If we use the normal form of Laplacian in cylindrical coordinates, we have only 3 terms summing each other up. But if we see the component theta of velocity in the Navier-Stokes Equation, in cylindrical coordinates, there are two additional terms: -Vtheta/r^2 + 2/r^2*dVtheta/dtheta. How are these terms obtained?
The difference is due to the rules of covariant differentiation and the fact that V_{\theta} is not a scalar quantity but a component of a vector.
Check this Wikipedia page, and compare the row for the "Laplace operator" to the row describing the "vector Laplacian":
https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
Consider, from a physical point of view, that the Laplacian corresponds to the calculation of the fluxes across the faces of the elementary cell (divided the volume of the cell) and that the area of the face (r * theta) varies with the radius. For example in 2D and considering only the r direction: (r2 * theta * Vr2 - r1 * theta * Vr1) / (r * theta * dr) --> (1/r) * d(r*Vr)/dr = dVr/dr + Vr/r (additional term).
Thanks to all for your answers that helped a lot. Now I am going to make the mathematical proof for Navier-Stokes equations in different coordinate systems.
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