gamma =1.5$ and $\gamma =8$.

Figure 18

Pattern formed on plate in LHSC with the hole for $\gamma =1.5$, (a), (d), (g) for $\mathrm{Ca}/{\left(h^{*}\right)}^3=1$, (b), (e), (h) for $\mathrm{Ca}/{\left(h^{*}\right)}^3={10}^2$, and (c), (f), (i) for $\mathrm{Ca}/{\left(h^{*}\right)}^3={10}^4$. Three time interval (0 s, 10 s, 24 s) results are shown for different $\mathrm{Ca}/{\left(h^{*}\right)}^3$, contour shows the pressure distribution. Nondimensional time for panels (a), (d), (g) is (0, 0.0096, 0.023), respectively, (0, 0.044, 0.106) for panels (b), (e), (h), and (0, 0.206, 0.494) for panels (c), (f), (i). Grid Size is $N\times N=675\times 675$, $M_{\text{in}}=2000$, $M_{\text{out}}=3000$.

Figure 19

Pattern formed on plate in LHSC with the hole for $\gamma =8$ (a), (d), (g) for $\mathrm{Ca}/{\left(h^{*}\right)}^3=1$, (b), (e), (h) for $\mathrm{Ca}/{\left(h^{*}\right)}^3={10}^2$, (c), (f), (i) for $\mathrm{Ca}/{\left(h^{*}\right)}^3={10}^4$. Three time interval (0 s, 10 s, 24 s) results are shown for different $\mathrm{Ca}/{\left(h^{*}\right)}^3$, contour shows the pressure distribution. Nondimensional time for panels (a), (d), (g) is (0, 0.0096, 0.0229), respectively, (0, 0.044, 0.106) for panels (b), (e), (h), and (0, 0.206, 0.494) for panels (c), (f), (i). Grid Size is $N\times N=675\times 675$, $M_{\text{in}}=375$ for $\mathrm{Ca}/{\left(h^{*}\right)}^3=1$, $M_{\text{in}}=1500$ for $\mathrm{Ca}/{\left(h^{*}\right)}^3={10}^2$ and ${10}^4$, $M_{\text{out}}=3000$.