Here the surface may be curved but its radius of curvature is assumed to be much larger than the boundary-layer thickness. We shall refer to the solution of the irrotational flow outside the boundary layer as the outer problem and that of the boundary-layer flow as the inner problem

According to the boundary-layer theory, when a fluid flows past an object, frictional effects are significant only in a thin region close to the wall, where large transverse gradients of velocity exist. Within this thin boundary layer, the velocity rises rapidly from zero at the wall to the freestream value at its edge

For a thin boundary layer that is contiguous to the solid surface on which it has formed, the full equations of motion for a constant-density constant-viscosity fluid,

Continuity Equation (Incompressible flow condition): \( \nabla \cdot \mathbf{u} = 0 \)
Navier-Stokes Equation (Momentum equation for a Newtonian fluid): \( \frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u} \)

may be simplified

Let \( \bar{\delta} (x) \) be the average thickness of the boundary layer at downstream location \( x \) on the surface of a body having a local radius of curvature \( R \). The steady-flow momentum equation for the surface-parallel velocity component, \( u \), is

\( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \)

which is valid when \( \bar{\delta} / R \ll 1 \)

The normal pressure gradient \( \frac{\partial p}{\partial y} \) is negligible due to the boundary layer thickness \( \delta \):

\( -\frac{1}{\rho} \frac{\partial p}{\partial y} = \mathcal{O}(\delta) \)

This is a key assumption in boundary layer theory