Boundary layer flow over a stretching sheet with variable thickness

In this work, the boundary layers over a continuously stretching sheet with a power law surface velocity were revisited for a sheet with variable thickness. Based on the boundary layer assumptions, the similarity equation governed by two parameters, namely the velocity power index and the wall thickness parameter, was obtained and solved numerically. Theoretical analysis was conducted for special conditions and analytical solutions were derived for the velocity power indices $m=‐\frac{1}{3}$ and $m=‐\frac{1}{2}$ . Numerical techniques were used to solve the similarity equation for general conditions. Quite different and interesting flow behavior was found for negative power indices. Multiple solutions were obtained for certain wall thickness parameter and velocity power indices. Velocity overshoot near the wall was observed for certain solution branches. It is found that the non‐flatness of the stretching surface has significant impacts on the boundary layer development along the wall, on the velocity profiles, and on the shear stress distribution in the fluid. When the velocity power index is less than one, the non‐flatness introduces a mass suction effect; while when it is greater than one, the non‐flatness leads to a mass injection effect. The results for a non‐flat stretching sheet offer quite interesting nonlinear behaviors and further enrich the solution and understanding of boundary layers.