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.

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 Modeling of heat and mass transfer processes in non-linearly viscous fluid film flows

2019 ◽  
Author(s):  
I. S. Tonkoshkur
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Film Thickness ◽  
Liquid Film ◽  
Heat And Mass Transfer ◽  
Boundary Layers ◽  
The Body ◽  
System Of Differential Equations ◽  
And Diffusion

The problem of heat and mass transfer in a liquid film of a nonlinearly viscous fluid flowing down the surface of a body of revolution under the influence of gravity is considered. The axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. It is assumed that the thermal and diffusion Prandtl numbers are large and the main changes in the temperature and diffusion fields occur in thin boundary layers near the solid wall and near the free surface separating the liquid and gas. A curvilinear orthogonal coordinate system (ξ, η, ζ) connected with the surface of the body is introduced. To describe the flow of a liquid film, a model of a viscous incompressible liquid is used, which is based on differential equations in partial derivatives - the equations of motion and continuity. As boundary conditions, the conditions of adhesion are used on the surface of a solid body, as well as the conditions of continuity of stresses and the normal component of the velocity vector - on the surface separating the liquid and gas. To simulate heat and mass transfer in a liquid film, the equations of thermal and diffusion boundary layers with boundary conditions of the first and second kind are used. To close the system of differential equations, the Ostwald-de-Ville rheological model is used. To simplify the system of differential equations, the small parameter method is used, in which the relative film thickness is selected. It is assumed that the generalized Reynolds number is of the order of unity. The solution of the equations of continuity and motion (taking into account the main terms of the expansion) is obtained in an analytical form. To determine the unknown film thickness, an initial-boundary-value problem is formulated for a first-order partial differential equation. The solution to this problem is found numerically using a running count difference scheme. To reduce the dimension of the problem for the equations of the boundary layer, the local similarity method is used. To integrate simplified equations, the finite-difference method is used.

scholarly journals The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

2021 ◽  
Vol 29 (2) ◽  
pp. 126-145
Author(s):  
Mohamed A. Bouatta ◽  
Sergey A. Vasilyev ◽  
Sergey I. Vinitsky
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Small Parameter ◽  
Asymptotic Method ◽  
Planck Equation ◽  
Singularly Perturbed ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Fokker Planck

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

scholarly journals Solving regularized equations in the form of polynomials

2020 ◽  
Vol 164 ◽  
pp. 02014
Author(s):  
Vera Petelina
Keyword(s):  
Differential Equations ◽  
Entire Functions ◽  
The Body ◽  
Body Motion ◽  
System Of Differential Equations ◽  
Rectangular Coordinates ◽  
Unified Algorithm ◽  
Special Points ◽  
The Right ◽  
Approximating Polynomials

The article is devoted to the determination of second-order perturbations in rectangular coordinates and components of the body motion to be under study. The main difficulty in solving this problem was the choice of a system of differential equations of perturbed motion, the coefficients of the projections of the perturbing acceleration are entire functions with respect to the independent regularizing variable. This circumstance allows constructing a unified algorithm for determining perturbations of the second and higher order in the form of finite polynomials with respect to some regularizing variables that are selected at each stage of approximation. The number of approximations is determined by the given accuracy. It is rigorously proven that the introduction of a new regularizing variable provides a representation of the right-hand sides of the system of differential equations of perturbed motion by finite polynomials. Special points are used to reduce the degree of approximating polynomials, as well as to choose regularizing variables.

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.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

scholarly journals Invariant measures for the flow of a first order partial differential equation

1985 ◽  
Vol 5 (3) ◽  
pp. 437-443 ◽  
Author(s):  
R. Rudnicki
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Invariant Measures ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order

AbstractWe prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

Exact Electroelastic Field of a Functionally Graded Piezoelectric Cantilever Beam Subjected to Pure Body Force Loading

2010 ◽  
Author(s):  
A. A. Emami ◽  
R. Hashemi ◽  
M. H. Kargarnovin ◽  
R. Naghdabadi
Keyword(s):  
Differential Equations ◽  
Body Force ◽  
Functionally Graded ◽  
The Body ◽  
Cantilever Beams ◽  
Exponential Distributions ◽  
Partial Differential ◽  
Numerical Studies ◽  
Piezoelectric Cantilever ◽  
Functionally Graded Piezoelectric

The electroelastic response of functionally graded piezoelectric cantilever beams which includes the effect of body force is presented in this paper. The material properties such as elastic compliance, piezoelectric and dielectric impermeability are assumed to be graded with different indices in the thickness direction according to exponential distributions. Systems of fourth order inhomogeneous partial differential equations (PDEs) which are satisfied by the stress and induction functions and involve the body force terms are derived. Spectral forms for electrical and mechanical variables in the x-axis are employed to convert the partial differential governing equations and the associated boundary conditions into sets of ordinary differential equations, and the resulting equations are solved in a closed form manner. Subsequently, in numerical studies, the effects of the material property graded indices are examined upon the electroelastic response of FGP cantilever beams under pure body force loadings.

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