  In this paper, we present a novel meshfree framework for fluid flow
simulations on arbitrarily curved surfaces. First, we introduce a new meshfree
Lagrangian framework to model flow on surfaces. Meshfree points or particles,
which are used to discretize the domain, move in a Lagrangian sense along the
given surface. This is done without discretizing the bulk around the surface,
without parametrizing the surface, and without a background mesh. A key novelty
that is introduced is the handling of flow with evolving free boundaries on a
curved surface. The use of this framework to model flow on moving and deforming
surfaces is also introduced. Then, we present the application of this framework
to solve fluid flow problems defined on surfaces numerically. In combination
with a meshfree Generalized Finite Difference Method (GFDM), we introduce a
strong form meshfree collocation scheme to solve the Navier-Stokes equations
posed on manifolds. Benchmark examples are proposed to validate the Lagrangian
framework and the surface Navier-Stokes equations with the presence of free
boundaries.
