Multiple solutions of two-dimensional and three-dimensional flows induced by a stretching flat surface

•New dual solutions for Crane stretching plate flow.•New dual solutions for Wang’s bi-orthogonally stretching plate flow.•Algebraic decay of dual solutions in the far field.

New solutions of flow induced by a biorthogonally stretching surface are reported. The flexible membrane has linear stretching rate a along the x-axis and b along the y  -axis. A similarity reduction of the Navier–Stokes equations yields a coupled pair of ordinary differential equations governed the single parameter α=b/aα=b/a. Dual solutions are found in the region αt<α⩽1αt<α⩽1, where αt=-0.2514αt=-0.2514. One of the two components of the dual solutions exhibits algebraic decay in the far field. It appears that no self-similar solutions exist for α<αtα<αt. It is also shown that the exact solution for flow induced by a unilaterally stretching sheet due to Crane has dual solutions with algebraic decay in the far field.