Wave propagation analysis in buried pipe conveying fluid

A study of wave propagation in buried pipe conveying fluid is presented in the paper. The Flüggle shell model is adopted for pipe and surrounding solid is modeled as elastic matrix by using Winkle model. Wave dispersion curves of a buried vacant pipe and a pipe conveying fluid are obtained numerically by considering coupling conditions. Results show that wave velocity exhibits sharp drop points in dispersion curves, and remains to an identical values before and after the points for both of vacant pipe and pipe conveying fluid. Effects of wall thickness, elastic matrix properties and fluid velocity are also discussed.     

Strain gradient beam model for dynamics of microscale pipes conveying fluid

Based on the strain gradient theory, we present a microstructure-dependent Bernoulli–Euler model to analyze the vibration and stability of microscale pipes conveying fluid. The equation of motion and boundary conditions are derived using Hamilton’s principle. The proposed strain gradient beam model contains three material length scale parameters to capture the size effect. This new model may be reduced to the modified couple stress beam model when two of these three material length scale parameters vanish and may be reduced to the classical beam model in the absence of all the material length scale parameters. From the numerical calculations for micropipes with both ends positively supported, it is found that the natural frequency and the critical flow velocity are size-dependent. The results show that the microscale pipe displays remarkable size effect when its outside diameter becomes comparable to the material length scale parameter, while the size effect is almost diminishing as the diameter is far greater than the material length scale parameter. Moreover, the size effect predicted by the current strain gradient beam model is stronger than that predicted by the modified couple stress beam model, since two other material length scale parameters have been accounted for in the former.     

Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

Based on the theories of thermal elasticity mechanics and nonlocal elasticity, an elastic Bernoulli-Euler beam model is developed for thermal-mechanical vibration and buckling instability of a single-walled carbon nanotube (SWCNT) conveying fluid and resting on an elastic medium. The finite element method is adopted to obtain the numerical solutions to the model. The effects of temperature change, nonlocal parameter and elastic medium constant on the vibration frequency and buckling instability of SWCNT conveying fluid are investigated. It can be concluded that at low or room temperature, the fundamental natural frequency and critical flow velocity for the SWCNT increase as the temperature change increases, on the other hand, while at high temperature the fundamental natural frequency and critical flow velocity decrease as the temperature change increases. The fundamental natural frequency for the SWCNT decreases as the nonlocal parameter increases, both the fundamental natural frequency and critical flow velocity increase as elastic medium constant increases.     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

A differential quadrature analysis of unsteady open channel flow

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of unsteady open channel flow. To the best of authors’ knowledge, this is the first attempt to use the DQM in open channel hydraulics. The Saint-Venant equations and the related nonhomogenous, time dependent boundary conditions are discretized in spatial and temporal domain by DQ rules. The unknowns in the entire domain are computed by satisfying governing equations, boundary and initial conditions simultaneously. By employing DQM, accurate results can be obtained using dramatically less grid points in spatial and time domain. The stability of DQM solution is not sensitive to choosing time step or Courant number unlike other methods. Although numerical problems such as instability, oscillation and underestimation near critical depth can be seen by using other methods but DQM solution is smooth and accurate in this case. The results are sensitive to grid distribution in time domain. In light of this, Chebyshev–Gauss–Lobatto distribution performance is excellent. To validate the DQM solutions, the obtained results are compared with those of the characteristic method. In conclusion, DQM is a potential powerful method with minimum computational effort for unsteady flow simulation.     

Free vibration analysis of stepped beams by using Adomian decomposition method

The Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of a stepped Euler-Bernoulli beam consisting of two uniform sections. Each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions, step ratios and step locations are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.     

Application to differential transformation method for solving systems of differential equations

In this paper, we present an analytical solution for different systems of differential equations by using the differential transformation method. The convergence of this method has been discussed with some examples which are presented to show the ability of the method for linear and non-linear systems of differential equations. We begin by showing how the differential transformation method applies to a non-linear system of differential equations and give two examples to illustrate the sufficiency of the method for linear and non-linear stiff systems of differential equations. The results obtained are in good agreement with the exact solution and Runge–Kutta method. These results show that the technique introduced here is accurate and easy to apply.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

Shifted Legendre Polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model

The dynamic analysis of viscoelastic pipes conveying fluid is investigated by the variable fractional order model in this article. The nonlinear variable fractional order integral-differential equation is established by introducing the model into the governing equation. Then the Shifted Legendre Polynomials algorithm is first presented for dealing with this kind of equations. The convergence analysis and numerical example verify that the algorithm is an effective and accurate technique for addressing this type complicated equation. Numerical results for dynamic analysis of viscoelastic pipes conveying fluid show the effect of parameters on displacement, acceleration, strain and stress. It also indicates that how dynamic properties are affected by the variable fractional order and fluid velocity varying. Most of all, the proposed algorithm has enormous potentials for the problem of high precision dynamics under the variable fractional order model.     

Application of the differential transformation method to the free vibrations of strongly non-linear oscillators

This paper adopts the differential transformation method to obtain the free vibration behavior of an oscillator with fifth-order non-linearities. The principle of differential transformation is briefly introduced, and is then applied in the derivation of a set of difference equations for the free vibration oscillator problem. The solutions are subsequently solved by a process of inverse transformation. The time responses of the oscillator are presented under different parameter conditions, and the current results are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the proposed method. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the proposed differential transformation method is a powerful and efficient tool for solving non-linear problems.     

Integrodifferential approaches to frequency analysis and control design for compressible fluid flow in a pipeline element

In this study, modelling, frequency analysis, and optimization of control processes are considered for the fluid flow in pipeline systems. A mathematical model of controlled pipeline elements with distributed parameters is proposed to describe the dynamical behaviour of compressible fluid which is transported in a long rigid tube. By exploiting specific functions representing cross-sectional forces and effective displacements as well as linear approximations of fluidic resistances, the original problem with non-uniform parameters is reduced to a partial differential equation (PDE) system with constant coefficients and homogeneous initial and boundary conditions. Three numerical approaches are applied to an efficient analysis of natural vibrations and reliable control-oriented modelling of pipeline elements. The conventional Galerkin method is compared with the method of integrodifferential relations based on a weak formulation of the constitutive laws. In the latter approach, the original initial-boundary value problem is reduced to the minimization of an error functional which provides explicit energy estimates of the solution quality. A novel projection approach is implemented on the basis of the Petrov–Galerkin method combined with the method of integrodifferential relations. This technique benefits from the advantages of the above-mentioned projection and variational approaches, namely sufficient numerical stability, a lower differential order, and an explicit quality estimation. Numerical optimization procedures, making use of a modified finite element technique, are proposed to obtain a feedforward control strategy for changing the pressure and mass flow inside the pipeline system to a desired operating state. At this given finite point of time, residual elastic oscillations inside the pipeline are minimized. Numerical results, obtained for ideal as well as viscous fluid models, are analysed and discussed.     

Free vibration and stability analysis of a multi-span pipe conveying fluid using exact and variational iteration methods combined with transfer matrix method

In this paper, a new formulation based on the variational iteration method (VIM) is applied to investigate the dynamic behavior and stability of a multi-span pipe conveying fluid. Transfer matrix method (TMM) is used to assemble the system of equations resulting from applying the boundary conditions. The natural frequencies of the pipe system are obtained for different flow velocities. Results from VIM are compared with those predicted by the exact solution method and also with published literature. The influence of the number of spans on the VIM convergence is investigated. Also, the effects induced by varying the value and location of an intermediate elastic support on the critical velocity and stability are studied. It is shown that using VIM yields highly accurate results that are in very well agreement with the exact solution. The main advantage of the VIM is that it successfully overcomes well-known computational difficulties that are usually encountered during complex root finding step maintaining high precision as well.     

Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins

In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.     

On the unbounded increase in vorticity and velocity circulation in stratified and homogenous fluid flows

We consider the Oberbeck-Boussinesq system without dissipation (ideal convection) in a horizontal layer and in a “barrel” with flat bottom and flat cover. It is shown that the velocity circulation along a fluid contour consisting of two fluid curves on the bottom and on the cover connected by two isothermic fluid curves can be calculated explicitly and is a linear function of time. The serre result stating that the azimuthal component of vorticity in rotationally symmetric ideal fluid flows between coaxial cylinders increases linearly is generalized to the case of stratified fluids. It is proved that all plane and axially symmetric isothermic flows in a layer or in a barrel are unstable with respect to nonisothermic perturbations and in the case of a homogenous fluid all axially symmetric flows between coaxial cylinders are unstable in the sense of Lyapunov with respect to perturbations of the azimuthal component of the velocity in any metric including the maximum of magnitude of vorticity. Translated fromMaternaticheskie Zametki, Vol. 68, No. 4, pp. 627–636, October, 2000.     

Performance of turbulence models for flows through rough pipes

The Reynolds-averaged Navier–Stokes (RANS) equations were solved along with turbulence models, namely k–ε, k–ω, Reynolds stress models (RSM), and filtered Navier–Stokes equations along with Large Eddy Simulation (LES) to study the fully-developed turbulent flows in circular pipes roughened by repeated square ribs with various spacings. Solutions of these flows were obtained using the commercial computational fluid dynamics (CFD) software Fluent. The numerical results were validated against experimental measurements and other numerical data published in literature. The performance of the turbulence models was compared and discussed. All the RANS models and LES model were observed to perform equally well in predicting the time-averaged flow statistics. However no instantaneous information can be obtained from the RANS results. Therefore, when a rough overview of the flow process in a pipe roughened by repeated ribs is needed, any one of the RANS models can be of value. On the other hand, the instantaneous as well as time-averaged flows could be studied with more insight using LES, albeit at a cost of CPU effort at least one order higher.     

Application of the differential transformation method to the solution of a damped system with high nonlinearity

This paper employs the differential transformation method to investigate the response of a damped system with high nonlinearity. The accuracy and validity of the proposed method are demonstrated through its application to the solution of a dynamic system with various values of power-form nonlinearity. The numerical results indicate that the amplitude decays more rapidly at higher values of nonlinearity, particularly at higher values of the damping coefficient. It is shown that the current results are in excellent agreement with those provided by the Runge–Kutta method. Therefore, the method presented in this study provides an effective scheme for determining the solutions of a damped vibration problem with high nonlinearity.     

Dynamical analysis of the transmission of seasonal diseases using the differential transformation method

The aim of this paper is to apply the differential transformation method (DTM) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models where the solutions exhibit periodic behavior due to the seasonal transmission rate. These models describe the dynamics of the different classes of the populations. Here the concept of DTM is introduced and then it is employed to derive a set of difference equations for this kind of epidemic models. The DTM is used here as an algorithm for approximating the solutions of the epidemic models in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the DTM is accurate, easy to apply and the calculated solutions preserve the properties of the continuous models, such as the periodic behavior. Furthermore, it is showed that the DTM avoids large computational work and symbolic computation.     

Accurate vibration analysis of skew plates by the new version of the differential quadrature method

An accurate free vibration analysis of skew plates is presented by using the new version of the differential quadrature method (DQM). Eight combinations of simply supported (S), clamped (C) and free (F) boundary conditions are considered. Detailed solution procedures are given and key points for success by using the DQM are emphasized. A way to simplifying the programming in using the DQM is proposed. Convergence study is made for the simply supported skew plate with a large skew angle. Good convergence of frequencies is observed. The DQ results agree very well with the existing first known accurate upper bound solutions, obtained by using Ritz method taking into considerations of the bending stress singularities occurred at corners having obtuse angles. Since slight discrepancy between the DQ data and the known accurate solutions is observed for plates with large skew angles, the DQ results are also compared with data obtained by using finite element method with very fine meshes to verify their accuracy.     

Spectral approximation for a nonlinear partial differential equation arising in thin film flow of a non-Newtonian fluid

Start-up thin film flow of fluids of grade three over a vertical longitudinally oscillating solid wall in a porous medium is investigated. The governing non-linear partial differential equation representing the momentum balance is solved by the Fourier-Galerkin approximation. The effect of the porosity, material constants as well as oscillations on the drainage rate and flow enhancement is explored and clarified.