It is found experimentally that all the components of fluid velocity (not just thenormal component) vanish at a wall. No matter how small the viscosity, the large velocity gradients near a wall invalidate Euler’s equations. Prandtl proposed that viscosity has negligible effect except near a thin region near a wall. Prandtl’s equations simplify the Navier-Stokes equation in this boundary layer, by ignoring one dimension. They have an unusual scale invariance in which the distances along the boundary and perpendicular to it have different dimensions. Using this symmetry, Blasius reduced Prandtl’s equations to one dimension. They can then be solved numerically. A convergent analytic approximation was also found by H. Weyl. The drag on a flat plate can now be derived, resolving d’Alembert’s paradox. When the boundary is too long, Prandtl’s theory breaks down: the boundary layer becomes turbulent or separates from the wall.
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