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.

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,

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,

Scaling the film thickness as _{0} with leading order dimensionless interfacial temperature (Oron et al., 1997) given as

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.

In Equation 6, the term _{T}^{3}_{XXX}]_{X}. The capillary effect of a porous substrate, as modeled by the Darcy flow equation, is captured by the term _{pm}_{XX}. The non-dimensional quantities _{pm}

Equation 6 was solved with periodic boundary conditions, with a period of 2

The value of _{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

The numerical solution was obtained using Wolfram

Thermocapillary instabilities affecting a liquid film on an isothermal solid substrate

To visualize the similarity in film dynamics and related spatial modes at different time instances with different substrates, a recurrence matrix was calculated and plotted. The recurrence matrix (Marwan et al., 2007), which is a sparse matrix, was calculated using Equation 8.

Where ɛ is a properly chosen cut-off distance, _{j} and _{i} are states of the dynamical system at closely spaced, immediate spatial locations _{j} – _{i}‖ is the Euclidean distance (or L-2 Norm) between states _{j} and _{i}. When the recurrence matrix was plotted as an array or matrix plot, a recurrence plot (RP) was available. This RP revealed all the times when the phase space trajectory of the dynamical system visited roughly (with proximity

The evolution of a liquid film on the IS-sub in zero gravity is shown in Figure 3. A cut-off distance ɛ of 0.0005 signifies that the relative (spatial) difference of 0.05% between a current state and the previous state is treated as a recurrence or a recurrent state. Physically, this cut-off distance corresponds to ≈1 μm or less. This distance was nearly equal to rupture thickness in our simulations.

In contrast to Figure 3, the recurrence plot for a liquid film on the IP-sub in zero gravity is depicted in Figure 4. Higher spatial modes were damped when a porous substrate was used. This was seen as a lack of gaps in the diagonals and off-diagonal segments of the matrix plot. These gaps existed when an IP-sub was used (Figure 3), which in this case meant that there was a continuing development of destabilizing thermocapillary modes.

A difference in dynamics between liquid films on the two different substrates is visualized through a cross-recurrence plot (CRp, Equation 9) in Figure 5. In Equation 9,

A larger value of the cut-off distance, ɛ=0.05, was used, and the closeness in film dynamics of a film on the IS-sub versus the film on the IP-sub is depicted in Figure 6. In other words, Figure 6 captured the recurrence of spatial-temporal states within 5% of each other. To quantify this closeness in dynamics, a rate of recurrence (RR%), which is one of the many recurrence quantification parameters (Marwan et al., 2007) was calculated and tabulated in Table 1. The rate of recurrence is simply the percentage of ‘1’ in the sparse recurrence matrix. A greater percentage of ‘1’ suggests a greater rate or percentage of recurrence.

Rates of recurrence comparing similarity of liquid films dynamics on IS-sub vs. IP-sub. Two different cut-off distances, ɛ, were used to describe the similarity between dynamics of liquid films. IS-sub, isothermal solid substrate; IP-sub, isothermal porous substrate.

0.0 | 100.0 | Similar dynamics | 0.0 | 100.0 | Similar dynamics |

400.0 | 100.0 | Similar dynamics | 400.0 | 33.66 | Dynamics within 33.66% of each other |

800.0 | 45.94 | Dynamics within 45.94% of each other | 800.0 | 6.86 | Dynamics within 6.86% of each other |

1650.0 (time rupture of film on IS-sub) | 39.79 | Dynamics within 39.79% of each other | 1650.0 | 5.29 | Dynamics within 5.29% of each other |

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.

#### Rates of recurrence comparing similarity of liquid films dynamics on IS-sub vs. IP-sub. Two different cut-off distances, ɛ, were used to describe the similarity between dynamics of liquid films. IS-sub, isothermal solid substrate; IP-sub, isothermal porous substrate.

0.0 | 100.0 | Similar dynamics | 0.0 | 100.0 | Similar dynamics |

400.0 | 100.0 | Similar dynamics | 400.0 | 33.66 | Dynamics within 33.66% of each other |

800.0 | 45.94 | Dynamics within 45.94% of each other | 800.0 | 6.86 | Dynamics within 6.86% of each other |

1650.0 (time rupture of film on IS-sub) | 39.79 | Dynamics within 39.79% of each other | 1650.0 | 5.29 | Dynamics within 5.29% of each other |

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