Flow along a diverging channel

This paper treats the two-dimensional steady flow of a viscous incompressible fluid driven through a channel bounded by two walls which are the radii of a sector and two arcs (the 'inlet' and 'outlet'), with the same centre as the sector, at which inflow and outflow conditions are imposed. The computed flows are related to both a laboratory experiment and recent calculations of the linearized 'spatial' modes of Jeffery-Hamel flows. The computations, at a few values of the angle between the walls of the sector and several values of the Reynolds number, show how the first bifurcation of the flow in a channel is related to spatial instability. They also show how the end effects due to conditions at the inlet and outlet of the channel are related to the spatial modes: in particular, Saint-Venant's principle breaks down when the flow is spatially unstable, there being a temporally stable steady flow for which small changes at the inlet or outlet create substantial effects all along the channel. The choice of a sector as the shape of the channel is to permit the exploitation of knowledge of the spatial modes of Jeffery-Hamel flows, although we regard the sector as an example of channels with walls of moderate curvature.