Therefore, to calculate net forces on bodies ( such as wings ), viscous flow equations must be used : inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.
22.
The problem arises because lift on an airfoil in inviscid flow requires Circulation and the Kutta-Joukowski theorem " below ), but a single potential function that is continuous throughout the domain around the airfoil cannot represent a flow with nonzero circulation.
23.
Inviscid flow was further analyzed by various mathematicians ( Leonhard Euler, Jean le Rond d'Alembert, Joseph Louis Lagrange, Pierre-Simon Laplace, Sim�on Denis Poisson ) and viscous flow was explored by a multitude of engineers including Jean L�onard Marie Poiseuille and Gotthilf Hagen.
24.
However, using his hypothesis ( and backed up by experiments ) Prandtl was able to derive an approximate model for the flow inside the boundary layer, called " boundary-layer theory "; while the flow outside the boundary layer could be treated using inviscid flow theory.
25.
The approximation states that, for a sufficiently high Reynolds number the flow over a surface can be divided into an outer region of inviscid flow unaffected by viscosity ( the majority of the flow ), and a region close to the surface where viscosity is important ( the boundary layer ).
26.
Referring to the adjacent diagram, using Bernoulli's equation in the special case of steady, incompressible, inviscid flows ( such as the flow of water or other liquid, or low speed flow of gas ) along a streamline, the theoretical pressure drop at the constriction is given by:
27.
The lift per unit span ( L') acting on a body in a two-dimensional inviscid flow field can be expressed as the product of the circulation ? about the body, the fluid density " ? ", and the speed of the body relative to the free-stream "'V " '.
28.
Prandtl discovered that when the Reynolds number, defined as Re = \ frac { \ rho V _ { \ infty } c _ A } { \ mu } \,, is large enough and the angle of attack is small enough, then the flow around a thin enough airfoil is composed of a narrow viscous region called the boundary layer near the body and an inviscid flow region outside.
29.
While the no-slip condition is used almost universally in modeling of viscous flows, it is sometimes neglected in favor of the'no-penetration condition'( where the fluid velocity normal to the wall is set to the wall velocity in this direction, but the fluid velocity parallel to the wall is unrestricted ) in elementary analyses of inviscid flow, where the effect of boundary layers is neglected.
