### Tensorial Equations for Three-Dimensional Laminar Boundary Layer Flows

#### Abstract

For a laminar and attached steady flow along a smooth three-dimensional surface of an incompressible, isothermal and viscous fluid of Newtonian type, the boundary layer equations written in orthogonal curvilinear coordinates and involving curvature terms are relatively well known, but a covariant formulation that has not neither of them is rather not further considered. The more conventional boundary layer equations are written as two-dimensional partial differential equations over the body surface itself but, since in many cases these surfaces are not flat, the curvature tensor or the two principal curvatures of the flow domain must be into account. On the other hand, the present work is focused on a covariant formulation of the three-dimensional boundary layer equations without any curvature terms. The formulation also uses an orthogonal curvilinear coordinates given by the two surface coordinates plus a third normal to the body surface. The lack of curvature terms is due to that the boundary layer equations are rewritten as three-dimensional partial differential equations in an Euclidean domain and, since this is a flat space, the flow domain has not curvature terms and only remains the surface metric tensor in the continuity equation. In particular, it is shown that the developed equations are covariant under a linear coordinate transformation on the two surface coordinates, and a scaling one along the normal coordinate to the body surface. As a practical use of them, the boundary layer flow on the surface of a sphere in an axisymmetrical flow is numericaly computed using a pseudo-spectral approach.