1.3. Interface tracking

As we found in Eq. (1.10), the interface moves with the fluids. The interface can be written with a scalar function \(S(\mathbf{x}, t)\) as \(S(\mathbf{x}, t) = 0\). The fluid particle on the interface always stay on it, so that,

(1.30)\[ \frac{DS}{Dt} = 0,~~~~\frac{\partial S}{\partial t} + \mathbf{v}_{int} \cdot \nabla S = 0\]

By using the definition \(\mathbf{n} = \nabla S / |\nabla S|\), we may rewrite the second expression as

(1.31)\[ \frac{1}{| \nabla S |} \frac{\partial S}{\partial t} = - \mathbf{v}_{int} \cdot \mathbf{n}\]

This form clearly shows that the interface shape is evolved by the normal velocity component; consider a rotating sphere with a finite rotational (tangential) velocity. The fluid particles are moving along the interface, but the interface shape does not change.