Dynamic Stability of a Fluid-Coveying Cantilevered Pipe on an Additional Combined Support

Based on an analytical study, a numerical analysis is made of the dynamic stability of a cantilevered steel pipe conveying a fluid. The pipe is modeled by a beam restrained at the left end and supported by a special device (a rotational elastic restraint plus a Q-apparatus) at the right end. The numerical analysis reveals that the critical velocity of the fluid depends on the governing parameters of the problem such as the ratio of the fluid mass to the pipe mass per unit length and the rotational elastic constant at the right end     

Transverse isotropic poroelastic osteon model under cyclic loading

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. This cylinder models an osteon, basic unit of cortical bone. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. Fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed as it could influence bone remodeling. Strain rate of loading is shown to play a significant role in mass flux in the porous material.     

Simultaneous extension, inflation, and shear of a cylindrical tube

Summary Rivlin's solution [1] of the title problem for Mooney materials is generalized by dropping the assumption that the hydrostatic pressure is a function of the radial distance only. Due to this generalization the normal tractions over the cylindrical surfaces are not constant as in the previous solution but vary linearly along the axis of the tube. Further, the longitudinal forces per unit length of the deformed tube, over the cylindrical surfaces, are no longer equal. Solutions for a solid cylinder are then deduced from the general solution. The effect of self-weight on the solutions is briefly dealt with.     

Force Action of a Body with a Permeable Boundary on a Wall in a Fluid

The ideal fluid flow due to fluid penetration through the boundary of an infinitely long solid cylinder in contact with a solid wall is determined. A formula is derived according to which the force exerted by a finite-length part of the cylinder on the wall is directed into the wall and can thus have an arbitrarily large absolute value. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 82–84, January–February, 2006.     

Plane waves in a heavy two-layer liquid excited by a cylinder moving at an angle to the horizontal

A solution of an initial-boundary-value problem for a system of integrodifferential equations which describes the plane waves excited in an initially stationary heavy two-layer ideal fluid by a cylinder moving at an angle to the horizontal is investigated. The homogeneous fluid fractions of different densities are assumed to be separated by an evolving fluid interface (horizontal plane, if the liquid is at rest). An approximate solution of two problems for the waves excited by a cylinder moving with a constant acceleration and an oscillating cylinder is constructed analytically. Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–152, July–August, 1998.     

Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid

We study infinitesimal deformations of a porous linear elastic body saturated with an inviscid fluid and subjected to conservative surface tractions. The gradient of the mass density of the solid phase is also taken as an independent kinematic variable and the corresponding higher-order stresses are considered. Balance laws and constitutive relations for finite deformations are reduced to those for infinitesimal deformations, and expressions for partial surface tractions acting on the solid and the fluid phases are derived. A boundary-value problem for a long hollow porous solid cylinder filled with an ideal fluid is solved, and the stability of the stressed reference configuration with respect to variations in the values of the coefficient coupling deformations of the two phases is investigated. An example of the problem studied is a cylindrical cavity leached out in salt formations for storing hydrocarbons.     

Asymmetric transient response of a hollow cylinder enclosing a compressible fluid

The asymmetric transient response of a hollow cylinder confining a compressible fluid is analyzed. The cylinder is excited by radial displacement prescribed over a rectangular footprint on the cylinder’s outer surface. The special case of plane-strain is also analyzed. A comparison of dilatational stress in the solid cylinder and fluid pressure in the fluid-filled cylinder reveals how a projectile may decelerate faster in the latter.     

Reduction in drag and vortex shedding frequency through porous sheath around a circular cylinder

A numerical study on the laminar vortex shedding and wake flow due to a porous‐wrapped solid circular cylinder has been made in this paper. The cylinder is horizontally placed, and is subjected to a uniform cross flow. The aim is to control the vortex shedding and drag force through a thin porous wrapper around a solid cylinder. The flow field is investigated for a wide range of Reynolds number in the laminar regime. The flow in the porous zone is governed by the Darcy–Brinkman–Forchheimer extended model and the Navier–Stokes equations in the fluid region. A control volume approach is adopted for computation of the governing equations along with a second‐order upwind scheme, which is used to discretize the convective terms inside the fluid region. The inclusion of a thin porous wrapper produces a significant reduction in drag and damps the oscillation compared with a solid cylinder. Dependence of Strouhal number and drag coefficient on porous layer thickness at different Reynolds number is analyzed. The dependence of Strouhal number and drag on the permeability of the medium is also examined. Copyright © 2009 John Wiley & Sons, Ltd.     

The EHD-driven fluid flow and deformation of a liquid jet by a transverse electric field

The aim of this study is to investigate the effect of a uniform transverse electric field on the steady-state behavior of a liquid cylinder surrounded by another liquid of infinite extent. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electrical and hydrodynamic stresses acting on the interface separating the two fluids is presented, and an expression is found for the interface deformation for small distortions from a circular shape. The electrical stresses acting on the interface of two leaky-dielectric liquids are compared with those acting on an interface separating a perfect dielectric or infinitely conducting core fluid cylinder from a surrounding perfect dielectric fluid. A comparison is made between the results of this study and those of a similar study for fluids with permeable interfaces and the classical results for liquid drops.     

A general approach for defining the macroscopic free energy density of saturated porous media at finite strains under non-isothermal conditions

A general approach is proposed for defining the macroscopic free energy density function (and its complement, the free enthalpy) of a saturated porous medium submitted to finite deformations under non-isothermal conditions, in the case of compressible fluid and solid constituents. Reference is made to an elementary volume treated as an ‘open system’, moving with the solid skeleton. The proposed free energy depends on the generalised strains (namely an appropriate measure of the strain of the solid skeleton and the variation in fluid mass content) and the absolute temperatures of the solid and fluid phases (which are assumed to differ from each other for the sake of generality). This macroscopic energy proves to be a potential for the generalised stresses (namely the associated measure of the total stress and the free enthalpy of the pore fluid per unit mass) and the entropies of the solid and fluid phases. In contrast with mixture theories, the resulting free energy is not the simple sum of the free energies of the single constituents. Two simplified cases are examined in detail, i.e. the semilinear theory (originally proposed for isothermal conditions and extended here to non-isothermal problems) and the linear theory. The proposed approach paves the way to the consistent non-isothermal-hyperelastic-plastic modelling of saturated porous media with a compressible fluid and solid constituents.     

Spin-up and spin-down dynamics of a liquid metal driven by a single rotating magnetic field pulse

This paper presents a study concerning the transient dynamics of the flow field inside a liquid metal filling a finite cylindrical container: The flow is created by applying a rotating magnetic field (RMF) in the form of a single pulse. The flow structure is governed by an impulsive spin-up from the rest state which is followed by a spin-down phase, with the fluid in a state of inertia. The pulse length has been found to have a distinct influence on the transient fluid flow. Two cases are considered: an enclosed cavity and a cavity with a free surface, in order to show that in both cases the recirculating flow in the radial-meridional plane displays periodical reversals. This phenomena is especially pronounced if the pulse length of the electromagnetic forcing corresponds to the so-called initial adjustment phase as defined by Nikrityuk, Ungarish, Eckert, Grundmann [P.A. Nikrityuk, M. Ungarish, K. Eckert, R. Grundmann, Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder. A numerical and analytical study, Phys. Fluids 17 (2005) 067101–0671016].     

Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

Cylindre transverse isotrope poroélastique chargé cycliquement : application à un ostéonCyclic loading of a transverse isotropic poroelastic cylinder: a model for the osteon

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. This cylinder is considered as a model for an osteon, the basic unit of cortical bone. The fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed, as this could influence bone remodeling. To cite this article: A. Rémond, S. Naili, C. R. Mecanique 332 (2004).     

Vortex and wake-induced vibrations of a tandem arrangement of two flexible circular cylinders with near wake interference

Results showing the dynamic response of a tandem arrangement of two vertical high aspect ratio (length over diameter) and low mass ratio (mass over mass of displaced fluid) flexible cylinders vibrating at low mode number are presented in this paper. Two circular cylinder models were aligned with the flow, so the downstream or trailing cylinder was immersed in the wake of the leading one. Centre-to-centre distances from 2 to 4 diameters were studied. The models were very similar in design, with external diameters of 16 mm and a total length of 1.5 m. Reynolds numbers up to 12 000 were achieved with reduced velocities, based on the fundamental natural frequency of the downstream cylinder in still water, up to 16. The trailing model had a mass ratio of 1.8 with a combined mass-damping parameter of 0.049, whilst the corresponding figures for the leading cylinder were 1.45 and 0.043, respectively. The dynamic response of the trailing model has been analysed by studying cross-flow and in-line amplitudes, dominant frequencies and modal amplitudes. The dynamic response of the leading one is analysed by means of its cross-flow amplitudes and dominant frequencies and it is also related to the motion of the trailing cylinder by studying the synchronisation between their instantaneous cross-flow motions. Planar digital particle image velocimetry (DPIV) was used to visualise the wake. Different response regimes have been identified based on the type of oscillations exhibited by the cylinders: vortex-induced (VIV), wake-induced (WIV) or combinations of both.     

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.     

Heat transfer in a second-order fluid for flow around a circular cylinder

The temperature distribution within the thermal boundary layer region due to the flow of an incompressible second-order fluid around a heated circular cylinder, maintained at a constant temperature higher than that of the fluid at infinity, has been obtained near the forward stagnation point by series expansion. The graph of the Nusselt number Nu for the Prandtl number P = 25 and the Eckert number E = 0.1 indicates that the non-Newtonian effect is to increase the heat flux from the cylinder to the liquid in the region 0° ? θ ? 15.7° and to decrease it in the region 15.7° < θ ? 0₀ where θ is the angular distance on the cylinder measured from the forward stagnation point. The critical point θ₀ at which Nu = 0, that is, where the effect of the frictional heating balances the effect of the temperature difference and there is no heat flux either from the cylinder or from the liquid, shifts towards the forward stagnation point with the increase of non-Newtonian effects.     

Nonisothermal steady flow of a viscoplastic liquid between two coaxial rotating cylinders

Rotational viscosimeters are widely used to determine liquid viscosity. The technique for processing the experimental data is based fundamentally on the analytic solution of the problem of isothermal flow of a viscous liquid between two rotating cylinders.If in the course of the experiment the heat released due to the internal friction leads to significant heating, then the processing of the experimental results using the equations obtained on the assumption of isothermocity of the flow may lead to large errors. The dissipative heating may be reduced by reducing the angular velocity of rotation of the cylinder; however extensive reduction of the angular velocity is not desirable, since this leads to an increase of the measurement relative error. In addition, there is the possibility of conducting the experiments with a wide variation of the angular velocities in order to identify the structural-rheological peculiarities of the liquid. In the latter case we must be able to separate the purely thermal effects from the influence of the rheological factors. All these questions are discussed in detail in [1]. The authors of [1] obtained the solution of the problem of nonisothermal flow of a Newtonian fluid between two rotating cylinders and gave a technique for processing the experimental data which takes account of the dissipative heating of the fluid. The present paper pursues the same objective for a visco-plastic fluid.An attempt to solve the problem of nonisothermal flow of a viscoplastic fluid was made by Dzhafarov in [2], where the problem was solved in two versions. In the first version it was considered that the viscosity varies hyperbolically with the temperature and the gap between the cylinders is small in comparison with the radius of the inner cylinder. As a result of the linearization of the equations of motion and heat balance, it turned out that in fact the problem of Couette flow of a viscoplastic fluid was solved rather than the original problem. In this case, naturally, such a characteristic of the flow of a viscoplastic fluid in an annular gap as the possibility of the formation of an elastic zone was not covered. In the second version the problem was solved under the assumption that the viscosity is independent of the temperature and the angular velocity is small.In the present study the problem is solved without the limitations discussed above on the angular velocity, the fluid visoosity, and the gap between the cylinders. In this case we consider two types of temperature boundary conditions: a) constant temperatures are specified on the surfaces of the cylinders, which in the general case may be different; and b) a constant temperature is given on the surface of the outer cylinder and the inner cylinder is thermally insulated.     

Analysis of Annular Liquid Membranes and Their Singularities

The singularities of the equations governing the fluid dynamics of steady, axisymmetric, annular liquid membranes subject to gravity are analyzed by means of two techniques based on the membranes's slope and curvature, and the membrane's mean radius, mass per unit length, and axial and radial velocity components, respectively. It is shown that no singularity is possible at or downstream from the nozzle exit for Weber numbers greater than unity because of the gravitational pull. For a Weber number equal to one, a singularity at the nozzle exit appears and the flow slope there is undetermined; however, the slope acquires a finite value if the liquid is assumed to leave the nozzle at angle different from that of the annular orifice. It is also shown that, for Weber numbers smaller than one, a singularity may occur downstream from the nozzle exit which may also be removed, and that the shapes of annular liquid membranes for Weber numbers equal to or less than one take a rounded form which is in agreement with experimental observations. An asymptotic analysis shows that, to leading order, the shapes of capillary, annular liquid membranes are arcs of circumferences, and this result is again in accord with available experimental findings.     

Diffusion toward a cylinder in the case of an arbitrary viscous flow. Diffusive boundary layer approximation

Steady convective diffusion of a dissolved substance toward the surface of a cylinder (optionally circular) in a viscous flow is examined. An analytical solution is obtained in [1, 2] for the case of laminar flow around a curved cylinder when the freestream flow is straight and uniform. More complex hydrodynamical problems are examined in [3, 4]. In the present work an approximate analytical expression is obtained for diffusive flow of a substance toward the surface of a solid cylinder in the case of an arbitrary two-dimensional flow. Formulas are given for calculating the mass transfer at a circular cylinder in some shear flows of a viscous, incompressible fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–166, September–October, 1976.The authors thank Yu. P. Gupalo and Yu. S. Ryazantseva for formulating the problem and their attention to the work.     

Internal resonance of pipes conveying fluid in the supercritical regime

This paper treats nonlinear vibration of pipes conveying fluid in the supercritical regime. If the flow speed is larger than the critical value, the straight equilibrium configuration becomes unstable and bifurcates into two possible curved equilibrium configurations. The paper focuses on the nonlinear vibration around each bifurcated equilibrium. The disturbance equation is derived from the governing equation, a nonlinear integro-partial-differential equation, via a coordinate transform. The Galerkin method is applied to truncate the disturbance equation into a two-degree-of-freedom gyroscopic systems with weak nonlinear perturbations. The internal resonance may occur under the certain condition of the supercritical flow speed for the suitable ratio of mass per unit length of pipe and that of fluid. The method of multiple scales is applied to obtain the relationship between the amplitudes in the two resonant modes. The time histories predicted by the analytical method are compared with the numerical ones and the comparisons validate the analytical results when the nonlinear terms are small.