Capillary flow in V-grooves
6.7 Numerical validation
6.7.1 Numerical validation methodology
The true (non-circular) cross-sectional fluid interface and corresponding electric field and flux values were computed using the finite element method in comsol (Com, 2017). Geometry corresponding to Figure 6.11 was implemented and given a triangular mesh by comsol, in which elements on the interface and within the liquid domain were chosen to have length at most7.5%of the center fluid thickness, and elements in the vacuum domain were allowed length at most5%of the total domain height b.
The initial geometry was set up with the liquid domain having a circular interface, as would be the case without an electric field. The electrostatic equation ∇2Ψ = 0 was set up in the vacuum domain, with boundary conditions Ψ|walls, interface = 0 and Ψ|y=b = Ψelectrode. At the top corners of the vacuum domain, the electrode with potential Ψelectrode ̸= 0 meets the wall with potential Ψ = 0. This was resolved only at the level of the smallest finite element, with length at most 0.05b; finer resolution of these interior corners is not expected to have a significant effect on the electric field far from the corners.
In order to naturally produce an interface in which the sum of capillary pressure and Maxwell pressure remains constant in x, a time-dependent non-inertial 2D incompressible fluid equa- tion was solved in the liquid domain. Using comsol’s built-in arbitrary Lagrangian-Eulerian (ALE) moving mesh algorithm. Specifically, a 2D incompressible fluid satisfying∇ ·[U, V] = 0,
∇2[U, V] = −∇P was solved, with no-slip wall boundary conditions (U|wall = V|wall = 0).
The interface boundary conditions were set by 2ˆn·(∇[U, V])·ˆn−P = 2K+ We(∇Ψ)2 and nˆ·(∇[U, V] +∇[U, V]T)·tˆ= 0 at the fluid interface, where nˆ and ˆt represent normal and tangent unit vectors at the fluid interface andK is the interface mean curvature (see Chapter 2
for a review of fluid mechanics boundary conditions). A kinematic condition (see Chapter 2 for a review) was set, imposing that the fluid interface must move according to the velocity of the fluid, specifically, ∂TΣ(X, T) +U ∂XΣ(X, T) =V at the fluid interface, whereY = Σ(X, T) represents the interface. comsolwas directed to simultaneously solve the electrostatic equa- tion in the vacuum region (with the updated fluid interface) and the fluid equations in the fluid region. The result was that the fluid interface was continually adjusted until the capillary and Maxwell pressures were constant on the interface.
Once the interface shape was determined, the Poisson equation for the streamwise velocity, Equation (6.13), was computed in the liquid domain to determine the flux factorΓnum.=Q/H4. The cross-sectional area Anum. was also computed. While the interface was no longer circular, H was still taken to be the midline thickness of the fluid.
In order to compare the numerical results to the theoretical model, it was necessary to numerically determine the constantχ0. This was accomplished by fitting all numerically computed Maxwell pressures (with all values of h and θ) for a given groove angle α onto the line χ0Hm, as in Figure 6.12. χ0is the only parameter which was fit from the numerical results (more specifically, C, the deviation of the Maxwell pressure from the annular wedge solution, is the value which isb fit, and in turn determinesχ0). Note that it is because of this fitting methodology that systems with very small and very large values ofθappear to have a larger error than those with middling values of θ. The error could be reduced by fitting a separate χ0 for each value ofθ. Thus, if one knows what material and what groove angle will be used, as well as the external electric field (or shape of the electrode) a priori, one can compute a more accurateχ0 and have a lower error in the theoretical model.
Numerical validation results
A parameter sweep was performed for thin films varying betweenh/b= (d/b)H ∈[10−4,10−1], with α =∈ {15◦,30◦,45◦,60◦,75◦} and θ ∈ {10◦,15◦,30◦,45◦,60◦}. In each case, the elec- trode potential was set to Ψelectrode = 1/√
m0.1m. This electric field strength was chosen in order to make hthresh. ≈ 0.1b. That is, the electric field was set to approximately the highest value accessible for (h/b) = 0.1 under the constraints of the model.4
First, the Maxwell pressure, −PMaxwell, at the fluid surface was computed. Two example plots comparing PMaxwell to h/b are shown in Figure 6.12, for the extreme values of α = 15◦ and α = 75◦. In each case, it can be seen that the numerical results (X’s) indeed confirm the dominant P ∝ Hm scaling of the theoretical model (solid lines). Note that, due to the large (1030) range of Figure 6.12 (a), no difference is visible between the different values ofθ. However, they are not exactly the same; PMaxwell(α= 15◦, θ= 60◦)/PMaxwell(α= 15◦, θ= 10◦)≈1.38.
4The highest electric field value, and hencehthresh., were not known a priori due to the fluid interface shape being unknown. The choiceh= 0.1bturned out to range from around1to1.3timeshthresh.. It is for this reason that Figures 6.13 and 6.14 show some data points with values ofH/Hthresh.exceeding 1.
(a) (b)
Figure 6.12: Log-log plots of rescaled Maxwell pressure,PMaxwell, against rescaled film midline thickness, h/b, forα = 15◦ (left) andα = 75◦ (right). X’s denote data computed numerically in a cross-sectional domain with a flat electrode of constant potential (Figure 6.11) using the finite element method software comsol, as described in Section 6.7.1. Solid lines denote the theoretically predicted power lawPMaxwell∝(h/b)m, with m= (π/α)−2.
(a) Interior groove half angleα = 15◦, with contact angles θ∈ {10◦,15◦,30◦,45◦,60◦}. Note that the results for different contact angles are indistinguishable. Forα= 15◦,m= 10.
(b) Interior groove half angleα= 75◦, with contact anglesθ= 10◦ (larger values ofθ were not considered because of the necessity of satisfying the Concus-Finn condition,α+θ < π/2 = 90◦).
For α= 75◦,m= 0.4.
For the parameter range considered, the error in the approximation of the flux factor Γ as compared to the numerically computed flux factor Γnum. = Q/H4 and in the approximation of the cross-sectional area AHb 2 compared to the numerically computed Anum. are shown in Figure 6.13. Plots in the left column fix the wetting angleθ= 10◦ and show results for various α, while plots in the right column fix the internal groove half angle at α = 15◦ and vary θ. Even when H > Hthresh., the error in the flux factor remains below 1.1%, while the error in the cross-sectional area reaches at most 1.7%. Thus, Γ and Ab appear to be fairly good approximations.
Next, the top row of Figure 6.14 displays the relative error in the pressure approximationP =
−1/H−χHm as compared to the numerically computed pressure Pnum.. Again, the plot on the left fixes θ = 10◦ and varies α, while the plot on the right fixes α = 15◦ and variesθ. So long as H < Hthresh., the error remains below approximately 3%. While it quickly grows for H > Hthresh., especially for large wetting angles, at that point the film thickness is outside of the model regime.
The plots in the bottom row of Figure 6.14 display the Maxwell pressure as a fraction of the total pressure. These results reveal that the small errors in the pressure approximation are not due simply to the Maxwell pressure being small compared to the capillary pressure. Indeed, for θ = 10◦ and α = 60◦ (left column, purple), the Maxwell force contributes up to half of the pressure, but the error is <1%.
We therefore conclude that the model approximations are indeed reasonable for the regime in which (h/b)≤0.1.
(a) (b)
(c) (d)
Figure 6.13: Flux factor (Γ) and cross-sectional area (A) approximation errors relative to nu-b merical results computed in a cross-sectional domain with a flat electrode of constant potential (Figure 6.11) using the finite element method software comsol, as described in Section 6.7.1.
Such errors arise due to neglecting the effect of a non-circular fluid interface in the derivation of the equations of motion.
The horizontal axis tracks H/Hthresh. as the electric potential strength Ψelectrode is held fixed, whereHthresh.is the threshold fluid thickness defined in Equation (6.44). In each simulation, the point of maximal H is h= 0.1b(H=h/d= 0.1b/d), whereb is the distance to the electrode.
Plots in the left column [(a),(c)] fix the contact angleθ= 10◦ and vary the interior groove half angleα; plots in the right column [(b),(d)] fix α= 15◦ and vary θ.
Top row [(a)-(b)]: Relative flux factor error, (Γ−Γnum.)/Γnum., due to approximating the cross-sectional flux parameter asΓ(α, θ)(see Figure 6.3 and Chapter 4) without considering the effect of a non-circular fluid interface. The numerical result,Γnum.=Q/H4, with Qbeing the numerically computed streamwise flux.
Bottom row [(c)-(d)]: Relative cross-sectional area error,(AHb 2−Anum.)/Anum.,due to approx- imating the cross-sectional area as A(α, θ)Hb 2. The numerical result, Anum., is determined by numerically integrating the cross-sectional domain.
(a) (b)
(c) (d)
Figure 6.14: Quantities related to pressure computed in a cross-sectional domain with a flat electrode of constant potential (Figure 6.11) using the finite element method softwarecomsol, as described in Section 6.7.1.
The horizontal axis tracks H/Hthresh. as the electric potential strength Ψelectrode is held fixed, whereHthresh.is the threshold fluid thickness defined in Equation (6.44). In each simulation, the point of maximal H is h= 0.1b(H=h/d= 0.1b/d), whereb is the distance to the electrode.
Plots in the left column [(a),(c)] fix the contact angleθ= 10◦ and vary the interior groove half angleα; plots in the right column [(b),(d)] fix α= 15◦ and vary θ.
Top row [(a)-(b)]: Relative pressure error, (−1/H −χ0Hm −Pnum.)/Pnum., due to approx- imating the pressure as −1/H−χ0Hm without considering the effect of a non-circular fluid interface. Pnum. is the numerically computed rescaled pressure (including both capillary and Maxwell pressure).
Bottom row [(c)-(d)]: Relative contribution of Maxwell pressure to total pressure in the numer- ically computed system: PMaxwell,num./Pnum. =PMaxwell,num./(Pcapillary,num.+PMaxwell,num.).