A new class of integro - partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian equations of motion of a many-particle system incorporating widely used inelastic particle-particle force formulas. By using Taylor series expansions, these models can be approximated by a system of partial differential equations of the Navier-Stokes type. Solutions of the new models for granular flows down inclined planes and in vibrating beds are compared with known experimental and analytical results and good agreement is obtained.
Theorems on the existence and uniqueness of a solution to the granular flow dynamical system are proved in the Faedo-Galerkin method framework. A class of one-dimensional models describing the dynamics of thin granular layers and some related problems of fluid mechanics was studied from the Liouville-Lax integrability theory point of view. The integrability structures for these dynamical systems were constructed using Cartan's calculus of differential forms, Grassman algebras over jet-manifolds associated with the granular flow dynamical systems, the gradientholonomic algorithm and generalized Hamiltonian methods. By proving the exact integrability of the systems, the quasi-periodicity of the solutions was explained as well as the observed regularity of the numerical solutions.
A numerical algorithm based on the idea of higher and lower modes separation in the theory of approximate inertial manifolds for dissipative evolutionary equations is developed in a finite-difference framework. The method is applied to the granular flow dynamical system. Numerical calculations show that this method has several advantages compared to standard finite-difference schemes.
A numerical solution to the granular flow in a hopper is obtained using the finite difference scheme in curvilinear coordinates. By making coefficients in the governing equations functionally dependent on the gradient of the velocity field, we were able to model the influence of the stationary friction phenomena in solids and reproduce in this way experimentally observable results.
Some analytical and numerical solutions to the dynamical system describing granular flows in vibrating beds are also presented. We found that even in the simplest case where we neglect the arching phenomena and surface waves, these solutions exhibit some of the typical features that have been observed in simulation and experimental studies of vibrating beds. The approximate analytical solutions to the governing system of equations were found to share several important features with actual granular flows. Using this approach we showed the existence of the typical dynamical structures of chaotic motion. By employing Melnikov theory the bifurcation parameter values were estimated analytically. The vortex solutions we obtained for the perturbed motion and the solutions corresponding to the vortex disintegration agree qualitatively with the dynamics obtained numerically.