I have read that the surface tension gradient operator is $ \nabla_{s} = (I - nn). \nabla$ , where n is the unit normal to the surface given by : $$ n = (-\frac{\partial h}{\partial r},1) \frac{1}{\sqrt{1+ (\frac{\partial h}{\partial r})^2}}$$ in a cylindrical coordinate system with an axisymmetric assumption ($\partial/\partial\theta = 0$).
I am trying to write an equation of particle diffusion tangential to an interface and need to calculate$ \nabla_{s}^{2} \tau$ where $\tau$ is a scalar quantity. I had no problems finding $(\nabla_s\tau)$ but i repeatedly got the same result of $ \nabla_{s}. (\nabla_{s}\tau)= \nabla_s.(V) = \nabla V - n(n.\nabla V) = \nabla V - \nabla V = 0$ where V is the vector from the initial gradient operation. I have read elsewhere that this definition of surface gradient only supports scalar quantities, although i cannot rule out that i have made a mistake in expressing the laplacian, in particular overlooking some key simplifying steps, as I am relatively new to vector calculus.
I must also shamelessly add out of desperation that I am currently writing my undergraduate dissertation and have been set back due to this obstacle. But I persist.