Damping of Thermocapillary Destabilization of a Liquid Film in Zero Gravity Through the Use of an Isothermal Porous Substrate

Dynamics of and patterning in thin liquid films are central to many important problems in engineering, geophysics, biophysics, etc. (Craster and Matar, 2008). A liquid film on an isothermal substrate is destabilized in zero gravity through the long wave instability of thermocapillary fingering. The lack of gravitational acceleration leads to quicker destabilization than under the influence of terrestrial or non-zero gravitational acceleration (Narendranath et al., 2014). A schematic of a liquid film on an isothermal solid substrate is depicted in Figure 1, along with some of the stabilizing and destabilizing effects.

Figure 1

A nonlinear evolution (partial differential) equation—derived from the Navier-Stokes equations and the energy conservation equation through a scaling and long wavelength approximation (Burelbach et al., 1988; Ajaev, 2012)—is solved numerically to study dynamics of liquid films in zero gravity (zero gravitational acceleration) and a porous substrate-induced stabilizing mechanism. This equation allows for the study of the influence of individual mechanisms by engaging required nonlinear terms through non-zero dimensionless coefficients (see Abbreviations). The rupture of liquid films (local thinning and exposure of underlying substrate) on isothermal substrates is governed by a balance between destabilizing thermocapillarity and stabilizing surface tension and gravity. In the absence of gravity, long wave instability-driven rupture occurs when thermocapillarity overwhelms surface tension (Narendranath et al., 2014).

Following the model put forth previously (Davis and Hocking,1999) or more recently (Liu et al., 2017), the effect of an isothermal porous substrate on zero-gravity evolution of non-evaporating liquid films, without disjoining pressures, was studied. Here, we only considered the case where the substrate was fully saturated with the liquid film and, hence, the effect of wettability; contact angles within the substrate and partial saturation on film dynamics were not considered. Evolution of spatial modes was visualized using recurrence plots and recurrence rate quantification. Other accounts in literature have studied an axisymmetric droplet spreading under the effect of injection or suction of fluid through a slot (Momoniat et al., 2010), the dynamic control of falling films through injection and suction (Thompson et al., 2016), and the spreading of droplets on a deep porous substrate (Liu et al., 2017). However, these did not account for the thermocapillary-driven instabilities in the absence of gravitational acceleration. The current study includes the effect of thermocapillarity, which was excluded from previous studies on films on porous substrates.

The dimensionless, nonlinear time evolution of a non-evaporating liquid film's interface thickness, H(H, T), in zero gravity is provided by the nonlinear partial differential Equation 6. A subscript signifies a partial derivative. The derivation of the evolution equation is summarized below. Inclusion of vital interfacial normal and shear stress balances has been discussed in greater detail in literature (Ajaev, 2012). Starting with the global mass conservation equation: Equation (1)∂h∂t+∂Q∂x=w{{\partial h} \over {\partial t}} + {{\partial Q} \over {\partial x}} = w where quantities with dimensions h, x, t, and w are film thickness h=h(x,t), lateral coordinate, time, and percolation Darcy velocity, respectively. Q is the interfacial normal and shear stress balance leading to surface tension driven stabilization and thermocapillary led destabilization and fingering. Equation (2)Q=σ3μ∂2h∂x2−γt2μh2∂T*i∂xQ = {\sigma \over {3\mu }}{{{\partial ^2}h} \over {\partial {x^2}}} - {{\gamma t} \over {2\mu }}{h^2}{{\partial {T^{*i}}} \over {\partial x}}

The following scaling arguments are used to remove dimensionality from the governing equations of fluid dynamics in the lubrication approximation limit of small aspect ratio, h0L≪1{\raise0.7ex\hbox{${{h_0}}$} \!\mathord{\left/ {\vphantom {{{h_0}} L}}\right.}\!\lower0.7ex\hbox{$L$}} \ll 1 . Equation (3)x∼X h0Ca−1/3x \sim X\,{h_0}\,C{a^{ - 1/3}}Equation (4)t∼T tνt \sim T\,{t_\nu }

Scaling the film thickness as h ~ Hh₀ with leading order dimensionless interfacial temperature (Oron et al., 1997) given as Ti=1(1+Bi H){T^i} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {(1 + Bi\,H)}}}\right.}\!\lower0.7ex\hbox{${(1 + Bi\,H)}$}}Equation (5)∂T*i∂x=∂∂x(ΔTTi+T∞)=ΔT∂Ti∂x=ΔT∂∂x(11+Bi H){{\partial {T^{*i}}} \over {\partial x}} = {\partial \over {\partial x}}\left( {\Delta T{T^i} + {T_\infty }} \right) = \Delta T{{\partial {T^i}} \over {\partial x}} = \Delta T{\partial \over {\partial x}}\left( {{1 \over {1 + Bi\,H}}} \right)

In the derivation of the evolution equation (Equation 6), gravity-driven stabilization is neglected. Davis and Hocking (1999) use a surface tension time scale in their work. However, a viscous time scale is used to develop the evolution equation (Burelbach et al., 1988; Narendranath et al., 2014) to capture thermocapillary effects. Choice of a time scale based on a general characteristic velocity instead of a viscous scale yields the same evolution equation. Equation (6)∂TH+[MaBiH2(1+BiH)2HX+SH3HXXX]X=NpmSHXX{\partial _T}H + {\left[ {MaBi{{{H^2}} \over {{{(1 + BiH)}^2}}}{H_X} + S{H^3}{H_{XXX}}} \right]_X} = {N_{pm}}S{H_{XX}}

In Equation 6, the term ∂_TH tracks the time evolution of the film interface. Thermocapillary interfacial instabilities are captured by the nonlinear term, [MaBiH2(1+BiH)2]X{\left[ {MaBi{{{H^2}} \over {{{(1 + BiH)}^2}}}} \right]_X} . The stabilizing effect of surface tension is captured by [SH³H_XXX]_X. The capillary effect of a porous substrate, as modeled by the Darcy flow equation, is captured by the term N_pmSH_XX. The non-dimensional quantities Ma and Bi are the Marangoni and interface Biot number. The non-dimensional surface tension number is S, and the one-dimensional porosity of the substrate is captured in the non-dimensional number N_pm. These non-dimensional numbers (see Abbreviations) are set to: Ma=5.00, Bi=1.00, S=100.0, which are values routinely used in literature (Oron, 2000). The magnitude of these values relative to each other determines film interface evolution. We have simulated the dynamics of dichloromethane films previously (Narendranath et al., 2014) with a Ma-S ratio (m of 0.1092. The current value of (m) of 0.05 leads to a slower thermocapillary evolution as compared to m =0.1092.

Equation 6 was solved with periodic boundary conditions, with a period of 2 π/k and a small perturbation initial condition, viz., 1 − 0.1cos(kx). Here, k was the fastest growing wavenumber (reciprocal of wavelength) as determined from a linear stability analysis, shown in Equation 7. Equation (7)k=[1S{Ma(1+Bi)2−Npm}]0.5k = {\left[ {{1 \over S}\left\{ {{{Ma} \over {{{(1 + Bi)}^2}}} - {N_{pm}}} \right\}} \right]^{0.5}}

The value of N_pm is set to 0.001 to ensure very little divergence from the fastest growing wavelength for a porous substrate with respect to a non-porous substrate. This value of NpmS=10−5{{{N_{pm}}} \over S} = {10^{ - 5}} ensures that initial film dynamics are strongly affected by the surface tension term as compared to capillarity due to the porous substrate. In the subsequent section, NpmS=10−5{{{N_{pm}}} \over S} = {10^{ - 5}} is termed “weak porosity.”

The numerical solution was obtained using Wolfram Mathematica and its method of lines option along with the LSODA solver (Momoniat et al., 2010) and was validated with results in literature (Narendranath et al., 2014).

In this short communication, numerical simulations were conducted for a Newtonian liquid film on an isothermal solid (IS-sub) and isothermal porous substrate (IP-sub), in zero gravity. Here, the film was flat and parallel to the substrate. The porous substrate was treated as having one-dimensional depth that was fully saturated with the same fluid as that of the liquid film, and the substrate was sufficient to store the entire volume of the liquid film that rested on it. A nonlinear evolution equation derived using a long wave expansion was solved as an initial value problem with periodic boundary conditions. The liquid film was under the influence of stabilizing surface tension and destabilizing thermocapillarity, with zero gravitational acceleration. With the IS-sub, the film ruptured through the creation and growth of thermocapillary modes and concomitant local thinning.

Next, a weakly porous substrate was chosen to ensure that the fastest growing wavelength stayed nearly the same as that for the non-porous substrate. We observed that this had a damping effect on spatial thermocapillary modes and prolonged film lifespan without rupture. Rupture of the film on the isothermal porous substrate was not observed in the simulated time frame.

Recurrence plots were used to analyze the temporal evolution of spatial thermocapillary modes, with and without a porous substrate. An isothermal porous substrate led to the damping of higher spatial modes. Cross-recurrence plots (CRPs) were used to visualize a difference in states between a liquid film on an isothermal solid substrate and an isothermal porous substrate. A recurrence rate (RR%) measuring the nearness of dynamics of the film states with different substrates revealed that with a weakly porous substrate, the film dynamics—although similar at early stages of evolution—led to different states at later stages.

Future work will include studies of zero gravity liquid film/coating pattern fidelity using recurrence plots and recurrence quantification. Windowing of recurrence plots will be used to quantify temporal evolution of spatial points of interest.