Stability analysis of a Newtonian film flow over hydrophobic microtextured substrates

We consider the flow of a Newtonian liquid film over an inclined hydrophobic wall textured with periodical microgrooves, the depth of which is much longer than their width, which is of the order of the capillary length of the liquid. Due to their structure, these grooves can be likened to slits. The flowing liquid fails to thoroughly wet the topography forming a second liquid-gas interface, with air encapsulated inside the topographical features. Under these conditions, two possible flow configurations may arise: (1) the inner interface will be pinned at the edges of the slit (ideal Cassie-Baxter state) and (2) the film may partially wet the sidewalls of the slits, forming two additional contact lines with the substrate. We investigate both the steady flow and the stability of a Newtonian liquid film flowing over various substrates with such flow configuration. We solve the 2D Navier-Stokes equations and develop a finite element model to accurately describe the exact shape of all liquid-gas interfaces at a steady state. We determine the linear stability of the steady-state solutions when subjected to perturbations in the streamwise direction and employ the Floquet-Bloch theory to account for disturbances of arbitrary wavelengths, i.e., not necessarily matching the periodicity of the substrate. Through numerical simulations, we highlight the effect of inertia, viscous and capillary forces, and the mobility of the contact line on the stability of the fluid flow. We examine the impact of substrate wettability and orientation with respect to gravity and geometric characteristics of the substrate. It is demonstrated that when the film partially wets the sidewalls of the trench, multiple steady states may arise, which are analyzed for their stability characteristics. It is also shown that the second air-liquid interface and the air pockets inside the grooves of a structured hydrophobic surface may considerably stabilize the flow mainly by the capillary forces, which act as a damper, preventing the disturbances of the outer free surface to grow.

Figure 1

Cross section of a film flowing over an inclined substrate featuring periodic slits in (a) ideal Cassie-Baxter state and (b) partial wetting configuration. In both cases, the unit cell of the periodic structures has a length $L^{*},$ and $L_1^{*}$ and $L_2^{*}$ denote the inflow and outflow length of the unit cell, respectively; the length of the solid part of the substrate is $\left(L_1^{*}+L_2^{*}\right)$. The gap of the slit has a width $W^{*}$, while the film height at the inlet of the periodic domain is denoted by $H^{*}$. Note that $H^{*}$ is not constant since it depends on the flow rate and material parameters and will be calculated as part of the solution. The angle of the substrate inclination is denoted by $\alpha$, while ${\theta }_1$ and ${\theta }_2$ correspond to the contact angles between the solid wall and the inner liquid-air surface. In (b) ${H_1}^{*}$ and ${H_2}^{*}$ indicate the distance of the contact lines along the upstream and downstream slit walls from the corresponding edges.

Figure 2

Schematic of the contact line along the sidewalls depicting the side view of the base state and perturbed interface profile and definitions of $l_i$ and $l_{b,i}$ which denote the interfacial distance from the wall. The relation between $x_{c,i}$, $x_{cb,i}$ and $x_{cd,i}$ is $x_{c,i}=x_{cb,i}+\delta x_{cd,i}$.

Figure 3

Streamlines and the spatial variation of $v_x$ for (a) $\mathrm{Ka}=10$, (b) $\mathrm{Ka}=100$, and (c) $\mathrm{Ka}=1000$. The remaining parameter values are $\mathrm{Re}=5.5$, $\alpha ={10}^{\circ },$ and the film thickness for each liquid is $1.39$, $0.39$, and $0.11$, respectively.

Figure 4

The maximum interfacial velocity, $U_s$, normalized by the surface velocity of simple Nusselt flow as a function of $\mathrm{Ka}$ for various values of (a) the slit width, $W,$ for $L=1$, and (b) the length of the unit cell, $L$, for $\mathrm{Tr}=0.67$. The remaining parameter values are $\mathrm{Re}=5.5$ and $\alpha ={10}^{\circ }$.

Figure 5

Critical $\mathrm{Re}$ as a function of $\mathrm{Ka}$ for (a) various values of $\mathrm{Tr}$ and for $L=1$ and (b) various values of $L$ and for $\mathrm{Tr}=0.67$. The other parameter value is $\alpha ={10}^{\circ }$.

Figure 6

Free-surface disturbances for $Q=0.25$ which corresponds to an unstable mode of the Cassie-Baxter state for $\mathrm{Re}=10$ and $\mathrm{Ka}=100$. The corresponding eigenvalue is $\lambda =0.053+1.249i$. The geometric parameter is $\mathrm{Tr}=0.67$ and $L=1$, while the inclination angle is $\alpha ={10}^{\circ }$.

Figure 7

Stability diagrams for (a) $\mathrm{Ka}=10$ and different values of $\mathrm{Tr}$, (b) $\mathrm{Tr}=0.67$ and different values of $\mathrm{Ka}$. The remaining parameter values are $L=1$ and $\alpha ={10}^{\circ }.$

Figure 8

Effect of inclination angle in the stability diagrams for $\mathrm{Ka}=10$, $L=1$ and $\mathrm{Tr}=0.67$.

Figure 9

(a) Wetting length $H_1$ as a function of $\mathrm{Re}$, for $W=2,L_1=2,L_2=2,$ $\alpha ={60}^{\circ },$ and $\theta ={120}^{\circ }$. (b), (c) steady-state formation and streamline pattern that corresponds to the upper [(b), $\mathrm{Re}=1$] and lower branch [(c),$\mathrm{Re}=1.27$] of panel (a), respectively.

Figure 10

(a), (b) Eigenspectra for the corresponding steady states of Figs. 9 (9) and 9, respectively. $Q\in \left[0,0.5\right]$.

Figure 11

Steady (black lines) and perturbed (red dashed lines) film shape for $\mathrm{Re}=1.27$, $\mathrm{Ka}=10,L_1=L_2=2$, $W=2$ $\alpha ={60}^{\circ }$. The perturbed film shape results from the superposition of the steady film and the disturbance that corresponds to the most unstable mode, i.e., for $Q=0,\lambda =0.24+0.00i$.

Figure 12

(a) Wetting length $H_2$ as a function of $\mathrm{Re}$ for different values of $\mathrm{Ka}$ and $W=1,L_1=2,L_2=2,\alpha ={60}^{\circ }$ and $\theta ={120}^{\circ }$. Panels (b) and (c) correspond to indicative flow configurations shown with points in panel (a).

Figure 13

Wetting length $H_2$ as a function of $\mathrm{Re}$ for different (a) slit-widths, W, for $L_1=L_2=2$, (b) inflow and outflow lengths of the unit cell for $W=1$ and $\mathrm{Ka}=10$, $\alpha ={60}^{\circ },$ and $\theta ={120}^{\circ }$, (c) substrate wettabilities for $W=1,L_1=L_2=2$, $\mathrm{Ka}=10$, and $\alpha ={60}^{\circ }$, and (d) inclination angles for $W=1,L_1=L_2=2$, $\mathrm{Ka}=10$, and $\theta ={120}^{\circ }$.

Figure 14

Eigenspectra for $Q\in \left[0,1\right)$ corresponding to the steady states shown in Fig. 12 for $\mathrm{Ka}=10$. Panels (a) and (b) depict the spectrum that corresponds to the upper solution branch for $\mathrm{Re}=5.8$ and $5.0$, respectively. Panels (c) and (d) depict the spectrum that corresponds to the lower solution branch for $\mathrm{Re}=5.8$ and $5.0$, respectively.

Figure 15

Stability diagrams for different values of $\mathrm{Ka}$ for $L_1=L_2=2,W=1,$ $\alpha ={60}^{\circ }$, and $\theta ={120}^{\circ }.$

Figure 16

Stability diagrams for different (a) inflow and outflow lengths of the unit cell and $\alpha ={60}^{\circ }$ and (b) inclination angles and for $L_1=L_2=2$. The remaining parameter values are $Ka=10,W=1,$ and $\theta ={120}^{\circ }.$

Figure 17

(a) Stability diagrams for substrates with different wettability. (b) Eigenspectrum for different values of ${\beta }_{cl}$, $\mathrm{Re}=5$. The remaining parameter values are $\mathrm{Ka}=10,W=1,\alpha ={60}^{\circ },$ and $\theta ={120}^{\circ }.$

Figure 18

Comparison of (a) the simulation in OpenFoam by Lampropoulos et al. [36], with (b) the predictions of our in-house FEM code, for $\mathrm{Re}=6\times {10}^{-4}$, $\mathrm{Ka}=13.9\times {10}^{-2}$, $a=89.{50}^{\circ }$, $\theta ={120}^{\circ }$, $L_1=L_2=10$, and $W=2.79$.

Figure 19

(a) Stability maps for Nusselt flow. The solid lines depict the predictions of our in-house FEM code, and the black dashed line depicts the predictions from the analytical expression (A1) for $\mathrm{Ka}=10$. (b) Mesh convergence test for the computed eigenspectrum of the ideal Cassie-Baxter state and for $\mathrm{Re}=10$, $\mathrm{Ka}=100$, $L=1$, and $\mathrm{Tr}=0.67$. The mesh properties can be found in Table 2.