ABSTRACT
Consider the two-dimensional laminar viscous flow over a semi-infinite flat plate. The family of similar solutions of the incompressible boundary layers was first obtained by Falkner and Skan [106] in 1931. Let x denote distance from the leading edge of a semi-infinite flat plate and y distance normal to the plate, U the velocity of the fluid in the mainstream, ν the kinematic viscosity, and u and v the components of the velocity of the fluid in the directions of x, y respectively. Falkner and Skan [106] demonstrated that, if U ∝ xκ, where κ is a constant, there exist solutions of the boundary layer equation
f ′′′(η) + f(η)f ′′(η) + β[1− f ′2(η)] = 0, (15.1) subject to the boundary conditions
f(0) = f ′(0) = 0, f ′(+∞) = 1, (15.2) where
β = 2κ
κ + 1 , η = y
√ (1 + κ)U
2νx (15.3)
and the prime denotes differentiation with respect to the similarity variable η. The components u, v of the fluid velocity are given by
u = Uf ′(η), v = [f(η)− (κ− 1)ηf ′(η)] √
νU
2(κ + 1)x . (15.4)
Note that f(η) depends on the physical parameter β only. When κ ≥ 0, from (15.3), it is easy to see that
0 ≤ β ≤ 2. When κ < 0, the mainstream velocity U ∝ xκ is singular at x = 0 so that Falkner-Skan’s solution f(η) cannot be taken right back to x = 0. This is a general difference between the solutions with a positive and negative β. In 1937 Hartree [107] numerically solved the Falkner-Skan’s equations. For large η, Hartree [107] provided the asymptotic expression
1− f ′(η) ≈ A exp(−f2/2) f−(2β+1) + B f2β , (15.5)
condition f ′(+∞) it is clear that f ∼ η as η → +∞ so that f → +∞ as η → +∞.
