Micropolar medium in a funnel-shaped crusher

Mariia Fomicheva,Wolfgang H Müller,Elena N Vilchevskaya,Nikolay Bessonov

doi:10.1007/s00161-021-00973-w

Abstract

In this paper, the solution to a coupled flow problem for a micropolar medium undergoing structural changes is presented. The structural changes occur because of a grinding of the medium in a funnel-shaped crusher. The standard macroscopic equations for mass and linear momentum are solved in combination with a balance equation for the microinertia tensor containing a production term. The constitutive equations of the medium describe a linear viscous material with a viscosity coefficient depending on the characteristic particle moment of inertia, the so-called microinertia. A coupled system of equations is presented and solved numerically in order to determine the distribution of the fields for velocity, pressure, viscosity coefficient, and microinertia in all points of the continuum. The numerical solution to this problem is found by using the implicit finite difference method and the upwind scheme.

Highlights

The microinertia tensor of a continuum particle, J, plays an important role in context with its rotational degrees of freedom, in combination with the angular velocity vector, ω, assigned to the continuum element
The microinertia tensor obeys a kinematic constraint in form of a rate equation, which expresses the possibility of material continuum particles to undergo rigid body rotations
It will be demonstrated that this extended theory allows for the modeling of processes accompanied by a considerable structural change characterized by a changing microinertia within a representative volume element

Summary

Introductory remarks

The microinertia tensor of a continuum particle, J, plays an important role in context with its rotational degrees of freedom, in combination with the angular velocity vector, ω, assigned to the continuum element. The main feature of our study consists in that a set of coupled equations for mass, linear momentum, and the balance of microinertia tensor with a production term is presented This system is solved numerically for the first time in order to determine the distribution of the fields for velocity, pressure, viscosity coefficient, and the moment of microinertia. Besides that, another novelty of this paper is that the constitutive equations of the medium describe a linear viscous material with a viscosity coefficient depending on the characteristic particle moment of inertia. That means that a coupled problem is analyzed, in which the moment of inertia and the viscosity coefficient can change from position to position

The balances of micropolar media

Problem description

Results

Conclusions and outlook