Elastic Cylindrical Tube Research Articles

Viscous fluid-thin cylindrical elastic body interaction: asymptotic analysis on contrasting properties

ABSTRACTA periodic, axially symmetric time-dependent model of interaction between a viscous fluid and a thin cylindrical elastic tube is considered. The problem depends on a small parameter , representing the ratio of the thickness of the wall and the radius of the cylinder. We consider Young’s modulus of the elastic medium and its density great or small parameters equal to some powers of . An asymptotic expansion is constructed for various magnitudes of the rigidity and of the density of the elastic tube. The generality of the considered model requires to take into account six different combinations of the three parameters depending on . We obtain in this way different limit problems which are Stokes equations with some special boundary conditions. The expansion is justified by the high-order estimates for the difference of the exact solution and truncated asymptotic approximation.

The effect of damage on small-amplitude radial oscillations of an incompressible isotropic tube under pressure

Abstract This study investigates the small-amplitude radial oscillations of a damaged incompressible isotropic homogeneous limited elastic cylindrical Gent tube under constant inflation pressure. The proposed work focuses on incorporating damage into an infinitesimal stability analysis, mainly for a class of materials characterized by their limited molecular-chain extensibility. These materials mimic the responses of soft biological tissues. A stability analysis of a damaged radially oscillating cylindrical tube is necessary for studying angioplasties, surgical procedures used to correct some inflammation-based cardiac disorders. Stability theory is inferred both from the mathematical formulas presented and the graphical results obtained for pressure–stretch and frequency–stretch curves. Finally, a stability analysis for an incompressible damaged Gent cylindrical membrane is presented, and universal relations are obtained. Infinitesimal stability of a long, damaged cylindrical Gent cavity is further explored, and all results are illustrated graphically.

Slow rotation of a spherical particle inside an elastic tube

In this paper, we present an analytical calculation of the rotational mobility functions of a particle rotating on the centerline of an elastic cylindrical tube whose membrane exhibits resistance towards shearing and bending. We find that the correction to the particle rotational mobility about the cylinder axis depends solely on membrane shearing properties while both shearing and bending manifest themselves for the rotational mobility about an axis perpendicular to the cylinder axis. In the quasi-steady limit of vanishing frequency, the particle rotational mobility nearby a no-slip rigid cylinder is recovered only if the membrane possesses a non-vanishing resistance towards shearing. We further show that for the asymmetric rotation along the cylinder radial axis, a coupling between shearing and bending exists. Our analytical predictions are compared and validated with corresponding boundary integral simulations where a very good agreement is obtained.

A new analysis of stresses in arteries based on an Eulerian formulation of growth in tissues

The simple analysis of stresses in arteries considers the artery as an incompressible elastic circular cylindrical tube. When the intact artery is unloaded and cut transmurally, it springs open to a nearly circular cylindrical sector. The resulting nonzero opening angle indicates the existence of residual stresses in the unloaded intact artery. This paper presents a new analysis of stresses in arteries based on an Eulerian formulation of elastic deformations in soft tissues that models the inelastic process of remodeling (homeostasis) towards its homeostatic state. Within the context of this new analysis it is not necessary to model details of homeostasis. Instead, it is possible to directly determine the loaded homeostatic state, which is considered as the reference configuration of the artery, and to specify the geometry and associated values of the circumferential and axial stresses by matching measurements of the geometry of the unloaded open artery. The predicted geometry of the unloaded intact artery is shown to be accurate relative to the measurements. Also, predictions of the stress distributions for remodeling due to hypertension are presented.

Deformation profiles of elastic cylindrical tubes filled with granular media under an overload

The deformation of a thin-walled vertical tube, filled with a liquid or a cohesionless granular material is investigated theoretically and experimentally. Experiments with an overload and without it were made with latex tubes filled with water or spherical glass beads and the results were compared with the theoretical profile derived from the Janssen model. The results suggest that the soft elastic tubes could provide a simple and convenient means to investigate the forces that arise in different materials.

Evaluation of the Buckling Modes of a Rotating Neo-Hookean Cylinder by WKB Method

The deformation of a circular cylindrical elastic tube of finite wall thickness under rotation about its axis is considered. The angular speed (\(\omega\)) is analyzed as a function of an azimuthal deformation parameter at fixed axial extension for incompressible, isotropic, neo-Hookean elastic strain-energy function. Our aim is to investigate the bifurcation of the tube for the case of the prismatic bifurcation mode. We used the asymptotic WKB method for the solution of the eigenvalue problem. Although the dependency of \(\lambda _a\) (the ratio of the deformed inner radius to the un-deformed case) on the wall thickness (A/B) has a boundary layer structure, it depends on the mode numbers and the magnitude of the axial extension (\(\lambda _z\)). Finally, the eigenfunctions are obtained for \(\lambda _z=0.2,\,1,\,5\) and \(m=10,\,15,\,20\). We find that our derived asymptotic results are very similar to the counterpart numerical data.

Stability of inhomogeneous micropolar cylindrical tube subject to combined loads

In the present paper, the stability of a nonlinear elastic cylindrical tube made of micropolar material is analyzed. It is assumed that the elastic properties of the tube vary through the wall thickness. The problem is studied for the case of axial compression of the tube under internal and external hydrostatic pressure. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the tube. By solving these equations numerically the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative axial compression, relative internal or external pressure). Using these results, the influence of elastic properties, as well as the size of the tube, on the loss of stability is studied. Special attention has been paid to the analysis of how the pattern of change in elastic parameters affects the stability of a cylindrical tube made of micropolar material.

A study of self-propelled elastic cylindrical micro-swimmers using modeling and computation

We study propulsion of micro-swimmers in 3D creeping flow. The swimmers are assumed to be made of elastic cylindrical hollow tubes. The swimming is generated by the contractions of the tube's elastic membrane walls producing a traveling wave in the form of a “step-function” traversing the swimmer from right to left, propelling the swimmer from left to right. The problem is motivated by medical applications such as drug delivery. The influence of several non-dimensional design parameters on the velocity of the swimmer is investigated, including the swimmer aspect ratio, and the amplitude of the traveling wave relative to the swimmer radius. An immersed boundary method based on a finite element method approach is successfully combined with an elastic spring network model to simulate the two-way fluid–structure interaction coupling between the elastic cylindrical tube and the flow of a 3D viscous, incompressible fluid. To gain a deeper insight into the influence of various parameters on the swimmer speed, a reduced 1D fluid–structure interaction model was derived and validated. It was found that fast swimmers are those with large tube aspect ratios, and with the amplitude of the traveling wave which is roughly 50% of the reference swimmer radius. It was shown that the speed of our “optimal swimmer” is around 1.5 swimmer lengths per second, which is at the top of the class of all currently manufactured micro-swimmers swimming in low Reynolds number flows ( Re = 10 − 6 ), reported in [11] .

Time-fractional effect on pressure waves propagating through a fluid filled circular long elastic tube

The pressure waves propagating through an incompressible inviscid fluid that moves in a circular cylindrical long elastic tube are considered. The reductive perturbation method is used to derive the KdV equation from the hydrodynamic equations of the system. The Euler–Lagrange variational technique described by Agrawal has been applied to formulate the time-fractional KdV equation. The derived time-fractional KdV equation is solved by employing the variational-iteration method represented by He. The effects of the tube and fluid parameters and the time fractional order on the propagation of pressure waves are investigated.

Navier–Stokes flow in converging–diverging distensible tubes

We use a method based on the lubrication approximation in conjunction with a residual-based mass-continuity iterative solution scheme to compute the flow rate and pressure field in distensible converging–diverging tubes for Navier–Stokes fluids. We employ an analytical formula derived from a one-dimensional version of the Navier–Stokes equations to describe the underlying flow model that provides the residual function. This formula correlates the flow rate to the boundary pressures in straight cylindrical elastic tubes with constant-radius. We validate our findings by the convergence toward a final solution with fine discretization as well as by comparison to the Poiseuille-type flow in its convergence toward analytic solutions found earlier in rigid converging–diverging tubes. We also tested the method on limiting special cases of cylindrical elastic tubes with constant-radius where the numerical solutions converged to the expected analytical solutions. The distensible model has also been endorsed by its convergence toward the rigid Poiseuille-type model with increasing the tube wall stiffness. Lubrication-based one-dimensional finite element method was also used for verification. In this investigation five converging–diverging geometries are used for demonstration, validation and as prototypes for modeling converging–diverging geometries in general.

A thick-walled fluid-solid-growth model of abdominal aortic aneurysm evolution: application to a patient-specific geometry.

We propose a novel thick-walled fluid-solid-growth (FSG) computational framework for modeling vascular disease evolution. The arterial wall is modeled as a thick-walled nonlinearly elastic cylindrical tube consisting of two layers corresponding to the media-intima and adventitia, where each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component. Blood is modeled as a Newtonian fluid with constant density and viscosity; no slip and no-flux conditions are applied at the arterial wall. Disease progression is simulated by growth and remodeling (G&R) of the load bearing constituents of the wall. Adaptions of the natural reference configurations and mass densities of constituents are driven by deviations of mechanical stimuli from homeostatic levels. We apply the novel framework to model abdominal aortic aneurysm (AAA) evolution. Elastin degradation is initially prescribed to create a perturbation to the geometry which results in a local decrease in wall shear stress (WSS). Subsequent degradation of elastin is driven by low WSS and an aneurysm evolves as the elastin degrades and the collagen adapts. The influence of transmural G&R of constituents on the aneurysm development is analyzed. We observe that elastin and collagen strains evolve to be transmurally heterogeneous and this may facilitate the development of tortuosity. This multiphysics framework provides the basis for exploring the influence of transmural metabolic activity on the progression of vascular disease.

Elastocapillary flows in flexible tubes

Interactions between capillary and elastic effects are relevant to a variety of applications from micro- and nano-scale manufacturing to biological systems. In this work, we investigate capillary flows in flexible, millimeter-scale cylindrical elastic tubes. We demonstrate that surface tension can cause sufficiently flexible tubes to collapse and coalesce spontaneously through non-axisymmetric buckling, and develop criteria for the initial deformation and complete collapse of a circular tube. Experimental results for capillary rise and evaporation of a liquid in a flexible tube are presented, and several regimes are seen for the equilibrium state of a flexible tube deforming under capillary pressure. Deformations of the tube walls are measured in different regimes and compared with a shell theory model. Analysis and experimental results show that despite the complex and non-axisymmetric deformed shapes of cylindrical structures, the elastocapillary length used in previous literature for flat plates and sheets can be used to predict the behavior of flexible tubes.

Azimuthal shear of doubly fibre-reinforced, non-linearly elastic cylindrical tubes

A doubly fibre-reinforced, non-linearly elastic tube subject to azimuthal shear is analysed. The materials involved are neo-Hookean models augmented with a function that accounts for the existence of two unidirectional reinforcements. This function endows the material with its anisotropic character and quantifies the effect associated with the two preferred material directions. The nature of the anisotropy considered has a particular influence on the shear response of the material, in contrast to previous analyses in which the effect of the anisotropy was taken to depend only on the stretch in the fibre directions. The inner boundary of the tube is fixed while the outer boundary is subject to a given shear traction. Non-smooth solutions arise inside the tube at a critical value of the shear stress, which can be obtained with the well-known Maxwell line analysis.

Mathematical modeling of unsteady blood flow through elastic tapered artery with overlapping stenosis

A nonlinear two-dimensional pulsatile blood flow through a stenosed artery is investigated by treating the deformable vascular wall as an elastic cylindrical tube containing the Newtonian fluid. In order to establish a resemblance to the in vivo conditions, the mathematical model of an improved shape of the time-variant overlapping stenosis is considered in the tapered arterial lumen. By applying a suitable coordinate transformation, the tapered cosine-shaped artery becames a non-tapered rectangular and rigid artery. The continuity and the nonlinear momentum equations are numerically solved under the appropriate physically realistic prescribed boundary conditions. In order to solve the resulting simultaneous equation system, the finite difference approximation code is developed and utilized. The effects of the taper angle, wall deformation, and severity of the stenosis within its fixed length on velocity profiles, volumetric flow rate, and resistive impedance are studied considering their dependencies with time. The present results are found in agreement with similar data from the literature.

Viscous fluid-thin cylindrical elastic layer interaction: asymptotic analysis

This article represents a generalization of our previous work. We consider a periodic, non-steady, axially symmetric, creeping flow of a viscous incompressible fluid that fills a cylindrical elastic hollow tube. We study the interaction problem “viscous fluid-thin cylindrical elastic layer” when the thickness of the tube wall, , tends to zero, while the density and the Young’s modulus of the elastic material are of order and , respectively. We construct a complete asymptotic expansion when tends to zero. The error between the exact solution and the asymptotic one is evaluated in order to justify the asymptotic construction.

Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes

Cylindrical tubes and membranes are universal structural elements found in biology and engineering over a wide range of scales. Working in the framework of nonlinear elasticity, we consider the possible deformations of elastic cylindrical shells reinforced by one or two families of fibres. We consider both small and large deformations and the reduction from thick cylindrical shells (tubes) to thin shells (cylindrical membranes). In particular, a number of universal parameter regimes can be identified where the response behaviour of the cylinder is qualitatively different. This include the possibility of inversion of twist or axial strain when the cylinder is subject to internal pressure.

Effect of size‐distributed bubbles on acoustic properties of an elastic tube with polymeric liquid

PurposeThe objective of this study is to model the influence of free gas, in the form of size‐distributed fine bubbles, on sound attenuation and dispersion in a thin‐walled elastic cylindrical tube filled with viscoelastic polymeric liquid.Design/methodology/approachSound wave propagation in the system is described within a three‐phase interaction scheme, based on a quasi‐homogeneous approach to liquid‐gas mixture dynamics in the wave. Coupled equations of tube wall deformations and viscoelastic liquid dynamics in the tube are solved using a long‐wave approximation. The dissipative losses, stemming from flow gradients in the wave, as well as from non‐equilibrium bubble‐liquid interaction, are accounted for. The dispersion equation for the waveguide is obtained and studied numerically.FindingsThe results of the study indicate that bubble‐size distribution in viscoelastic liquid has an essential impact on sound propagation in the tube at sufficiently high frequencies. The frequency range in which the mixture heterogeneity influences the acoustic properties of the system is sensitive to both the distribution parameters and the rheological properties of the liquid. As distinct to polydispersity features, the viscoelastic properties of liquid are also relevant in the low‐frequency range, where they lead to an increase of the wave speed and a decrease of its attenuation.Originality/valueA model of sound wave propagation in a tube filled with a heterogeneous viscoelastic liquid‐bubble mixture is formulated. The study provides a basis for modeling transient processes in tubes filled with polymeric liquids containing free gas, and for acoustic control of certain processes in polymer technologies.

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.

Effects of magnetic field, porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics

A mathematical model for blood flow through an elastic artery with multistenosis under the effect of a magnetic field in a porous medium is presented. The considered arterial segment is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid representing blood. An artery with mild local narrowing in its lumen forming a stenosis is analyzed. The effects of arterial wall parameters represent viscoelastic stresses along the longitudinal and circumferential directions Tt and Tθ, respectively. The degree of anisotropy of the vessel wall γ, total mass of the vessel, and surrounding tissues M and contributions of the viscous and elastic constraints to the total tethering C and K respectively on resistance impedance, wall shear stress distribution, and radial and axial velocities are illustrated. Also, the effects of the stenosis shape m, the constant of permeability X, the Hartmann number Hα and the maximum height of the stenosis size δ on the fluid flow characteristics are investigated. The results show that the flow is appreciably influenced by surrounding connective tissues of the arterial wall motion, and the degree of anisotropy of the vessel wall plays an important role in determining the material of the artery. Further, the wall shear stress distribution increases with increasing Tt and γ while decreases with increasing Tθ, M, C, and K. Transmission of the wall shear stress distribution and resistance impedance at the wall surface through a tethered tube are substantially lower than those through a free tube, while the shearing stress distribution at the stenosis throat has inverse characteristic through totally tethered and free tubes. The trapping bolus increases in size toward the line center of the tube as the permeability constant X increases and decreases with the Hartmann number Ha increased. Finally, the trapping bolus appears, gradually in the case of non-symmetric stenosis, and disappears in the case of symmetric stenosis. The size of trapped bolus for the stream lines in a free isotropic tube (i.e., a tube initially unstressed) is smaller than those in a tethered tube.

Theoretical and experimental analysis of a thin elastic cylindrical tube acting as a non-Hookean spring

We analyze the (large) deformation and energy of a thin elastic cylindrical tube compressed between two plates parallel to the tube axis. The deformation is studied theoretically using a numerical calculation and the variational approach. The results are used to interpret the experimental data obtained by pressing tubes made from plastic-foil transparencies. We obtain a universal scaling relation that characterizes the response of the tube. Our results may serve as a benchmark for the application of variational methods to thin-walled nanoscale systems in order to obtain functional relations between the energy and the deformation.