Mathematical modeling of film flow of a liquid on a surface of a body of a rotation ; Математическое моделирование пленочных течений жидкости по поверхности тела вращения ; Математичне моделювання плівкових течій рідини по поверхні тіла обертання

The problem of the spatial nonwave stationary flow of the viscoplastic fluid on the surface of the body of rotation under the action of gravity 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 dis-tance along the normal to the surface. To describe the flow of a liquid film, a viscous in-compressible 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. For the closure of a system of differential equations, the Schulman rheological model is used, which is a gener-alization of the Ostwald-de-Ville power model and the Shvedov-Bingham viscoplastic model. 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 according to the proposed method for the circular cone located at a certain angle to the vertical are presented. ...