Some Exact Mathematical Solutions for Granular Stock Piles and Granular Flow in Hoppers

Abstract

The flow of granular materials in the presence of gravity in hoppers and the storage of granular materials as a stock pile occur in many industrial situations. The governing ordinary differential equations for two-dimensional wedges and three-dimensional cones are highly nonlinear and there are no known general solutions, apart from that we have given for a special angle of internal friction. Here, we give the overall picture relating to those special cases which give rise to analytical solutions for the two problems of granular flow through hoppers and the stress distributions at the base of stock piles. These equations are fundamental to granular mechanics and previously only some special isolated exact solutions have been known. We list here a number of new exact analytical solutions applying for the two special cases of β= ±1, noting that β= sin φ where φ is the angle of internal friction. The case β= −1 corresponds to a non-physical material, but there are materials such as silica and alumina cake which do indeed exhibit large angles of internal friction and the case β= 1 is by no means unrealistic. However, all the solutions presented are meaningful mathematical solutions of the governing equations and constitute the only known general solutions of these important equations. For certain cases, a full independent numerical solution has been obtained and shown to coincide with the appropriate exact analytical solution.

Some exact velocity profiles for granular flow in converging hoppers

“…This gives rise to the notion of the Haar-von Karman regimes, and in particular, either that an exact parametric solution can only be determined for the case of the angle of internal friction equal to minus ninety degrees, which is evidently non-physical. However, when the hoop stress is equal to the maximum principal stress, we find from Cox and Hill [9] that an exact parametric solution can be determined for the special case of the angle of internal friction equal to ninety degrees. Thus, we assume that the hoop stress is equal to the maximum principal stress, or in other words we suppose = 1.…”

“…We also note that if we eliminate G from (2.21) then we may deduce the following single second order ordinary differential equation for Ψ(Θ), where throughout the paper primes denote differentiation with respect to Θ. Again we note that this formulation formed the basis for the exact analysis for δ = π/2 given in [9], and it is also the basis for an independent numerical scheme. Now, the stress distribution is symmetrical about the vertical axis, and we observe from the equilibrium equations (2.15) that σ RR , σ ΘΘ and σ ΦΦ must be even functions of Θ, while σ RΘ must be an odd function or skew-symmetric.…”

“…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.…”

“…Recently, these equations have been shown to admit exact parametric solutions for the special cases of β = ±1, where β = sin δ and δ is the angle of internal friction (see Hill and Cox [8] and Cox and Hill [9]). These solutions are the first exact solutions of the highly nonlinear coupled ordinary differential equations which involve two arbitrary constants.…”

“…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.…”