Navier-Stoke vs Prandtl lift calculation
The Navier-Stokes equations represent a highly non-linear set of equations. Actually, although the have almost 2 centuries, they are one of the Millennium Prize Problems. Most of the interesting phenomenas represented by these equations include turbulence.
In specific conditions and hypothesis, analytical solutions have been demonstrated which is allowing us to understand the problem and physicis by reducing the complexity of the equations.
The law of Poiseuille appeared also after Navier-Stokes, and event the Darcy equation is empirical, not derived from Navier Stokes.
The interest of aerodynamics started to increase at the end of sXIX (first wind tunnel was constructed on 1871), so, in front of the complexity of this phenomena and the challenge the Navier-Stokes equations are some simplified equations appeared with some specific conditions. One of them was the the Prandtl lifting line theory.
One of the advantages of this theory is that relates the aerdoynamic properties of a wing to its geometry, without the need to calculate the complete flow field... think about that in CFD you have to calculate the complete flow field during hours (days?) in order to understand the properties and behaviour of a surface, this theory gives an analytical answer...
Still this theory is useful today, as helps to understand the behaviour of the wing.