Stokes flow between two intersecting circular arcs

Extract

This paper considers a family of viscous flows closely related to the exact Jeffery-Hamel solution ((l), (2)) of the two-dimensional Navier-Stokes equations, for diverging or converging flow in a channel. It is known that if the walls of the channel intersect at an angle less than π then there is a unique solution of the Navier-Stokes equations in which the streamlines are straight lines issuing from the point of intersection of the walls and the flow is everywhere diverging or everywhere converging. The flow parameters depend on the total fluid mass M emitted at the point of intersection and the angle 2α between the walls. By taking the Reynolds number R = M/ν, where v is the kinematic viscosity, the stream function can be expanded in a power series in R in which the leading term is a Stokes flow. Alternatively the solution can be developed by perturbing the Stokes flow and is one of very few examples known in which a Stokes flow can be regarded as a uniformly valid first approximation everywhere in an infinite fluid region. The class of flows to be considered is a generalization of the Jeffery–Hamel flow by taking the flow region to be finite and bounded by two circular arcs which intersect at an angle less than π At one point of intersection fluid is forced into the region and an equal amount is absorbed out at the other point. It is found to the first order that the flow at the two points of intersection corresponds to the zero Reynolds number limit for diverging and converging flow, respectively. Now since the flow at these points can be developed by perturbing the Stokes flow solution it is reasonable to assume that the zero Reynolds number flow in the entire finite region bounded by the arcs is a Stokes flow since the most likely region in which this approximation becomes invalid is locally at the points of intersection but here the validity of the approximation is ensured. A comparison of the convection terms with the viscous terms verifies that this conclusion is borne out.