@misc{Piven_M._Planned,
author={Piven, M.},
publisher={The Lublin Branch of the Polish Academy of Sciences},
abstract={In the article the equations have been worked making it possible to model the motion of freerunning
grain mixture flow on a flat sloping vibrating sieve within the framework of shallow water theory.
Free-running grain mixture is considered as a heterogeneous system consisting of two phases, one of
which represents solid particles and the other one gas.
The mixture is brought into a state of fluidity by means
of high-frequency vibration imposition. Coefficients of
internal and external friction and dynamic-viscosity decrease by exponential law as the fluctuation intensity
is increased.
When considering grain mixture dynamics, the
following assumptions are put forward: we ignore the
air presence in space between particles, we consider the
density of particles to be constant, the free-running
mixture is similar to Newtonian liquid.
The basic system of equations of grain mixture
dynamics is due to the laws of continuum mechanics.
The equation of continuity is issued from the law of
conservation of mass, and the dynamic equations are
issued from the law of variation of momentum.
The stress tensor equals to the sum of the
equilibrium tensor and the dissipative tensor. The
equilibrium part of the stress tensor is represented by
the spherical tensor, which is found to conform to
Pascal law for liquids, and the dissipative part, which is
responsible for viscous force effect and defined by
Navier-Stokes law.
Boundary conditions on the surfaces (restricting the
capacity of the free-running grain mixture) have been
researched. The distributions of apparent density and
velocity field are assigned at the inlet and outlet flow
sections of the mixture. The normal velocity component
of the grain mixture on the side frames and on the sieve
becomes zero, which meets the no-fluid-loss condition
of the medium through the frame. Beyond that point at
this time we satisfy dynamic conditions, which
characterize the mixture sliding down the hard frame,
motion flow resistance force is represented as average
velocity linear dependence. A kinematic condition and
two dynamic ones are stipulated on the free surface
layer. One of the conditions states mass flow continuity
across the free surface, the other one states the stress
continuity while passing through the free surface.
The basic premise of planned motion equations is
the condition of small size of flow depth in comparison
with its width. With the use of shallow water theory the
basic principles of the equations of flow dynamics are
simplified and for their solving a Cauchy problem can
be set.},
title={Planned Motion Equations of Free-running Grain Mixture Flow},
type={ArtykuĹ‚y / Articles},
keywords={shallow water theory, planned flow motion, free-running mixture, stress tensor, strain velocity, viscous friction},
}