An exact parametric solution for granular flow in a converging wedge

Abstract

The flow of granular materials in the presence of gravity through converging wedges and cones arises in many industrial situations. For both wedges and cones, and assuming an ideal cohesionless granular material which satisfies the Coulomb-Mohr yield condition, the number of simple analytical solutions is limited and generally the governing coupled ordinary differential equations need to be solved numerically. Here we show that for plane wedge flow, an exact parametric solution may be determined for the special case of the angle of internal friction φ is assumed to be π/2 . This is the only known exact solution of these important equations involving two arbitrary constants. A general numerical solution obtained by previous authors is shown to coincide with the special exact solution. Mathematics Subject Classification (2000). 73B10, 73B99, 73Q05.

Some exact velocity profiles for granular flow in converging hoppers

“…This latter formulation formed the basis for the exact analysis for δ = π/2 given in [8], and it is also the basis for an independent numerical scheme. Now, due to the geometry of the hopper, we assume that the stress distribution is symmetrical around the vertical axis.…”

“…In this section, we briefly state the two and three-dimensional exact parametric solutions given in [8] and [9] for the special case of an angle of internal friction equal to ninety degrees and for gravity flow through a converging hopper. We utilize these known solutions to determine the corresponding velocity fields applying to flow through converging hoppers, according to the non-dilatant double-shearing theory of granular materials.…”

“…The two-dimensional exact parametric solution for the special case of δ = π/2 was first derived in Hill and Cox [8] for the determination of the stress distribution in a hopper, and later utilized in Hill and Cox [13] to solve the problem of determining the exact stress distribution beneath a granular heap and in Cox et al [14] to find the stress distribution in a sloping rat-hole. Here, following Spencer and Bradley [6], we utilize the non-dilatant double-shearing theory to determine the corresponding velocity field for converging wedge flow.…”

“…Here, following Spencer and Bradley [6], we utilize the non-dilatant double-shearing theory to determine the corresponding velocity field for converging wedge flow. Hill and Cox [8] show that equation (2.8) for the special case of δ = π/2 admits the following exact parametric solution for ψ(θ),…”

“…While of course the special case of β = −1 corresponds to a non-physical material, there are however many granular materials which do indeed exhibit large angles of internal friction such as those shown in Table 1, and the case β = 1 is physically plausible and constitutes an ideal limiting theory. In this paper, we utilize the non-dilatant double-shearing theory of granular materials to determine the corresponding two and three-dimensional associated velocity fields for the exact parametric solutions determined by Hill and Cox [8] and Cox and Hill [9] for the special case of δ = π/2. Here, for convenience and for purposes of comparison, we follow the notation adopted by Spencer and Bradley [6], including adopting the unusual convention of the x-axis being vertical.…”