The Lie-group shooting method for multiple-solutions of Falkner–Skan equation under suction–injection conditions

Abstract

For the Falkner–Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t ∈ [ 0 , 1 ] , and moreover, the initial slope can be expressed as a closed-form function of r ∈ ( 0 , 1 ) , where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge–Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.