Bits of the roller cone type are now very common in drilling columns and machines for geological exploration and mining purposes.

Consider a horizontal, fairly wide channel; the influence of vertical walls can be ignored. Then we can assume that there is a movement of a two-component medium between two parallel horizontal infinite planes. The weight of the gas will be neglected, and the weight of the particles will be taken into account...

Let us consider a two-component medium, for example, a mixture of gas or liquid and suspended particles. We will take into account the interaction between them according to the linear or quadratic law. The fluid motion is assumed to be one-dimensional (vertical), steady and laminar. The flow extends wide enough in horizontal directions, so that no vertical walls are considered. Of particular importance is the choice of continuity equations. In the author's works, an attempt was made to use continuity equations specially derived for the case under consideration, but the difficulties of extending them to three-dimensional space forced the author to switch to the generally accepted continuity equations.

Motion of particles and fluid bounded by parallel planes. We have already considered the problem of motion of suspended particles in the particular case of absence of obstacles. In this paper we consider the one-dimensional motion of particles and gas bounded by two parallel planes - walls.

Предположим, что имеется движущаяся среда — жидкость или газ — в условиях, когда газ можно считать несжимаемой жидкостью. В этой среде находится большое количество малых частиц. Частицы будем считать приближенно совершенно одинаковыми по весу и объему твердыми телами, имеющими сферическую форму. Траектории, скорости и ускорения частиц отличны от движения среды в любой точке пространства.

When designing a dredge and selecting the main parameters that determine its operation, an important issue is the choice of angular speed of rotation of the motor and the length of the bottom of the scoop. For normal operation of the dredge it is necessary to select these parameters so that the material, pouring out of the scoop, falls into the receiving device. To solve this issue, the movement of material along the bottom of the scoop should be studied.

Suppose we have a moving medium - a liquid or a gas under conditions when the latter can be considered an incompressible liquid. In this medium there are a large number of small particles, which we consider as solid bodies - material points, the position of which we will define by their coordinates, attributing to each particle some small mass and some small volume common to all particles. For definiteness we will assume that all particles have a spherical shape and motion different from the motion of the above-mentioned medium. The mass distribution in the medium will be continuous, while the mass distribution of the particles will be clearly discrete.