The Lie-Group Shooting Method for Multiple-Solutions of Falkner-Skan Equation under Suction-Injection Conditions

摘要

For the Falkner-Skan equation, including the Blasius equation as a special case, we will develop a new numerical technique, transforming the governing equation into a nonlinear second-order ordinary differential equation by a new transformation technique, and then solving it by the Lie-group shooting method. The second-order equation is singular, which is however much saving computational cost than the original equation denned in an 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 boundry condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under boundary conditions imposed. 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 multiple-solutions under very complex conditions of boundary suction or injection, and the motion of plate. The boundary layer flows studied here have broad applications in liquid film condensation, geophysical and oceangraphy contexts.