The Lie-group shooting method for boundary-layer problems with suction/injection/reverse flow conditions for power-law fluids

For power-law fluids we propose a Lie-group shooting method to tackle the boundary-layer problems under a suction/injection as well as a reverse flow boundary conditions. The Crocco transformation is employed to reduce the third-order equation to a second-order ordinary differential equation, and then through a symmetric extension to a boundary value problem with constant boundary conditions  , which can be solved numerically by the Lie-group shooting method. However, the resulting equation is singular, which might be difficult to solve, and we propose a technique to overcome the initial impulse caused by the singularity using a very small time-step integration at the first few time steps. Because we can express the missing initial condition through a closed-form formula in terms of the weighting factor r∈(0,1)r∈(0,1), the Lie-group shooting method is very effective for searching the multiple-solutions of a reverse flow boundary condition.

► Boundary layer problems of power-law fluids were treated by a Lie-group shooting method.
► Singularity appears by using the Crocco transformation.
► A new technique was developed to overcome the initial impulse.
► The missing initial condition could be expressed in terms of a weighting factor.
► Multiple solutions of a reverse flow were found.