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We consider the nonlocal multiscale model for surface tension\citep{Tartakovsky2018} as an alternative to the (macroscale) Young-Laplacelaw. The nonlocal model is obtained in the form of an integral of amolecular-force-like function with support $\varepsilon$ added to theNavier-Stokes momentum conservation equation. Using this model, we calculateanalytical forms for the steady-state equilibrium pressure gradient andpressure profile for circular and spherical bubbles and flat interfaces in twoand three dimensions. According to the analytical solutions, the pressurechanges continuously across the interface in a way that is quantitativelysimilar to what is observed in MD simulations. Furthermore, the pressuredifference $P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies theYoung-Laplace law for the radius of curvature greater than $3\varepsilon$ anddeviates from the Young-Laplace law otherwise (i.e., $P_{\varepsilon, in} -P_{\varepsilon, out}$ goes to zero as the radius of the curvature goes to zero,where $P_{\varepsilon, out}$ is the pressure outside of the bubble at thedistance greater than $3\varepsilon$ from the interface and $P_{\varepsilon,in}$ is the pressure at the center of the bubble). The latter indicates thatthe surface tension in the proposed model decreases with the decreasing radiusof curvature, which agrees with molecular dynamics simulations and laboratoryexperiments with nanobubbles. Therefore, our results demonstrate that thenonlocal model behaves microscopically at scales smaller than $\varepsilon$ andmacroscopically, otherwise.

Analytical steady-state solutions for pressure with a multiscale non-local model for two-fluid systems