Compressible material flow in cylindrical channel with variable cross section

In the mathematical model of the flow of compressible material the effect of friction and the slip velocity of the material at the side boundary surfaces are considered. The dependence of the slip velocity on the average velocity at the entrance of the channel is built.


Introduction
Mathematical modelling of the compressible frictional flow of material is important theoretical [1,2,3] and practical task [4,5]. Corresponding models are applying to describe and to control technological processes of composite material formation. For example, solid-phase plunger extrusion is a process of one-sided compression under press and extrusion. A material is moved from cylindrical chamber in the calibre across conical domain.

Problem statement
The process of the compressible material flow in a cylindrical channel with variable cross section under the effect of one-sided pressure P=P(t) applied to the end surface ( fig. 1) is considered. The movement of the material occurs in three areas: cylindrical channel of radius 1 R (I), transition zone (II), the channel of radius 2 R (III). A friction force Ffr between the channel wall and the material counteracts the flow. In general, the movement of the material is described by the system consisting the equations for the density determined by the ratio , 1 ρ ρ ⋅ where 1 ρ is density of the incompressible base of the material, ; , ,ϕ ρ ρ = is relative density, the flow velocity and the stress tensor here (1) is the continuity equation, (2) is the Navier-Stokes equation [6], F are mass forces acting on the material. The Newton's law of viscosity (3) is accepted as the differential equation of the state, where ξ µ, are the variable dynamic and volume viscosity depended on density, I is unit tensor, Φ is strain rate tensor.

Statement of the problem in cylindrical coordinate system
A cylindrical coordinate system z r , ,ϕ is used to describe the problem. The axis of the channel symmetry is assumed as the axis z which positive direction is opposite to the direction of the material movement. At the time t the right-hand boundary of zone I is at the position the boundary of zone III is denoted as . Transition zone II has a conical shape with a taper angleα , its boundaries are constant and denoted as 0 = z and 2 z z = . The radius of the cross section of the transition zone is defined by the formula ( ) Thus the movement of the material occurs in a region with variable boundaries.
Due to the symmetry of the channel, the functions do not depend on the angular position and the tangential velocity ϕ v , shear stresses ϕ σ r and ϕ σ z are equal to zero.
In view of the Reynolds number smallness: viscosity of the incompressible base of the material is large, the equation of motion (2) is replaced by the conditions of equilibrium. It is assumed that external forces F do not exist, pressure p is equal to zero. The motion in zone III is supposed to be unobstructed.
Under these assumptions the equations (1)-(3) in the cylindrical coordinate system with the appropriate initial and boundary conditions in the area I can be written as follows The boundary conditions in the transition zone are written as follows:

Solution of the problem
To avoid the difficulties associated with the consideration of two-dimensional model (4)-(25) and to analyze the results, the averaging method [2,7], is applied to the system. Average value of function ( ) Let us write the averaged system in region I, using previous designations:     decreases. This fact, for example, indicates the properties of composite products. Velocity, similarly to the stress, decreases from the boundary surface to the hole ( fig. 3), the distribution is almost linear. The material becomes more compacted and the velocity values decrease. This effect is accorded with the results obtained in [1,3,5]. Due to the changes in the geometric shape and the material density in the region II the flow of the material is slowed down substantially the velocity is equal to the slip velocity at the walls.
which coincides with the results obtained, for example, in [8]. This polynomial can be used to specify the functional dependence of the slip velocity from the velocity on the boundary layer of the material.

Conclusions
In the research the mathematical model of the composite material flow in cylindrical channel with variable cross section is presented. The form of the boundary conditions is suggested, the numerical experiment was carried out. The dependence estimate of the slip velocity on the average velocity of the material at the entrance is obtained.