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Moving contact line dynamics calculations include two models: precursor film models and slip models. The lubrication approximation method is used to establish a three-dimensional mathematical model to analyze the droplet thermocapillary migration behavior on a non-uniformly heated solid substrate with a wettability track. The contact line dynamics in the slip model and the disjoining pressure effect in the precursor model are proposed to regulate the substrate wettability. Both models are numerically implemented to investigate droplet spreading for three cases: free spreading on an isothermal substrate, thermocapillary migration on a uniform wettability substrate, and thermocapillary migration on a wettability-confined track. For the case of free spreading on an isothermal substrate, the three-dimensional results of the slip and precursor contact line models are essentially consistent with two-dimensional slip model results. For the case of thermocapillary migration on a uniform wettability substrate, the results of the two models essentially agree with the experimental results. Decreasing the thermal gradient reduces the discrepancies between the two models that result from the coordinate transformation method used in the slip model, which reduces the contact angles measured in the y-direction and enlarges the advancing contact angle in the migration direction. For the case of thermocapillary migration on a wettability-confined track, the slip model gradually shows a "dynamic-pinning" behavior with increasing equilibrium contact angle in the hydrophobic region. By contrast, the precursor film model maintains a stationary pinning behavior but separates a residual liquid outside the track. The precursor film model is preferred over the slip model in lubrication approximations for three-dimensional fluids when calculating complex moving contact dynamics caused by wettability differences. However, the precursor film model must be further optimized to prevent numerical instability. [ABSTRACT FROM AUTHOR]

Understanding the dynamics and inherent mechanisms of sphere impact on suspended films is important for improving sphere-film separation techniques. In this study, we conducted experiments to investigate the dynamics of sphere impact on suspended films and examine typical phenomena. We revealed the effects of dynamic viscosity and surface tension of films by altering the glycerol content (G) and the relative surfactant concentration (C*) and elucidated the characteristics of film deformation, sphere trajectory (hs), and contact time (tc). Moreover, we obtained the influences of sphere and film properties on bubble volume (Vbub) by analyzing force balance. The results indicate that three modes are observed and divided using the dimensionless energy parameter E* = Ek0/(ΔEfs + Evis) based on energy analysis, considering the sphere kinetic energy (Ek0), film surface energy increment (ΔEfs), and viscous dissipation (Evis): satisfying E* < 1, retention occurs; satisfying 1 < E* < 127.7(Ds/Df)2 (where Ds is the sphere diameter, Df is the film diameter), bubble entrainment passing appears; satisfying E* > 127.7(Ds/Df)2, non-bubble entrainment passing emerges. During retention, increasing G and C* causes film surface elasticity and hs to present a trend of first rising and then falling. For passing, the increase in G reduces deformability, leading hs to decrease, while increasing C* makes the film more susceptible to deformation, causing hs to increase. In addition, a film vibration period (τf) is introduced to measure tc, satisfying tc > 2τf for retention, while satisfying tc < τf/3 for passing. Inspection of the relationship between film deformation and falling height indicates that Vbub enlarges with increasing Ds and C* but shrinks with increasing G and release height Hs0. [ABSTRACT FROM AUTHOR]

Evaporation of binary mixture droplets (BMDs) is a ubiquitous natural phenomenon with numerous industrial applications. In this study, a theoretical model of BMD evaporation under forced convection is established by considering the influence of evaporative cooling, thermal and Marangoni effects, convection, and a Stefan flow. The dynamic evaporation of a binary ethanol–water droplet on a heated substrate is simulated, and the internal and external flow structures of the droplets and their interactions are investigated. The influence of temperature-dependent physical properties on the evaporation dynamics is analyzed, and the effect of the forced convection intensity on the exclusion distance and Marangoni instability is explored. Our findings reveal that, during the stable flow stage, a single vortex flow pattern prevails, characterized by a circulating zone with low ethanol concentration within the droplets. However, in the Marangoni instability-driven flow (MIF) stage, a complex multi-vortex flow appears inside the droplets, with a heterogeneous ethanol distribution. Under the action of the Stefan flow, external forced convection cannot directly affect the flow inside the droplets through viscous shear but indirectly impacts the internal flow through heat and mass transfer. The temperature-dependence of physical properties significantly influences the internal flow and delays the onset of the MIF stage. Forced convection affects the heat and mass transfer by changing the thickness of the thermal and concentration boundary layers. Compared with BMD evaporation under natural convection, the heat and mass transfer rate are significantly higher under forced convection, particularly in the MIF stage. [ABSTRACT FROM AUTHOR]

In this study, we investigate the linear and weakly nonlinear stability of a liquid film flowing down an inclined plane with an insoluble surfactant. First, the nonlinear evolution equations of a liquid film thickness and surfactant concentration are derived using the long-wave expansion method at a moderate Reynolds number (0 < Re ≤ 20). The linear stability of the flow is examined using the normal-mode method, and the linear stability criterion and critical Reynolds number Rec are obtained. The results reveal the destabilizing nature with increasing Reynolds number Re and the stabilizing nature with increasing Marangoni number M. Second, the nonlinear equations described by the complex Ginzburg–Landau equation are obtained using the multiple-scale method to investigate the weakly nonlinear stability of the system. The results show that a new linear instability region appears above the neutral stability curve caused by the solute-Marangoni effect, which develops into a supercritical stable zone under the influence of nonlinear factors. Increasing M generally improves the stability of the flow but continuing to increase M under the condition of M > Mc (critical Marangoni number) improves the nonlinear instability in the region and transforms part of the unconditional stability zone into a subcritical instability zone. The increase in Re extends an explosive unstable zone and reduces the threshold amplitude in the subcritical unstable zone. In contrast, the unconditional stable zone decreases and disappears after increasing Re to a specific value, which reflects the destabilizing effect of Re on the nonlinear zone of the flow. [ABSTRACT FROM AUTHOR]