Tensorial equations for three-dimensional laminar boundary layer flows

Tensorial equations are derived for a laminar and attached boundary layer flow with null pressure gradient in the normal direction to a smooth three-dimensional surface. An incompressible, isothermal and viscous fluid of Newtonian type is assumed. Covariant derivatives in the three dimensional Euclidean domain are employed, where the surface curvature terms are implicitly included in the Christoffel symbols with the aim of writing the boundary layer equations in an invariant form irrespective to the particular choice of the coordinate system. These equations are covariant under a linear coordinate transformation on the two surface coordinates, and a scaling along the normal direction to the surface. As a test case, the boundary layer near a sphere in an axisymmetrical steady flow is numerically computed using a pseudo-spectral approach.