When two Newtonian liquid droplets are brought into contact on a solid substrate, a highly curved meniscus neck is established between the two which transforms the bihemispherically shaped fluid domain to a hemispherically shaped domain. The rate at which such topological transformation, called coalescence phenomenon, evolves results from a competition between the inertial force which resists the transformation, the interfacial force which promotes the rate, and the viscous force which arrests it. Depending on the behaviour of these forces, different scaling laws describing the neck growth can be observed, predicted theoretically, and proved numerically. The twofold objective of the present contribution is to propose a simple theoretical framework which leads to an Ordinary Differential Equation, the solution of which predicts the different scaling laws in various limits, and to validate these theoretical predictions numerically by modelling the phenomenon in the commercial Finite Element software COMSOL Multiphysics.

The evolution of a thin liquid film subject to a volatile solvent source and an air-blow effect which modifies locally the surface tension and leads to Marangoni-induced flow is shown to be governed by a degenerate fourth order nonlinear parabolic h-evolution equation of the type given by ∂ t h = − div x M 1 h ∂ x 3 h + M 2 h ∂ x h + M 3 h , where the mobility terms M 1 h and M 2 h result from the presence of the source and M 3 h results from the air-blow effect. Various authors assume M 2 h ≈ 0 and exclude the air-blow effect into M 3 h . In this paper, the authors show that such assumption is not necessarily correct, and the inclusion of such effect does disturb the dynamics of the thin film. These emphasize the importance of the full definition t → · grad γ = grad x γ + ∂ x h grad y γ of the surface tension gradient at the free surface in contrast to the truncated expression t → · grad γ ≈ grad x γ employed by those authors and the effect of the air-blow flowing over the surface.

In a generalized topological space Tg = (Ω, Tg), generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, are merely two of a number of generalized primitive operators which may be employed to topologize the underlying set Ω in the generalized sense. Generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, are other generalized primitive operators by means of which characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) can be given without even realizing generalized interior and generalized closure operations first in order to topologize Ω in the generalized sense. In a recent work, the present authors have defined novel types of generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg and studied their essential properties and commutativity. In this work, they propose to present novel definitions of generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, a set of consistent, independent axioms after studying their essential properties, and established further characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) in Tg.

In a generalized topological space Tg = (Ω, Tg ) (Tg -space), various ordinary topological operators (Tg -operators), namely, int_g, cl_g, ext_g, fr_g, der_g, cod_g : P (Ω) −→ P (Ω) (T_g-interior, T_g-closure, T_g-exterior, T_g-frontier, T_g-derived, T_g-coderived operators), are defined in terms of ordinary sets (T_g-sets). Accordingly, generalized T_g-operators (g-T_g-operators), namely, g-Int_g, g-Cl_g, g-Ext_g, g-Fr_g, g-Der_g, g-Cod_g : P (Ω) −→ P (Ω) (g-T_g-interior, g-T_g-closure, g-T_g-exterior, g-T_g-frontier, g-T_g-derived, g-T_g-coderived operators) may be defined in terms of generalized T_g-sets (g-T_g-sets), thereby making g-T_g-operators theory in T_g-spaces an interesting subject of inquiry. In this paper, we present the definitions and the essential properties of the g-T_g-interior and g-T_g-closure operators g-Int_g , g-Cl_g : P (Ω) −→ P (Ω), respectively, in terms of a new class of g-T_g-sets which we studied earlier. The outstanding results to which the study has led to are: Firstly, (g-Int_g, g-Cl_g) : P (Ω) × P (Ω) −→ P (Ω) × P (Ω) is (Ω, ∅)-grounded, (expansive, non-expansive), (idempotent, idempotent) and (∩, ∪)-additive. Secondly, g-Int_g : P (Ω) −→ P (Ω) is finer (or, larger, stronger than int_g : P (Ω) −→ P (Ω) and g-Cl_g : P (Ω) −→ P (Ω) is coarser (or, smaller, weaker) than cl_g : P (Ω) −→ P (Ω). The elements supporting these facts are reported therein as sources of inspiration for more generalized operations.

Several specific types of generalized maps of a generalized topological space have been defined and investigated for various purposes from time to time in the literature of topological spaces. Our recent research in the field of a new class of generalized maps of a generalized topological space is reported herein as a starting point for more generalized classes.