What is the Blasius equation that describes the steady two-dimensional boundary layer stream function?

What is the Blasius equation that describes the steady two-dimensional boundary layer stream function?

• A. f''' + f'' = 0
• B. f''' + f f'' = 0
• C. f'''+ f' f = 0
• D. d^3f/dy^3 = 0

Answer: (B)

In steady, laminar two-dimensional boundary layer flow over a flat plate, a similarity transformation reduces the momentum equation to a single nonlinear third‑order ODE for a nondimensional stream-function f(η). The velocity components relate to f by u/U = f′(η) and the vertical velocity comes from the continuity of the transformed variables. Substituting these into the boundary-layer equations leads to the equation f‴(η) + f(η) f″(η) = 0. The primes denote derivatives with respect to the similarity variable η, which bundles the x and y behavior into one variable. This form captures the essential balance in the boundary layer: viscous diffusion (the third derivative) and nonlinear advection (the product f times the second derivative) interact to shape the velocity profile. The boundary conditions are f(0) = 0, f′(0) = 0, and f′(∞) = 1, reflecting no slip at the wall and the free-stream velocity far from the plate. Other options don’t arise from this similarity reduction. One is linear and lacks the nonlinear convective term; another places the nonlinearity in a different product that doesn’t come from the transformed momentum equation; and the last uses derivatives with respect to the physical coordinate y instead of the similarity variable η, which is not compatible with the Blasius similarity solution.

In steady, laminar two-dimensional boundary layer flow over a flat plate, a similarity transformation reduces the momentum equation to a single nonlinear third‑order ODE for a nondimensional stream-function f(η). The velocity components relate to f by u/U = f′(η) and the vertical velocity comes from the continuity of the transformed variables. Substituting these into the boundary-layer equations leads to the equation f‴(η) + f(η) f″(η) = 0. The primes denote derivatives with respect to the similarity variable η, which bundles the x and y behavior into one variable.

This form captures the essential balance in the boundary layer: viscous diffusion (the third derivative) and nonlinear advection (the product f times the second derivative) interact to shape the velocity profile. The boundary conditions are f(0) = 0, f′(0) = 0, and f′(∞) = 1, reflecting no slip at the wall and the free-stream velocity far from the plate.

Other options don’t arise from this similarity reduction. One is linear and lacks the nonlinear convective term; another places the nonlinearity in a different product that doesn’t come from the transformed momentum equation; and the last uses derivatives with respect to the physical coordinate y instead of the similarity variable η, which is not compatible with the Blasius similarity solution.