Flow of fluid of non-uniform viscosity in converging and diverging channels

Abstract

It is shown that the well-known Jeffery–Hamel solution of the Navier–Stokes equations admits generalization to the case in which the viscosity μ and density ρ are arbitrary functions of the angular co-ordinate θ. When |Rα| [Lt ] 1, where R is the Reynolds number and 2α the angle of divergence of the planes, lubrication theory is applicable; this limit is first treated in the context of flow in a channel of slowly varying width. The Jeffery–Hamel problem proper is treated in §§ 3–6, and the effect of varying the viscosity ratio λ in a two-fluid situation is studied. In § 5, results already familiar in the single-fluid context are recapitulated and reformulated in a manner that admits immediate adaptation to the two-fluid situation, and in § 6 it is shown that the singlefluid limit (λ → 1) is in a certain sense degenerate. The necessarily discontinuous behaviour of the velocity profile as the Reynolds number (based on volume flux) increases is elucidated. Finally, in § 7, some comments are made about the realizability of these flows and about instabilities to which they may be subject.