Blasius

Blasius refers to a foundational boundary-layer solution in fluid dynamics, derived in 1908 by a German physicist. The solution addresses steady, incompressible, laminar, two-dimensional flow over a flat plate aligned with a uniform free stream, under a zero pressure gradient along the plate.

Using a similarity transformation with the dimensionless variable η = y sqrt(U/(ν x)) and a stream-function form ψ = √(ν U x) f(η), the boundary-layer equations reduce to the third-order nonlinear ordinary differential equation f''' + (1/2) f f'' = 0. The boundary conditions are f(0) = 0, f'(0) = 0, and f'(∞) = 1. The dimensionless velocity is given by u/U = f'(η).

The Blasius solution is obtained numerically, with f''(0) ≈ 0.332. The wall-shear stress scales as τ_w = μ U/(√(ν x)) f''(0), and the skin-friction coefficient for a flat plate is Cf = 0.664 Re_x^(-1/2), where Re_x = U x/ν. The solution provides a practical model for predicting boundary-layer growth, skin friction, and related heat-transfer characteristics in laminar regimes.

Significance and limitations: Blasius represents a cornerstone of boundary-layer theory, offering the first workable similarity solution for a flat-plate flow with zero pressure gradient. It assumes steady, incompressible, laminar flow, a smooth flat plate, and no adverse pressure gradients. It does not apply to turbulent regions, significant pressure gradients, or complex geometries.