Open AccessModeling the gravitational flows of a viscoplastic fluid on a conical surface
Author(s) -
I. S. Tonkoshkur,
K. V. Kalinichenko
Publication year - 2019
Publication title -
pitannâ prikladnoï matematiki ì matematičnogo modelûvannâ
Language(s) - English
Resource type - Journals
ISSN - 2074-5893
DOI - 10.15421/321917
Subject(s) - curvilinear coordinates , generatrix , mechanics , surface of revolution , physics , streamlines, streaklines, and pathlines , classical mechanics , mathematical analysis , laminar flow , coordinate system , geometry , mathematics , conical surface , surface (topology)
рідка плівка,The problem of a stationary waveless gravitational flow of a viscoplastic fluid over the surface of a cone with an arbitrary smooth cross section is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the distance along the normal to the surface. To describe the flow of a liquid film, a viscous incompressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. To close the system of differential equations, the Shvedov-Bingham rheological model is used. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations by the proposed method for cones with a cross section in the form of a circle and a square with rounded corners are presented. Calculations show that the plasticity parameter and the cross-sectional shape significantly affect the velocity and distribution profiles of the thickness of the viscous layer over the surface of the body.