Local similarity solutions and their limitations

Abstract

Two problems exhibiting breakdown in local similarity solutions are discussed, and the appropriate asymptotic form of the exact solution is determined in each case. The first problem is the very elementary problem of pressure driven flow along a duct whose cross-section has a sharp corner of angle 2α. When 2α < ½π, a local similarity solution is valid, whereas, when 2α > ½π, the solution near the corner depends on the global geometry of the cross-section. The transitional behaviour when 2α = ½π is determined.The second problem concerns low-Reynolds-number flow in the wedge-shaped region |θ| < α when either a normal velocity proportional to distance from the vertex is imposed on both boundaries, or a finite flux 2Q is introduced or extracted at the vertex (the Jeffery–Hamel problem). It is shown that the similarity solution in either case is valid only when 2α < 2α_c ≈ 257·5^°; a modified problem is solved exactly revealing the behaviour when α > α_c, and the transitional behaviour when α = α_c (when the similarity solutions become infinite everywhere). An interesting conclusion for the Jeffery–Hamel problem is that when α > α_c, inertia forces are of dominant importance throughout the flow field no matter how small the source Reynolds number 2Q/ν may be.