RESUMEN
In turbulent free shear flows such as jets and wakes, and also in turbulent boundary layers, the turbulent region is bounded by a region of irrotational flow where the magnitude of the potential velocity fluctuations can be very high. This is particularly true close to the turbulent-nonturbulent interface layer (TNTI) that separates the regions of turbulent (rotational) and nonturbulent (irrotational) fluid motion in these flows. Previous works have shown that for distances from the TNTI x_{2} much bigger than the integral scale L in the nearby turbulent region (x_{2}â«L), the variance of the velocity fluctuations ãu_{i}^{2}ã (i=1,2,3) depends on the shape of the kinetic energy spectrum in the infrared region E(k)â¼k^{n} [O. M. Phillips, Proc. Camb. Phil. Soc. 51, 220 (1955)10.1017/S0305004100030073; Xavier et al., J. Fluid Mech. 918, A3 (2021)10.1017/jfm.2021.296]. Using rapid distortion theory, we derive the generalized scaling laws for the potential velocity fluctuations, at distances sufficiently far from the TNTI layer, for any value of n. While the cases n=4 (Batchelor turbulence) and n=2 (Saffman turbulence) have been previously derived, with ãu_{i}^{2}ãâ¼x_{2}^{-4} and ãu_{i}^{2}ãâ¼x_{2}^{-3}, for n=4 and n=2, respectively [O. M. Phillips, Proc. Camb. Phil. Soc. 51, 220 (1955)10.1017/S0305004100030073; Xavier et al., J. Fluid Mech. 918, A3 (2021)10.1017/jfm.2021.296.], we extend these results by including any other value of n. In particular, we obtain ãu_{i}^{2}ãâ¼x_{2}^{-2} and ãu_{i}^{2}ãâ¼x_{2}^{-4}, for n=1 and n≥5, respectively, while n=3 yields ãu_{i}^{2}ãâ¼x_{2}^{-4}ln(x_{2}). These theoretical results are confirmed by direct numerical simulations of turbulent fronts evolving into an irrotational flow region in the absence of mean shear.
RESUMEN
We have studied a single vertical, two-dimensional liquid bridge spanning the gap between two flat, horizontal solid substrates of given wettabilities. For this simple geometry, the Young-Laplace equation can be solved (quasi-)analytically to yield the equilibrium bridge shape under gravity. We establish the range of gap widths (as described by a Bond number [Formula: see text]) for which the liquid bridge can exist, for given contact angles at the top and bottom substrates ([Formula: see text] and [Formula: see text], respectively). In particular, we find that the absolute maximum span of a liquid bridge is four capillary lengths, for [Formula: see text] and [Formula: see text]; whereas for [Formula: see text] and [Formula: see text] no bridge can form, for any substrate separation. We also obtain the minimum value of the cross-sectional area of such a liquid bridge, as well as the conditions for the existence and positions of any necks or bulges and inflection points on its surface. This generalises our earlier work in which the gap was assumed to be spanned by a liquid film of zero thickness connecting two menisci at the bottom and top substrates.
RESUMEN
It is commonly assumed that the liquid making up a sessile bubble completely wets the surface upon which the bubble lies. However, this need not be so, and the degree of wetting will determine how well a collection of bubbles - a foam - sticks to a surface. As a preliminary to this difficult problem, we study the shape of a single vertical soap film spanning the gap between two flat, horizontal solid substrates of given wettabilities. For this simple geometry, the Young-Laplace equation can be solved (quasi-)analytically to yield the equilibrium shapes, under gravity, of the two-dimensional Plateau borders along which the film contacts the substrates. We thus show that these Plateau borders, where most of a foam's liquid resides, can only exist if the values of the Bond number Bo and of the liquid contact angle θc lie within certain domains in (θc,Bo) space: under these conditions the substrate is foam-philic. For values outside these domains, the substrate cannot support a soap film and it is foam-phobic. In other words, on a substrate of a given wettability, only Plateau borders of a certain range of sizes can form. For given (θc,Bo), the top Plateau border can never have greater width or cross-sectional area than the bottom one. Moreover, the top Plateau border cannot exist in a steady state for contact angles above 90°. Our conclusions are validated by comparison with both experimental and numerical (Surface Evolver) data. We conjecture that these results will hold, with slight modifications, for non-planar soap films and bubbles. Our results are also relevant to the motion of bubbles and foams in channels, where the friction force of the substrate on the Plateau borders plays an important role.
RESUMEN
We have calculated the equilibrium shape of the axially symmetric meniscus along which a spherical bubble contacts a flat liquid surface by analytically integrating the Young-Laplace equation in the presence of gravity, in the limit of large Bond numbers. This method has the advantage that it provides semianalytical expressions for key geometrical properties of the bubble in terms of the Bond number. Results are in good overall agreement with experimental data and are consistent with fully numerical (Surface Evolver) calculations. In particular, we are able to describe how the bubble shape changes from hemispherical, with a flat, shallow bottom, to lenticular, with a deeper, curved bottom, as the Bond number is decreased.