scholarly journals Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity

2021 ◽  
Vol 8 (2) ◽  
pp. 220-232
Author(s):  
Gennadi I. Mikhasev ◽  
Keyword(s):  
Differential Equation ◽  
Integral Form ◽  
Variable Cross Section ◽  
Nonlocal Theory ◽  
Variable Coefficients ◽  
Theory Of Elasticity ◽  
Variable Cross ◽  
Two Phase ◽  
Differential Model ◽  
Eigen Frequencies

Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed in the form of the superposition of a main solution and edge effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.

The Effect of Gas Evolution on Hydrodynamic Characteristics of Fluid Flow in Pipeline

2016 ◽  
Author(s):  
Geylani M. Panakhov ◽  
Eldar M. Abbasov ◽  
Sayavur I. Bakhtiyarov ◽  
Dennis A. Siginer
Keyword(s):  
Temperature Difference ◽  
Gas Flow ◽  
Variable Cross Section ◽  
Throughput Capacity ◽  
Hydrodynamic Characteristics ◽  
Variable Cross ◽  
Gas Generation ◽  
Two Phase ◽  
Maximum Value ◽  
Oil Gas

A relative motion of different phases leads to formation of certain forces at the interface of transported fluid and pipe walls. In the non-isothermal flow case a thermal interaction between the phases will affect the flow velocity, the pressure and the temperature distributions in variable cross section pipes. Laboratory experiments were conducted in order to study the effects of the gas generation at the pipe walls on the hydrodynamic characteristics of the two-phase oil/gas flow. It is shown that a throughput capacity of the pipe is affected by the temperature difference between the oil and the pipe walls. At certain temperature difference value (∼3°C) the pipe capacity reached a maximum value.

scholarly journals Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

2019 ◽  
Vol 41 (1) ◽  
pp. 1-12
Author(s):  
Józef Szybiński ◽  
Piotr Ruta
Keyword(s):  
Cross Section ◽  
Legendre Polynomials ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Beam Axis ◽  
Geometrical Parameters ◽  
Beam Model ◽  
Variable Cross ◽  
Thin Walled

AbstractThis article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

Application of the Three-Parameter Differential Model of Turbulence for Solving Problems of Flow and Heat Transfer in Channels of Variable Cross-Section. Part 2

2021 ◽  
Vol 12 (2) ◽  
pp. 89-106
Author(s):  
V. G. Lushchik ◽  
◽  
M. S. Makarova ◽  
A. I. Reshmin ◽  
◽  
...  
Keyword(s):  
Heat Transfer ◽  
Cross Section ◽  
Numerical Study ◽  
Narrow Channel ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Differential Model ◽  
Heat Transfer Intensification ◽  
Flow And Heat Transfer ◽  
Shear Turbulence

A description of the method of numerical study in the approximation of a narrow channel of the problems of flow and heat transfer in flat and circular channels of variable cross-section using a differential three-parameter model of shear turbulence is presented. The main results of numerous studies using the proposed method are described, one of the goals of which was to substantiate the possibility of using the narrow channel approximation. This review study is carried out in two parts. In the second part the results of the study of laminarization during flow in the con-fuser and the pipe, heat transfer intensification during flow in diffusers and in a plate heat exchanger with diffuser channels are presented.

scholarly journals Flexural Free Vibrations of Multistep Nonuniform Beams

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Guojin Tan ◽  
Wensheng Wang ◽  
Yubo Jiao
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Cross Section ◽  
Order Differential Equation ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
Exact Approach ◽  
Constant Coefficients ◽  
One Step

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as I(x)=α11+βxr+4 and m(x)=α21+βxr was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

scholarly journals Re-Reflection Effect On Shock Waves in Two-Phase Flows through a Tube of Variable Cross-Section

2018 ◽  
Vol 3 (12) ◽  
pp. 67-73
Author(s):  
Kanti Pandey
Keyword(s):  
Cross Section ◽  
Volume Fraction ◽  
Particle Volume Fraction ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Reflection Effect ◽  
Two Phase Flows ◽  
Particle Volume ◽  
Two Phase ◽  
Cross Sectional

In present paper Re-reflection  effect on  shock –waves in two-phase flows through a tube of variable cross-section is considered when particle  volume fraction appeared as an additional variable .It is concluded that re-reflected effects reduce the cross sectional  area .For two-phase flows when equilibrium is eventually established , presence of particle volume fraction , further reduce  the cross – sectional area. One dimensional area relation for a non – uniform , steady flow ahead of a shock   is obtained and concluded that  all the results are valid for the case   when  direction of the shock motion and the gas flow ahead of the  shock is same  .  In preparation of graphs Mathematica 7 is used .

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