An excellent closed-form approximate solution for laminar boundary-layer flow at a separation point

Abstract Laminar boundary-layer flow at the separation point is governed by the Falkner–Skan equation at β = −0.19884 and is difficult to represent with a compact analytical expression. This study develops a closed-form approximate solution for the dimensionless stream function, streamwise velocity, and shear stress across the boundary layer. An existing cubic velocity formulation is first examined; however, the separation-point condition forces the linear and cubic coefficients to vanish, reducing the approximation to a quadratic exponent and producing substantial disagreement with numerical data. A quartic velocity formulation is therefore proposed, f ′( η ) = 1 − exp(− bη 2 − dη 4 ), with b = 0.09942 and d = 0.0049421682. Corresponding analytical expressions are derived for the shear stress and stream function, the latter represented by a rapidly convergent series involving the lower incomplete gamma function; eight terms are sufficient for accurate evaluation. Comparison with published numerical solutions shows that the quartic formulation closely reproduces the dimensionless stream-function, streamwise-velocity, and shear-stress distributions from the wall to the freestream. The maximum Falkner–Skan residual is approximately 9.45 × 10 −3 at η = 3.1. The solution also predicts a velocity-profile inflection point at η = 2.33550, where the maximum dimensionless shear stress is 0.359528. The proposed quartic expression thus provides a compact, explicit, and accurate analytical approximation for laminar boundary-layer flow at separation.