Abstract
The problem of determining the forces of interaction of a viscous fluid with the cylindrical pipe wall is considered. It is assumed that near the pipe wall, the fluid motion is completely determined by viscous forces. The pipe moves along the streamline. The annular fluid element motion law is a special case of the Navier–Stokes equation in a cylindrical coordinate system. The equation is solved by the Fourier method in Bessel functions. Considering the orthogonality of the eigenfunctions, an equation for the squared norm is found. As an example, the case is considered when the pipe is subjected to vibration. Equations have been obtained for the velocities and viscous friction forces in the laminar sublayer. It has been found that when the pipe moves harmonically, the velocities and shear stresses at the pipe wall do not reach their maximum synchronously. The distribution of velocities and stresses in the section of the steel-pouring ladle gate channel has been considered for three vibration modes. The solution provided can be, in particular, used to determine the fluid–pipe wall interaction forces when the pipe is technologically affected by vibration, impulse, etc., as well as study moving joints such as piston, plunger, etc.
Highlights
In the steel casting, there is a problem of the skull formation in the gate channel
The paper provides a mathematical model of the axisymmetric fluid motion in a cylindrical pipe under the action of viscous forces
For the near-wall layer, a motion equation is obtained, the solution to which is sought in analytical form by the Fourier method
Summary
There is a problem of the skull formation in the gate channel. A way to reduce the skull formation rate is vibration action on the gate, which has earlier been established by studies performed on the physical model of the steel-pouring ladle gate [1]. In [11, 12], the dynamics of a viscous fluid under circular motions of a cylinder is studied using a numerical and analytical approach to solve the problem; the examples considered in the papers do not give a universal solution but are only particular case studies. Formulation of the problem Consider the liquid metal motion along the gate channel as a viscous liquid flow in a smooth cylindrical pipe. For the case of fluid flow in a cylindrical pipe considered, the boundary conditions have the form ( ) v R*,t = 0 ; v(R,t) = 0. For the convenience of solving the boundary value problem (1) with conditions (9), we rewrite equation (8) as follows.
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