Constant Circular Cross Section Research Articles

A Three-Dimensional Velocity Field Related to a Generalized Third-Grade Fluid Model

In this work, we propose a new three-dimensional constitutive equation related to a third-grade fluid. This proposal is based on experimental work for which the viscosity term and the terms related to viscoelasticity may depend on the shear rate, in accordance with a power-law type model. The numerical implementation of this fluid model is rather demanding in terms of computational calculation and, in this sense, we use the Cosserat theory related to fluid dynamics, which makes the transition from the three-dimensional fluid model to a one-dimensional fluid model for a specific geometry under study which, in this case, is a straight tube with constant circular cross-section. Based on this approximation theory, the one-dimensional fluid model is solved by assuming an ordinary differential equation involving: an unsteady mean pressure gradient; an unsteady volume flow rate; the Womersley number; and the viscosity and viscoelasticity parameters. Consequently, for specific data, and using the Runge–Kutta method, we can obtain the solution for the unsteady volume flow rate and we can present simulations to the three-dimensional velocity field.

One-dimensional model for the unsteady flow of a generalized third-grade viscoelastic fluid

Specific algorithms for non-Newtonian fluids flow depend on the governing constitutive equation. In this work, we present a constitutive equation for a third-grade fluid in which a specific normal stress coefficient depends on the shear rate. This new three-dimensional model is suitable for studies where phenomena like shear-thinning or shear-thickening occur. Using a function of power-law type, we apply the Cosserat theory to fluid dynamics, reducing the exact three-dimensional equations to a one-dimensional system depending only on time and on a single spatial variable. From this reduced system, we solve the unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, Womersley number, viscoelastic coefficients, and flow index over a finite section of the tube geometry with constant circular cross section. Attention is focused on numerical simulations of unsteady flows regimes by using a Runge–Kutta method.

A Spectral Approach to Peak Velocity Estimation of Pipe Flows from Noisy Image Sequences

Abstract Motivated by plane wave ultrasound image velocimetry (a.k.a. Echo PIV), we introduce a novel method to globally estimate the velocity field of a laminar and steady flow from the motion of tracer particles. Our approach exploits the fact that the equation of motion for pipe flows of constant circular cross-section is governed by a single global but unknown parameter. We connect this parameter to the Fourier spectrum of the input image sequence of the tracer particles and formulate our motion estimation problem as a spectral support estimation problem. The approach is validated on both simulated and experimental real data. It returns accurate velocity estimates after few seconds runtime and effectively copes with speckle patterns and noise.

On solitary wave diffraction by multiple, in-line vertical cylinders

The interaction of solitary waves with multiple, in-line vertical cylinders is investigated. The fixed cylinders are of constant circular cross section and extend from the seafloor to the free surface. In general, there are N of them lined in a row parallel to the incoming wave direction. Both the nonlinear, generalized Boussinesq and the Green–Naghdi shallow-water wave equations are used. A boundary-fitted curvilinear coordinate system is employed to facilitate the use of the finite-difference method on curved boundaries. The governing equations and boundary conditions are transformed from the physical plane onto the computational plane. These equations are then solved in time on the computational plane that contains a uniform grid and by use of the successive over-relaxation method and a second-order finite-difference method to determine the horizontal force and overturning moment on the cylinders. Resulting solitary wave forces from the nonlinear Green–Naghdi and the Boussinesq equations are presented, and the forces are compared with the experimental data when available.

Single-axis equilibrium orientations to the attracting center of a symmetrical prolate orbital gyrostat with an elastic rod

Under study in the restricted formulation is the motion of a symmetrical prolate stationary gyrostat along a Keplerian circular orbit in a central Newtonian field of forces. An elastic homogeneous rod, rectilinear in the undeformed state, is rigidly clamped by one end in the body of gyrostat along its axis of symmetry. There is a point mass at the free end of the rod. The inextensible elastic rod, for simplicity of constant circular cross-section, performs infinitesimal space oscillations in the process of system motion. In this case, we neglect the terms in the system’s tensor of inertia which are nonlinear with respect to displacements of the points of the rod.We consider the following (so-called semi-inverse) problem: Under what kinetic momentumof the flywheel, among the relative equilibriums of the system (the states of rest relative to the orbital coordinate system) does there exist an equilibrium such that the axis, arbitrarily chosen in the coordinate system associated with the gyrostat, is collinear with the local vertical? In the discretization of the problem, we present the values of the Lagrange coordinates that define the deformation of the rod for these equilibria and the value of gyrostatic moment providing the presence of the equilibrium in question.

Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection

This work is concerned with the flow of a Newtonian fluid through a pipe of constant circular cross section which is uniformly porous. The governing unsteady equations are solved analytically after converting them to a third order ordinary differential equation using similarity transformation method. Expressions for axial and radial velocity components and pressure distribution are obtained. The characteristics of complex axial velocity and complex radial velocity for different values of parameters are analysed. The effects of small suction and small injection are delineated through graphs. Results reveal that suction or injection has significant influence on the flow.

Cable equation for general geometry.

The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.

Asymptotic Analysis of the Curved-Pipe Flow with a Pressure-Dependent Viscosity Satisfying Barus Law

Curved-pipe flows have been the subject of many theoretical investigations due to their importance in various applications. The goal of this paper is to study the flow of incompressible fluid with a pressure-dependent viscosity through a curved pipe with an arbitrary central curve and constant circular cross section. The viscosity-pressure dependence is described by the well-known Barus law extensively used by the engineers. We introduce the small parameterε(representing the ratio of the pipe’s thickness and its length) into the problem and perform asymptotic analysis with respect toε. The main idea is to rewrite the governing problem using the appropriate transformation and then to compute the asymptotic solution using curvilinear coordinates and two-scale asymptotic expansion. Applying the inverse transformation, we derive the asymptotic approximation of the flow clearly showing the influence of pipe’s distortion and viscosity-pressure dependence on the effective flow.

Estimation of Small Amplitude Pressure Wave in Gaseous Pipeline Using a Linear Kalman Filter

Measurement of pressure distribution in gaseous pipeline is important from a view point of control or safe management. In the present paper, a linear Kalman filter is developed to estimate the pressure estimation of small amplitude pressure wave in a single straight gaseous pipeline with constant circular cross section. The Kalman filter uses the linear dynamic system of the pipeline derived by using the approximation method of the unsteady friction, high speed and accurate computing method. The inputs to the Kalman filter are inlet and outlet pressures, and the pressure at the point one third of the pipe entrance is used as the observation point for feedback compensation. The performance of the developed method is tested using two experimental results by comparing with calculation results without feedback compensation. The comparison demonstrated that the developed method has better performance in the presences of measurement noise and erroneous boundary condition than does the ordinary calculation. It also showed much higher ability in error reduction than the ordinary calculation in computation with an erroneous initial condition.

On the measurement of surface tension and contact angle

AbstractWe propose a new method which combines experiments with numerical calculations to determine the capillary constant and the capillary contact angle. We measure the ascent of the liquid at the interior and at the exterior of a circular transparent tube. In contrast to known methods we do not assume that the contact angle is known. Assuming the interior contact angle is equal to the exterior angle, we find the capillary constant of the given liquid from the fact that the surface tensions of the interior and the exterior surfaces are equal. Dipping a cylindrical wire with constant circular cross section into a liquid with known surface tension we can determine the contact angle between the given liquid and the material from which the wire is made.

Quasi-static analysis of a ferrofluid blob in a capillary tube

Ferrofluids have promising application potentials for biological, medical, subsurface, and many other industrial purposes. To bring the potentials to reality, it is of utmost importance to characterize the behavior of ferrofluids under different conditions, especially in the presence of more than one phase. In this study, the quasi-static behavior of a non-wetting incompressible and inviscid ferrofluid blob surrounded by a wetting non-magnetic fluid confined in a capillary tube is theoretically and computationally investigated when a uniform magnetic field is applied, assuming isothermal conditions. The effect of geometrical, hydrodynamic, and magnetic properties of the blob on its deformations when subject to a magnetic field is explained. Moreover, the effect of nonlinear magnetization on the ferrofluid blob evolution in the capillary tube is investigated in detail. In the case of a tube with constant circular cross-section, the behavior of the blob before the critical state of detachment is determined numerically, while the post-critical behavior is resolved analytically. To characterize the pre-critical blob deformations, the magnetic field distribution inside the blob for given external magnetic fields is calculated using a commercial finite element software and is employed to calculate the interfacial configuration from balances among magnetic, capillary, and hydrostatic forces. We show that increasing magnetic field intensity above a critical value causes the blob to detach from the tube wall under certain conditions and quantitatively characterize the critical magnetic field as well as blob configurations before and after detachment. Results show that there is a maximum blob length beyond which detachment does not occur, due to the nonlinear magnetization of the ferrofluid blob. This length depends on the relative magnitude of magnetic and capillary forces and also on the geometry of the confining capillary tube. Even if detachment occurs, the nonlinear magnetization prevents the blob from evolving infinitely after detachment. The conditions under which detachment may occur are analytically determined. The simulations confirm that, for sufficiently small magnetic fields, the linear approximate magnetization yields satisfactory results. However, with increasing magnetic field intensity, the deviation between the results with nonlinear magnetization and those with linear approximate magnetization increases significantly. In addition, this deviation is more pronounced for longer blobs. These findings emphasize the importance of incorporating the nonlinear magnetization for relatively large magnetic field intensities. Finally, in order to relax the simple confining geometry assumption, we show preliminary simulations using the level set method in complex solid geometries. The method was previously developed for capillarity and used for realistic rock geometries and now accounts for the magnetic pressures as well.

Laminar Flow in Helical Coils: A Parametric Study

In the present study, experimental observations on pressure drop measurements for fully developed, incompressible Newtonian fluid flowing through helical coils of constant circular cross section under laminar flow conditions are reported. To elucidate the effect of coil pitch and coil diameter on the friction factor, 5 types of coils with 12 different combinations of varying radii, such that 8.3 ≤ p/dt ≤ 66.7 and 11.7 ≤ Dc/dt ≤ 105.48, have been used. The experimental observations indicated that the Germano number (NGn) successfully signifies the combined effect of various coil parameters on the pressure drop. Hence, the data corresponding to low and high Germano numbers (NGn ≤ 70 and NGn > 70) have been treated separately, to yield suitable correlations for the laminar flow region. The correlations developed explain the present experimental data to within ±10% (in the range of 1 < NGn < 130) and the data available in the literature to within ±15%.

Numerical Investigation of Coil Configurations That Provide Ultrahigh Packing Density of Saccular Aneurysms

BACKGROUND: The long-term outcome of endovascular coiling of cerebral aneurysms is directly related to the packing density at time of treatment. In general, the highest packing density achievable is only about 45% due to the quasi-random distribution of currently available coils within aneurysms. We investigated whether packing densities could be maximized via more ordered coil configurations. METHODS: Three different coil configurations - circular loops, planar spirals, and spherical helices - were investigated. The packing densities achievable in maximally filling the volume of an exemplar human basilar aneurysm with each coil configuration were calculated numerically. Coil packing simulations were also carried out for aneurysms idealized as spheres over diameters ranging from 2 to 30 mm. RESULTS: The packing densities with the loop, spiral, and spherical helix configurations were 82%, 60%, and 73%, respectively, for the human aneurysm model; the numbers of coils required were 693, 34, and 13, respectively. Simulations within idealized aneurysms suggest that aneurysms cannot be packed to more than 91% with coils of constant circular cross-section. The spherical helix configuration provides a constant packing density (coefficient of variation 0.4%) over the range of aneurysm diameters studied as compared to the two other configurations (coefficients of variation 9% and 8%). CONCLUSIONS: Coil configurations that allow for ordered filling of cerebral aneurysms can potentially provide packing densities that are twice those currently achieved. The spherical helix configuration seems to be the most technically feasible and stable configuration of the three coil types investigated.

In-Plane Load-Deflection Behavior and Buckling of Pressurized Fabric Arches

This study focuses on the in-plane load-deflection and buckling response of pressurized fabric arches of constant circular cross section. These lightweight portable structural members support tents used for temporary medical facilities, disaster-relief shelters, and military applications. A quadratic Timoshenko beam element is developed for materially and geometrically nonlinear analysis based on a virtual work principle that includes work done by internal pressure due to deformation-induced volume changes. The inability of the arch fabric to carry compressive stress is addressed with a nonlinear moment-curvature relationship that captures the effect of fabric tensile stiffness, internal pressure, and axial load. A corotational formulation and cylindrical arc-length solver are used to permit large displacement analysis and the tracking of postbuckling softening response. A convergence study demonstrates the good performance of the element and the solver. Results of laboratory load tests on a 9.5 m-span semicircular inflated fabric arch are presented, and shown to agree well with the model predictions. Parametric studies are conducted to examine the significance of inflation pressure and lateral restraint on arch deflections and buckling under the action of code-specified snow loads. The numerical studies show that the work done by the confined air significantly increases arch load capacity, and that shear deformations of woven fabrics can significantly reduce arch capacity.

Trapped modes in bent elastic rods

We investigate the existence of trapped modes in elastic rods of constant circular cross-section that possess bends of arbitrary curvature and straighten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. An asymptotic model assuming smallness of dimensionless curvature is developed to describe the trapping. Existence conditions depending on Poisson’s ratio are offered, and the equations from which they derive are numerically validated. A physical explanation of why trapped modes should be expected is also given.

Average temperature of the rotor of high-speed rotating heat exchanger during the period of thermal start-up

The paper presents results of the calculations of the average temperature variations of model rotor of a high-speed rotating heat exchanger in the course of thermal start up. A model rotor adopted for the calculations has a porous structure formed by radially oriented ducts of a constant circular cross-section. The scope of this paper covers numerical calculations of temperature distribution in the material of the rotor versus its rotational speed, temperature difference of the gases at the inlet, and types of the rotor material. With the use of these results, graphs are elaborated representing the dependence of the rotor’s dimensionless temperature on the Fourier number, with the Biot number used as a parameter.

Description of Quasi-plane Nonlinear Standing Waves in Cylindrical Resonators

Many works have been dedicated to the problems of description of the nonlinear standing waves in the resonators of the circular cross-section. These works are restricted only to strictly plane waves. This paper connects to these works and extends them in the sense that it deals with quasi-plane nonlinear standing waves in the resonators of constant circular cross-section. Under certain suppositions it is possible to simplify the original Kuznetsov’s model equation and get the new much simpler model equation. This equation enables us to study the diffraction and lateral dispersion effects caused by the non-constant distributions of acoustic velocity on the exciting piston.

EFFECT OF END HEAT EXCHANGER PARAMETERS ON THE PERFORMANCE OF A NATURAL CIRCULATION LOOP

The effect of end heat exchanger parameters on the performance of a single phase rectangular natural circulation loop with a constant circular cross section is reported in the present paper. Based on one-dimensional governing equations, the mass flow rate of the coupling fluid has been determined as a function of physical properties and geometrical parameters of the loop. The relevant non-dimensional numbers for the loop performance have also been identified. The model developed has been used for parametric study to determine the circulation rate as a function of (Ntu*, Ntu max , CR h -CR c combinations and θ o ).

Periodic waveguide structures with long cells

In this paper, a class of periodic waveguide structures with stepwise constant circular cross section is analyzed by means of Hahn's method of modal field matching along transverse planes. It allows one to resolve algebraically the implicit determinant relation between the phase shift per cell and frequency and to obtain an explicit formula. Dispersion diagrams and field distributions are also calculated and plotted and convergence properties of the method are tested.

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part II: Experimental Analysis

In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the λ parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity es was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C1, C and C2) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.