I am reading a CFD paper http://www.cs.columbia.edu/cg/surfaceliquids/droplets.pdf. In page 5 paragraph "Tangential velocity", it says that in a liquid-air surface $\Gamma$, the pressure gradient $\nabla p$ always has the same direction as the normal vector $\bf n$ of surface $\Gamma$, where $p$ is the pressure term in the incompressible Euler equation. They argue that
For the scenarios we consider, air density is negligible compared to liquid density. Imagine that the pressure gradient $\nabla p$ has a tangential component: it would effect an infinite tangential air velocity, in turn instantly restoring equilibrium.
I am not convinced by this argument. I think it does not hold when we consider the external body force (such as gravity) and surface tension. Here is my counter-example: Imagine we have a cup of water. Due to the gravity, the pressure gradient on the surface is vertically downward. However, due to the surface tension, the direction normal vector of the liquid-air surface is not always vertical.
Am I wrong?