Converging flows of viscoplastic fluids, driven steadily through wedges and axisymmetric cones, are studied analytically and numerically. When the yield stress is relatively large, the bulk of the fluid flows plastically apart from within thin layers where the fluid is strongly sheared in order to achieve no slip at the boundary. Conversely, when the yield stress is relatively small, the motion is viscously dominated with weak corrections to the velocity and stress fields due to viscoplastic effects. For both regimes, visco-plasticity induces a weak angular velocity, directed away from the boundaries, and purely radial flow is not possible. The structure of the flow is calculated using asymptotic methods, confirmed by finite element numerical simulations. Flows of both Bingham and Herschel-Bulkley fluids are analysed, and both planar and axisymmetric geometries are considered. Although these cases differ in their details, they share the same qualitative structure. In particular, the viscoplastic boundary layers that emerge when the yield stress is relatively large, ensure not only that no slip is enforced, but also, through an intermediate matching layer, that the shear rates remain bounded.
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