  We consider the nonlocal multiscale model for surface tension
\citep{Tartakovsky2018} as an alternative to the (macroscale) Young-Laplace
law. The nonlocal model is obtained in the form of an integral of a
molecular-force-like function with support $\varepsilon$ added to the
Navier-Stokes momentum conservation equation. Using this model, we calculate
analytical forms for the steady-state equilibrium pressure gradient and
pressure profile for circular and spherical bubbles and flat interfaces in two
and three dimensions. According to the analytical solutions, the pressure
changes continuously across the interface in a way that is quantitatively
similar to what is observed in MD simulations. Furthermore, the pressure
difference $P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies the
Young-Laplace law for the radius of curvature greater than $3\varepsilon$ and
deviates 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 the
distance greater than $3\varepsilon$ from the interface and $P_{\varepsilon,
in}$ is the pressure at the center of the bubble). The latter indicates that
the surface tension in the proposed model decreases with the decreasing radius
of curvature, which agrees with molecular dynamics simulations and laboratory
experiments with nanobubbles. Therefore, our results demonstrate that the
nonlocal model behaves microscopically at scales smaller than $\varepsilon$ and
macroscopically, otherwise.
