J. Fluid Mech. (1982), vol. 117, pp. 283-304 Printed in Great Britain 283 w Flow of fluid of non-uniform viscosity in converging and diverging channels By ALISON HOOPER, School of Mathematics, Bristol University B. R. DUFFY AND H. K. MOFFATT Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge (Received 17 February 1981 and in revised form 9 July 1981) It is shown that the well-known Jeffery-Hamel solution of the Navier-Stokes equations admits generalization to the case in which the viscosity , U and density p are arbitrary functions of the angular co-ordinateo. When I Raj < 1, where R is the Reynolds number and 2a 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 $93-6, and the effect of varying the viscosity ratio h 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 single- fluid limit (A +- 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. 1. Introduction Flows with non-uniform viscosity, particularly two-fluid flows with a viscosity jump at the interface, arise in many processes of technological importance. Analysis of such problems is complicated (i) by the fact that the interface geometry in general changes with time even if the boundary conditions are steady, and (ii) by the fact that, if the interface intersects a solid boundary, the no-slip condition is (in general) inadequate in an immediate neighbourhood of the contact line, which may be observed to move relative to the solid surface. In attempting to approach problems in this category in a general way, it seems natural first to study situations in which these particular difficulties are avoided. Figure I shows one such situation, viz the flow of a fluid of non-uniform viscosity along a two-dimensional duct of slowly varying width. This elementary problem is analysed (under the lubrication approximation) in $ 2, in order to provide motivation for the subsequent study ($$ 3-7) of the Jeffery-Hamel configuration (figure 2), which is the main content of this paper. In $ 3, we show that the well-known exact solution of the Navier-Stokes equations (Jeffery 1915, Hamel 1916; see Batchelor 1967, 5 5.6) can be generalized to the situation in which the viscosity ,U and/or the density p are 10 FLM 117