  We propose a non-local model for surface tension obtained in the form of an
integral of a molecular-force-like function with support $3.5\varepsilon$ added
to the Navier-Stokes momentum conservation equation. We demonstrate
analytically and numerically that with the non-local model interfaces with a
radius of curvature larger than the support length behave macroscopically and
microscopically, otherwise. For static droplets, the pressure difference
$P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies the Young-Laplace law
for droplet radius greater than $3.5\varepsilon$ and otherwise deviates from
the Young-Laplace law. The latter indicates that the surface tension in the
proposed model decreases with decreasing radius of curvature, which agrees with
molecular dynamics and experimental studies of nanodroplets. Using the
non-local model we perform numerical simulations of droplets under dynamic
conditions, including a rising droplet, a droplet in shear flow, and two
colliding droplets in shear flow, and compare results with a standard
Navier-Stokes model subject to the Young-Laplace boundary condition at the
fluid-fluid interface implemented via the Conservative Level Set (CLS) method.
We find good agreement with existing numerical methods and analytical results
for a rising macroscopic droplet and a droplet in a shear flow. For colliding
droplets in shear flow, the non-local model converges (with respect to the grid
size) to the correct behavior, including sliding, coalescence, and merging and
breaking of two droplets depending on the capillary number. In contrast, we
find that the results of the CLS model are highly grid-size dependent.
