The velocity equations for dilatant granular flow and a new exact solution (Q1366338)

A new exact solution is presented to kinematically determined velocity equations for axially symmetric flow of dilatant granular materials. Two sets of three first-order partial differential equations are obtained from the stress equations in the limits of certain plastic regimes, assuming a Coulomb-Mohr plastic potential in the first case and dilatant double shearing in the second one. All known solutions of these partial differential equations are classified into four known functional forms. Using standard algebraic packages for the determination of Lie symmetries of main equation set, it is shown that the velocity equations admit a four-dimensional symmetry algebra. Then all above-mentioned functional forms of existing solutions are systematically obtained by identification of the so-called ``optimal systems'' of the invariant Lie algebra. This result indicates that there are no further solutions which can be generated from the classical symmetries. Any new families of solutions arise only from the so-called ``non-classical'' symmetries. Then a simple new solution of the velocity equations is obtained in closed form for a special case which has not been given in the literature. For this particular case, the integral curves (streamlines) and simple analytical approximations for particle paths are found and shown graphically.