Accounting Viscous Dissipation at Round Tubes Filling

2016 ◽  
Vol 685 ◽  
pp. 191-194
Author(s):  
E.I. Borzenko ◽  
O.Yu. Frolov ◽  
G.R. Shrager
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Dissipative Heating ◽  
Mathematical Basis ◽  
One Dimensional ◽  
Parametric Studies ◽  
Initial And Boundary Conditions ◽  
Formulation Of Problems

The fountain nonisothermal flow of a viscous fluid realized during circular pipe filling is investigated. The mathematical basis of the process is formed by equations of motion, continuity and energy with respective initial and boundary conditions with due account of the temperature dependence of viscosity, the presence of a free boundary and dissipation of mechanical energy. To solve the problem numerically a finite difference method is required. Depending on the values defining the dimensionless parameters the results of parametric studies in temperature, viscosity, dynamic and kinematic characteristics of the flow are shown. Flow patterns for the formulation of problems with different initial and boundary conditions are given. The separation of flow into the zone of spatial flow in the vicinity of the free surface and one dimensional flow away from it, and changing the shape of the free boundary, depending on the level of dissipative heating are demonstrated.

A Galerkin-Type Method to Solve One-Dimensional Telegraph Equation Using Collocation Points in Initial and Boundary Conditions

2018 ◽  
Vol 15 (05) ◽  
pp. 1850031 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Murat Karaçayır
Keyword(s):  
Approximate Solution ◽  
Boundary Conditions ◽  
Telegraph Equation ◽  
Inner Product ◽  
Present Scheme ◽  
One Dimensional ◽  
Collocation Points ◽  
Correction Technique ◽  
Initial And Boundary Conditions ◽  
Residual Correction

In this study, a Galerkin-type approach is presented in order to numerically solve one-dimensional hyperbolic telegraph equation. The method includes taking inner product of a set of bivariate monomials with a vector obtained from the equation in question. The initial and boundary conditions are also taken into account by a suitable utilization of collocation points. The resulting linear system is then solved, yielding a bivariate polynomial as the approximate solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution, is discussed briefly. The method and the residual correction technique are illustrated with four examples. Lastly, the results obtained from the present scheme are compared with other methods present in the literature.

Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain

2019 ◽  
Vol 7 (2B) ◽  
Author(s):  
Fernanda Tumelero ◽  
Celso M. F. Lapa ◽  
Bardo E. J Bodmann ◽  
Marco T. Vilhena
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Decomposition Method ◽  
Equation System ◽  
Delayed Neutron ◽  
Source Terms ◽  
One Dimensional ◽  
Homogeneous Domain ◽  
Initial And Boundary Conditions ◽  
Kinetic Diffusion

In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution.

Nonlinear Shell Dynamics—Intrinsic and Semi-Intrinsic Approaches

1983 ◽  
Vol 50 (3) ◽  
pp. 531-536 ◽  
Author(s):  
A. Libai
Keyword(s):  
Nonlinear Dynamics ◽  
Boundary Conditions ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Field Equations ◽  
Nonlinear Shell ◽  
Initial And Boundary Conditions ◽  
Shell Dynamics

The intrinsic approach to the nonlinear dynamics of shells, which was introduced in [6], is reviewed and extended by the addition of appropriate initial and boundary conditions of the dynamic and kinematic types to the field equations. The alternative semi-intrinsic velocity approaches (where the velocity components supply the connection between the equations of motion and the time rates of the metric and curvature) are also presented. Both linear and rotational velocity forms are included. The relative merits of these approaches to shell dynamics are discussed and compared with extrinsic approaches.

scholarly journals On the Implementation of Stream-Wise Periodic Boundary Conditions

2005 ◽  
Author(s):  
Steven B. Beale
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Equations Of Motion ◽  
Periodic Boundary Conditions ◽  
Constant Heat Flux ◽  
State Variables ◽  
Constant Heat ◽  
Mathematical Basis ◽  
Constant Wall Temperature ◽  
Primitive Variables

Fully-developed periodic boundary conditions have frequently been employed to perform calculations on the performance of typical elements of heat exchangers. Many such calculations have been achieved by transforming the equations of motion to obtain a new set of state variables which are cyclic in the stream-wise direction. In others, primitive variables, based on substitution schemes are employed. In this paper; a review of existing procedures is provided, and a new method is proposed. The method is based on the use of primitive variables with periodic boundary conditions combined with the use of slip values. Either pressure difference or mass flow rate may be prescribed, and both constant wall temperature and constant heat flux wall conditions may be considered. The example of an offset-fin plate-fin heat exchanger is used to illustrate the application of the procedure. The scope and limitations of the method are discussed in detail, and the mathematical basis by which the method may be extended to the consideration of problems involving mass transfer, with associated continuity, momentum, and species source/sinks is proposed.

scholarly journals Analytical and Experimental Vibration of Sandwich Beams Having Various Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Eshagh F. Joubaneh ◽  
Oumar R. Barry ◽  
Hesham E. Tanbour
Keyword(s):  
Finite Element Method ◽  
Boundary Conditions ◽  
Sandwich Panel ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Sandwich Beams ◽  
Various Boundary Conditions ◽  
Parametric Studies ◽  
Generalized Differential ◽  
Equations Of Motions

Generalized differential quadrature (GDQ) method is used to analyze the vibration of sandwich beams with different boundary conditions. The equations of motion of the sandwich beam are derived using higher-order sandwich panel theory (HSAPT). Seven partial differential equations of motions are obtained through the use of Hamilton’s principle. The GDQ method is utilized to solve the equations of motion. Experiments are conducted to validate the proposed theory. The results from the analytical model are also compared to those from the literature and finite element method (FEM). Parametric studies are conducted to investigate the effects of different parameters on the natural frequency and response of the sandwich beam under various boundary conditions.

On the statics and dynamics of granular-microstructured rods with higher order effects

2021 ◽  
pp. 108128652110099
Author(s):  
Nima Nejadsadeghi ◽  
Anil Misra
Keyword(s):  
Boundary Conditions ◽  
Mass Density ◽  
Equations Of Motion ◽  
Strong Dependence ◽  
Continuum Theory ◽  
Static Problem ◽  
Order Effects ◽  
One Dimensional ◽  
Second Gradient ◽  
Dynamic Length

Granular-microstructured rods show strong dependence of grain-scale interactions in their mechanical behavior, and therefore, their proper description requires theories beyond the classical theory of continuum mechanics. Recently, the authors have derived a micromorphic continuum theory of degree n based upon the granular micromechanics approach (GMA). Here, the GMA is further specialized for a one-dimensional material with granular microstructure that can be described as a micromorphic medium of degree 1. To this end, the constitutive relationships, governing equations of motion and variationally consistent boundary conditions are derived. Furthermore, the static and dynamic length scales are linked to the second-gradient stiffness and micro-scale mass density distribution, respectively. The behavior of a one-dimensional granular structure for different boundary conditions is studied in both static and dynamic problems. The effects of material constants and the size effects on the response of the material are also investigated through parametric studies. In the static problem, the size-dependency of the system is observed in the width of the emergent boundary layers for certain imposed boundary conditions. In the dynamic problem, microstructural effects are always present and are manifested as deviations in the natural frequencies of the system from their classical counterparts.

Axial Impact of Twisted Wire Cables

1977 ◽  
Vol 44 (1) ◽  
pp. 127-131 ◽  
Author(s):  
J. W. Phillips ◽  
G. A. Costello
Keyword(s):  
Boundary Conditions ◽  
Finite Length ◽  
Equations Of Motion ◽  
Laplace Transforms ◽  
Numerical Results ◽  
Longitudinal Impact ◽  
Axial Impact ◽  
Coupled Equations ◽  
Initial And Boundary Conditions ◽  
Twisted Wire

The nonlinear, coupled equations of motion governing the axial and rotational displacements of a straight, single lay, twisted wire cable are presented. Linearization of the equations of motion allows a solution by Laplace transforms which is valid for arbitrary initial and boundary conditions. The longitudinal impact of a finite-length cable fixed at one end is considered in detail, and numerical results for this case are presented.

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