Finding the trajectories of the movement of a material particle on the inner surface of a cone with a vertical axis with lateral feed of the material

The movement of material particles along the inner surface of the cone takes place in cyclones, the designs of which can have both cylindrical and conical parts. Aerodynamic processes occurring in a cyclone are complex in nature, therefore they cannot be accurately modeled on the basis of theoretical approaches. A number of simplifications were introduced during the research: air resistance is not taken into account, since the particle is fed into the cone together with the air, although later their directions of movement do not coincide (the particle damps the speed and falls down, and the air along the central part along the axis of the cone rises up and goes out); the influence of particles on each other, their size, etc. The purpose of the article is to study the motion of a material particle entering the inner surface of a vertical cone with a given initial velocity. If a material particle is directed with an initial velocity to the inner wall of the cone perpendicular to its generator, then its further motion will include both rotation around the axis of the cone and descent down under the action of its own weight. To find the trajectory of motion, a material point was taken as the vertex of the accompanying Frenet trihedron, which has three mutually perpendicular orthogonal planes. The second accompanying Darboux trihedron has a common orthogonal plane tangent to the trajectory with the Frenet trihedron. The balance of the acting forces in the projections onto the orthogonal planes of the Darboux trihedron was considered. This made it possible to determine the projections of the curvature of the curve onto the corresponding orthogonal planes of the Darboux trihedron. The differential geometry apparatus made it possible to find them through the first and second quadratic forms of the surface, which allows avoiding cumbersome transformations. Differential equations of motion of a material particle along the inner surface of a vertical cone were compiled. The equations were solved using the MatLab system. The equation of motion of a particle along the inner surface of the cone was obtained. Analyzing the trajectory of the particle, we can conclude that it is significantly different from the trajectory of motion along the inner surface of the cylinder. The graphs of changes in velocity also show the difference between the motion of a particle along a cone and the same motion along a cylinder. If, upon entering the surface of a cylinder, the particle damps its velocity to a certain limit, and then it begins to increase again, then during movement along a cone the velocity of the particle has a certain periodic character and approaches zero over time. In the absence of friction and air resistance, a material particle, after entering the inner surface of the cone at a certain angle to the generator (except zero), performs an oscillatory motion, alternately rising and falling along a trajectory in the form of a loop, moving for any length of time. Depending on the initial conditions, the particle can describe a finite number of branches of the loop, an infinite number of branches, move along a straight-line generator of the cone, or along an intermediate trajectory between a straight line and a loop. In the presence of friction, the particle will descend to the top of the cone, with possible local rises, the magnitude of which will depend on the initial velocity and the angle of inclination of the generating cone. The velocity in such a motion will damp out, while also having an oscillatory character.