scholarly journals MODELING THE INTERACTION OF A FILM OF A NONLINEAR VISCOUS LIQUID WITH A GAS FLOW

2021 ◽  
pp. 203-209
Author(s):  
Illia Tonkoshkur
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Viscous Liquid ◽  
Gas Flow ◽  
Flat Surface ◽  
Liquid Films ◽  
System Of Differential Equations ◽  
Layer Film ◽  
Gas Layer ◽  
Analytical Expressions

The problem of the interaction of a two-layer film of a nonlinear viscous liquid flowing down a flat surface with a gas flow directed vertically up or down is considered. To simplify the initial system of differential equations, the method of a small parameter is used, for which the relative thicknesses of the films and the gas layer were chosen. Analytical expressions are obtained for the profiles of the velocities and thicknesses of liquid films.

The nonwandering set of a special system of differential equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 35-48
Author(s):  
V. A. Pliss
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
System Of Differential Equations ◽  
Special System ◽  
Non Linear ◽  
Nonwandering Set

SynopsisIn the theory of non-linear oscillations there occur systems with a small parameter in the derivatives and discontinuous forcing terms. Here we study such a system.

scholarly journals Mathematical modeling of film flow of a liquid on a surface of a body of a rotation

2018 ◽  
Author(s):  
I. S. Tonkoshkur
Keyword(s):  
Differential Equations ◽  
Film Thickness ◽  
Small Parameter ◽  
Rheological Model ◽  
The Body ◽  
Viscoplastic Fluid ◽  
Parameter Method ◽  
Order Partial Differential Equation ◽  
System Of Differential Equations ◽  
Partial Differential

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. Calculations show that the parameters of nonlinearity and plasticity of this rheological model of a liquid can significantly affect the speed profiles and the distribution of the thickness of the viscous layer on the surface of the body

scholarly journals Periodic solutions of quasilinear non-autonomous systems with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 185-197 ◽  
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Small Parameter ◽  
Periodic Solutions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Image Position ◽  
The Given

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the formwhere x ∈ Rn, ε is a small parameter,Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

scholarly journals Modeling the gravitational flows of a viscoplastic fluid on a conical surface

2019 ◽  
Author(s):  
I. S. Tonkoshkur ◽  
K. V. Kalinichenko
Keyword(s):  
Differential Equations ◽  
Film Thickness ◽  
Small Parameter ◽  
Cross Section ◽  
The Body ◽  
Viscoplastic Fluid ◽  
Parameter Method ◽  
Order Partial Differential Equation ◽  
System Of Differential Equations ◽  
Partial Differential

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.

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