Chapter II: Cuspidal Formation in Thermocapillary Thin Liquid Films
2.5 Numerical Solution of Nonlinear Thermocapillary Equation
Substituting equation (2.86) into equation (2.85) for a perturbation of the form δH = ∂2H
∂X2 (2.87)
(note RΩδHdΩ= 0 is admissible) yields δ2F[H, P;δH, δP] =−
Z
Ω
∂2H
∂X2 d3Φ
dH3 H
∂H
∂X 2
dΩ
=− Z
Ω
d3Φ dH3 H
1 3
∂
∂X ∂H
∂X 3
dΩ
= 1 3
Z
Ω
d4Φ dH4 H
∂H
∂X ∂H
∂X 3
dΩ= 1 3
Z
Ω
∂H
∂X 4d4Φ
dH4 HdΩ.
(2.88) All boundary terms from integrations by parts vanish due to periodic boundary con- ditions. For the thermocapillary model described by the potential function (2.70), its fourth derivative (as plotted in figure 2.4)
d4Φ
dH4 =−2(1−2H)2+ 4H2
H3(1−H)4 <0 for 0< H <1 (2.89) is always negative. When substituted into equation (2.88), this yields the relation δ2F[H, P;δH, δP] < 0. This inequality assures that for every (if exists) nonuniform stationary state H such that ∂H/∂X is not identically zero everywhere, there always exists a neighboring state H+δH with same periodicity as H but of strictly lower free energy. Therefore we conclude that equation (2.80) cannot therefore support any energetically stable stationary periodic states, at least not any classical smooth solutions such that H > 0 everywhere. This analysis is quite general and can be modified and applied to many other thin film systems (even volume non-conserving systems) so long as the governing interface equation can be cast into the gradient flow equation (2.67).
formation process (in contrast with other thin film problems which involve asymptotic matching to regions described by a Stokes flow (Krechetnikov,2010)). A mixed Lagrange finite element method (COMSOL Multiphysics, Inc. V5.3, 2017) was used to evolve the solutions, subject to no-flux conditions at the boundaries of the lateral domain [0, Λmax/2]and initial condition
H(X, τ = 0) = 1 3
1 + 0.1 cos 2πX Λmax
(2.90) (withXreplaced byRfor cylindrical geometry). The restriction to a domain sizeΛmax/2 ensured that the dynamics of an individual cuspidal shape could be examined with high resolution without interference from similar adjacent shapes arising from the native linear instability (2.66). Quadratic elements numbering about 20000 and of minimum size4×10−8 ensured sufficient spatial resolution of the emerging cuspidal region. The mesh sizes were everywhere much smaller than|∇k2H|−1at all times. Integration in time relied on a second order backward difference scheme with small adaptive time stepping.
Typically, full evolution toward the asymptotic shapes required about 11000 integration steps. Simulations were terminated when the (dimensionless) distance between the virtual singularity at H = 1 and the liquid cusp apex Hapex(τ) = H(0, τ) reached a value of about 10−4.
Shown in figure2.5are far-field (a) and magnified views (b) of an evolving cusp capped by a conical tip. As expected from consideration of volume accumulation, the rectilinear geometry leads to a slightly thinner cusp for the same time interval. Inspection of the shape of the fluid tip reveals a conical protrusion with constant slope whose tip radius decreases rapidly in time. Plotted in figure 2.5(c) are the tip speed∂H/∂τ|apex
and magnitude of the tip curvature |∇k2H|apex as a function of the decreasing distance 1−Hapex(τ). The power law behavior observed persists for almost four decades in time indicating robust self-similar growth. The indicated asymptotic values for the slope and intercept values (in parentheses) of the lines shown were obtained from least square fits over the shaded (yellow) portion shown. This self-similar behavior confirms the relations
∂Hapex
∂τ ∼ 1
(1−Hapex)3, ∇k2Hapex ∼ 1
1−Hapex . (2.91) Introducing the singular time τs where the local film apex Hapex reaches H = 1—the singular point of equation (2.64)—yields the scaling relations governing the conical tip region, namely(1−Hapex/(τs−τ)∼(1−Hapex)−3and(1−Hapex)/X2 ∼(1−Hapex)−1. These reveal the self-similar variables characterizing this asymptotic regime, namely
X ∼1−Hapex∼(τs−τ)1/4, (2.92) which reflect the lack of an intrinsic spatial or temporal scale in the conical region.
As evident in figure 2.5(d), the shape of the conical tip undergoes collapse onto a
1.5 1.0 0.5 0 0.5 1.0 1.5 0.0
0.2 0.4 0.6 0.8 1.0
0 0.01 -0.01
1.0 0.99
0.4 0.2 0 0.2 0.4
0.80 0.85 0.90 0.95 1.00 1.05
10-5 10-4 10-3 10-2 10-1 100 1000
1002 1004 1006 1008 1010 1012 1014
1−Hapex(τ)
100 50 0 50 100
0 20 40 60 80 100
1 1-1
-0.5 Rectilinear(a)
profileH(X, τ) Axisymmetric
profileH(R, τ) τ
τ
R X
(b)
R X
−3.010(
−1.675)
−3.015(
−1.742)
−0.995(−0.056)
−0.993(−0.146)
Rectilinear (X) Axisymmetric (R)
∂H
∂τ apex
∇k2H|apex
(d)
(1−Hapex) ×|∇2 kH|apex
η η 0
η= 1−HXapex η= 1−HRapex slope 1.044
slope 0.764 τ
τ
τ τ (1−H)/(1−Hapex)
Figure 2.5: Self-similar formation of conical cusp from numerical solution of equation (2.64) for rectilinear (X) and axisymmetric (R) geometry. Arrows indicate increasing timeτ. (a) Far field view of cuspidal formation forHapex(τ) = 0.367, 0.4,0.5, ..., 0.8,0.9,0.9875. (b) Magnified view of conical tip forHapex(τ) = 1−0.2/2n showingn= 0(H),n= 1(),n= 2(),n= 3 (l) andn = 4 (). Inset: Magnified view of conical tip forHapex(τ) = 1−0.2/2n showing n= 5−10(N). (The last two curvesn= 9,10are indistinguishable.) (c) Power law behavior of∂H/∂τ|apexand|∇k2H|apexversus1−Hapex(τ). Slopes and intercept values (in parentheses) were obtained from least squares fits over the shaded (yellow) region. (d) Rescaled solutions (1−H)/(1−Hapex)showing self-similar collapse of the conical tip forHapex(τ) = 1−0.2/2n wheren= 0−10. Inset: Rescaled apex curvature(1−Hapex)(∇k2H)apex versusη.
0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1
-0.14 0 0.18 -1.46 0 0.43 -6.86 01.49 -157 1.4 Hmax = 0.18 Hmax= 0.41 Hmax= 0.82 Hmax= 0.99
Figure 2.6: Four images of the film thickness H(X, τ) (top panel) and interface curvature ∇2kH (bottom panel) from numerical simulation of equation (2.64) on a square periodic domain with edge length Λmax ≈ 3.02. The initial condition was H(X,0) ={1−0.05[cos(2πX/Λmax)+cos(2πY /Λmax)]+rand(X)}/6, whererand(X) denotes a uniformly distributed random variable between -0.2 and 0.2. The maximum film thickness is denotedHmax. The orange lines are the boundaries between regions of positive and negative curvature. The evolution times depicted are τ = 0.0,30.0, 50.5 and50.84552722.
common curve when both the vertical and lateral dimensions are normalized by the factor (1−Hapex). The extent of the collapsed region is observed to increase in time.
Shown in the inset of figure2.5(d) is the rescaled apical curvature(1−Hapex)(∇k2H)apex
versus η =X/(1−Hapex) or R/(1−Hapex), which also exhibits self-similar collapse.
The virtual singularity Hapex = 1 appears therefore to act as an attractor state for formation of the conical tip.
Two-dimensional periodic square domain
We also performed a full two-dimensional direct numerical simulation of equation (2.64) to demonstrate the robustness of cuspidal formation in a thermocapillary-driven thin film. The top panel shown in figure2.6represents 3D views of an evolving cusp with a conical tip at the four times designated, as obtained from finite element simulation of the full nonlinear equation (2.64). The square domain was discretized into 15872 triangular elements of quadratic order with 63746 degrees of freedom in total. Since the evolving cusp was centered about the origin of the domain, the nested mesh shown in figure2.7 was implemented in order to resolve details of the apical region with sufficient resolution.
The edge size ∆X of the smallest mesh element in the central was about 0.0004,
Figure 2.7: Image of progressively refined mesh used to resolve details in the apical region.
intentionally chosen to be smaller than the minimum value of |∇kH|−1 ∼ O(10−2).
The finite element simulation was run until the cusp apex height reached a maximum value Hmax that just exceeded 0.99.
The bottom panel displays the value of the curvature of the gas/liquid interface at every point within the computational domain. The orange curves delineate concave from convex regions. The last image in the bottom panel clearly reveals that the interface evolves into a cuspidal shape capped by a conical tip with shrinking radius flanked by a broader convex surface.
We verified that these results, i.e. rectilinear, axisymmetric and full two-dimensional simulations, converged upon mesh refinement and that the minimum mesh size chosen was sufficient to capture the dynamics of the evolving tip with high resolution. In particular, the scaling relations (2.92) noted above establish constraints on the minimum mesh size ∆X ∼ 1−Hapex ∼ 1/|∇k2H|apex required to resolve the curvature in the apical region. For the results shown in figure2.5, the simulations were terminated when 1−Hapex ∼ O(10−4). The minimum element size used of 4×10−8 ensured that O(∆X)O(1−Hapex). Therefore, the overall local error in was sufficiently small in our simulations to capture with high resolution the self-similar dynamics in the apical region before the simulations were terminated to prevent contact with the top colder substrate. Similar argument applies to the two-dimensional simulation on the square domain.