Converging and diverging flow in narrow conical passages

This paper deals with sink and source flows developed in cones with small apex angles and in narrow gaps formed by two concentric cones. Numerical and approximate analytical solutions to these flows are presented. An exact solution for the creeping flow in conical gaps in terms of Legendre polynomials is derived. The analytical solutions to the flow in cones, includes the linearized inertia terms in the momentum equations, and are given using Gauss' hypergeometric series. For low Reynolds numbers, both converging and diverging flows are shown to coincide and are similar to Poiseuille's flow. However, when inertia effects are included, these are found to be radically different. For a sink flow, the radial velocity flattens in the neighborhood of the mid-angle and, as Re increases, the plateau expands out towards the conical walls, tending to the inviscid flat profile throughout the entire flow field. Contrary to the accelerating flow, the maximum velocity of the decelerating flow is shown to increase with Re. As a first critical Reynolds number is approached, the shear stress reduces to zero on the cone walls or, for a conical gap, on the outer cone walls. A further increase in Re above this first critical value, is found to produce a flow reversal either near the wall or, for the case of a conical gap, in the proximity of the outer cone. Thus, when Re exceeds this second value, purely decelerating flow cannot exist. The results for accelerating (sink) flow indicate that the approximate analytic solution is an excellent representation of this flow, whereas, for decelerating (source) flows particularly near separation, the results indicate that the numerical approach is needed to properly capture all flow features.