I am looking for references which give vector calculus expressions in boundary layers around curved surfaces. My application is fluid dynamics, so I want to be able to write the Navier-Stokes equations in these coordinates $$ \partial_t u + u\cdot \nabla u + \nabla p - \nu \nabla^2 u = 0,$$ $$\nabla \cdot u = 0.$$

The specific coordinate system I am interested in uses the signed distance as a normal coordinate, and principal directions of curvature as the remaining basis vectors. More or less it's using a Darboux frame with the principal directions of curvature. The basics of this coordinate system are well known, and standard differential geometry textbooks go through the curvature of surfaces in 3D (such as do Carmo's textbook Differential Geometry of Curves and Surfaces (1976), specifically chapter 4.3).

However, I haven't found any references which give expressions for higher order differential operators in the vicinity of the surface for these coordinates (like the vector Laplacian at a finite distance from the boundary).

As for what I've found, Görtler has a fluid dynamics paper on boundary layer coordinates (Görtler, Henry. 1957. A New Series for the Calculation of Steady Laminar Boundary Layer Flows. Journal of Mathematics and Mechanics 6 (1): 1–66.), but it isn't really what I want. It's a bit too focussed on the equations and not the vector calculus itself.

There are more mathematical textbooks on these coordinates (like Giga, Surface Evolution Equations: A Level Set Approach (2006), or Gray, Tubes (2004)), but I haven't seen vector calculus identities I'm interested in.

If anyone knows of somewhere these identities are given, I would greatly appreciate them.




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