Instability of stationary liquid sheets

Abstract

The rupture of a 3D stationary free liquid film under the competing effects of surface tension and van der Waals forces is studied as a linearized stability problem in a purely irrotational analysis utilizing the dissipation method. The results of the foregoing analysis are compared with a 2D long-wave approximation that has given rise to an extensive literature on the rupture problem. The irrotational and long-wave approximations are here compared with the exact 2D solution. The exact solution and the two approximate theories give the same results for infinitely long waves. The problem considered depends on two dimensionless parameters, the Hamaker number and the Ohnesorge number. The Hamaker number is a dimensionless number defined as a measure of the ratio of van der Waals forces to surface tension. The exact solution and the two approximate solutions differ by <1% when the Hamaker number is small for all values of the Ohnesorge number. When the Ohhnesorge number is close to one, as in the case of water films separated by distance 100 Å, the long-wave approximation overestimates and the potential flow approximation underestimates the exact solution by similar small amounts. The high accuracy of the dissipation method shows that the effects of vorticity are small for small to moderate Hamaker numbers. thin film | irrotational flow | dissipation methodA stable free liquid layer in balance between stabilizing surface tension and destabilizing van der Waals molecular forces can become unstable and be driven to rupture. Molecular forces are important in thin films in the range 100-1,000 Å. Ultrathin "black films" may arise from the stabilizing effects of electric double-layer potentials discussed by Overbeek (1).The rupture of a film often occurs in many contexts, the coalescence of droplets and bubbles, in the creation of dry patches, in the problem of aerodynamic dissemination (2), and in the formation of holes in moving sheets from atomizers studied by Dombrowski and Fraser (3). Stability analysis based on the free energy of stationary films for the critical thickness at which spontaneous local thinning first occurs was given by Vrij (4) and Scheludko (5). Ruckenstein and Jain (6) introduced the fundamental analytic simplification by introducing the disjoining pressure as an "extra pressure" analogous to other prescribed body forces in potential form and constructed a dynamic analysis based on the linear theory of stability. The term "disjoining pressure" was introduced by Deryagin to designate the excess pressure in a thin layer compared with a thick one. If film thickness h is small enough, the stabilizing effects of surface tension are insufficient and the film will thin under the influence of the large disjoining pressure. For a planar film, molecular interactions are accounted for in a strict thermodynamic theory through the concept of a disjoining pressure by Ivanov and Tosev (7). The stability analysis gives the critical h, the wave length, and the maximum growth rate of the most unstable disturbance. He...

Transitions of bouncing and coalescence in binary droplet collisions

“…The results in figure 7 indicate a consensus with previous understanding that droplet merging occurs if the minimum film thickness is small enough to be dominated by vdW forces (Gopinath & Koch 2002;Pan et al 2008;Zhang & Law 2011;Li 2016;Chubynsky et al 2020). To further assess the critical thickness, h cr , in a quantitative manner, we consider the condition when the disjoining pressure given by the vdW force which tends to destabilize the interfaces becomes comparable to the capillary pressure of surface tension that is opposing the fluctuations (Vaynblat, Lister & Witelski 2001;Yoon et al 2007;Ardekani & Joseph 2009;Thete et al 2015). From the scaling relation, A H /6πh 3 cr ∼ 4σ/D, we obtain h cr and the dimensionless critical thickness:…”

“…It is also noted that, while more complications could be rendered by varying B, here we consider only head-on collisions due to the significance in marking off the formation of a FB in binary droplet collisions. In (3.11), the scaled values of h cr , which is independent of We, are of the same order of magnitude as that approximated in previous studies (Yoon et al 2007;Ardekani & Joseph 2009;Thete et al 2015), showing that droplet merging occurs when the minimum thickness of the gas film falls in the range 100-1000 Å. This critical thickness suggests a threshold for determination of bouncing.…”

“…where A H is the Hamaker constant and A * = A H /σ D 2 the dimensionless Hamaker constant (Erneux & Davis 1993;Yoon et al 2007;Ardekani & Joseph 2009). Critical thickness h cr has long been studied (Ivanov et al 1970;Ivanov 1988;Yoon et al 2007;Ardekani & Joseph 2009;Kaur & Leal 2009) and found to depend mainly on the dimensionless Hamaker constant (Yoon et al 2007;Ardekani & Joseph 2009;Kaur & Leal 2009).…”

The unique properties of the solid-like confined liquid films: A large scale molecular dynamics simulation approach

“…There are two relevant dimensionless numbers for such problem: the Hamaker number and the Ohnesorge number. The Hamaker number, Ha = A/ πγa 2 , is a dimensionless number defined as a measure of the ratio of van der Waals forces to surface tension [49] . The molecular length [50] can be derived from the Hamaker number as follows:…”

Viscous Potential Flow Analysis of Electroaerodynamic Instability of a Liquid Sheet Sprayed with an Air Stream

“…Awasthi and Agrawal [19] have studied the viscous potential flow analysis of Kelvin-Helmholtz instability of a cylindrical interface and found that the viscosity of the fluids stabilizes the interface. The rupture of a 3D stationary free liquid film under the competing effects of surface tension and van der Waals forces has been studied by Ardekani and Joseph [20] as a linearized stability problem in a purely irrotational analysis utilizing the dissipation method.…”