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Boundary Layer Approximation

The core simplification assuming that the flow gradients normal to the surface are much larger than the gradients parallel to the surface. This allows the Navier-Stokes equations to be reduced to simpler forms, valid when the boundary layer thickness δ\delta is small.
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The statement of the theorem

Consider the flow domain Ω\Omega with coordinates (x,y)(x, y), where xx is the streamwise direction and yy is the normal direction. The boundary layer approximation assumes that the characteristic length scale normal to the surface (δ\delta) is much smaller than the characteristic length scale parallel to the surface (LL), i.e., δ/L≪1\delta/L \ll 1. This allows the simplification of the full Navier-Stokes equations by neglecting terms involving second derivatives with respect to xx (or higher-order derivatives in xx) relative to those involving yy. Specifically, the assumption is that the flow gradients are dominated by the normal direction: ∣∂∂y∣≫∣∂∂x∣\left| \frac{\partial}{\partial y} \right| \gg \left| \frac{\partial}{\partial x} \right|. This leads to the reduction of the governing PDEs to a simplified set valid near the wall.