On a Powder Consolidation Problem
The problem of the consolidation of an aerated fine powder under gravity is considered. The industrial relevance of the problem is discussed and a mathematical model is introduced. The mathematical structure is that of a coupled system for three unknowns, pressure, stress, and height of the powder in the (axisymmetric) bunker containing it. The system itself consists of a parabolic PDE, an ODE, and an integral equation determining a free boundary corresponding to the height of the powder. Existence and uniqueness of a solution is established. A numerical method based on a formulation of the semidiscretized problem as an index 1 DAE is proposed and implemented. The feasibility of the approach is illustrated by computational results.
A theory of flow of particulate solids in converging and diverging channels based on a conical yield function
Abstract It has been evident for some time that the theory of flow developed by the author twenty-five years ago [1 – 9] is flawed. While the design method based on that theory is adequate in the design of mass-flow hoppers, it predicts incorrect channel flow angles in funnel flow and, hence, is useless in predicting the location of the level at which a funnel-flow channel meets the cylinder wall of a silo. That location is needed in the structural design of silos. The theory introduced in this paper appears to correct that flaw. The correction is obtained by replacing the modified Tresca yield pyramids of the original theory with conical yield surfaces and relating the strain rates to the stresses by the Levi flow rule. In order to obtain solutions applicable to a wide range of solids properties and channel geometries, only steady-state radial fields are considered. In this paper, conditions of flow are analyzed. Obstructions to flow as well as design recommendations to prevent arching and ratholing will appear in a future publication.
The complexity of plane hyperbolic incidence geometry is ∀∃∀∃
We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)