Pressure Boundary Research Articles

КРАЕВАЯ ЗАДАЧА ДЛЯ ОДНОЙ ПЕРЕОПРЕДЕЛЕННОЙ СИСТЕМЫ, ВОЗНИКАЮЩЕЙ В ДВУХСКОРОСТНОЙ ГИДРОДИНАМИКЕ

In the half-plane R2 , we consider a stationary system of two-velocity hydrodynamics with one pressure and homogeneous divergent and boundary conditions for two velocities. The system is overdetermined. The solution to this system is reduced to a consistent solution of the two boundary value problems: the Stokes problem for one velocity and pressure and the overdetermined system for a different velocity. For an appropriate choice of function spaces, the existence and uniqueness of the generalized solution with a corresponding stability estimate is proven.

Magnetite (Fe3O4) and nickel ferrite (NiFe2O4) zeta potential measurements at high temperature: Part I—Design, materials and preliminary characterization of an apparatus implementing the streaming potential method

An apparatus for measuring the streaming potential of particles at high temperature has been designed and tested in a recirculating water loop operating at temperatures up to 300 °C. The ultimate objective is to derive values of the zeta potential of corrosion-product particles under the conditions of the primary coolant of pressurized water reactors (PWRs), quantities that until now have been unavailable. They are required for modeling the transport and radioactivation of particulate oxides around the coolant system and the consequential growth of radiation fields around components. The experimental technique employed, measuring the potential difference across a packed bed of particles through which the coolant flows, involved fabricating a tube of temperature-resistant polymer that insulated the particles from the containing stainless steel pressure boundary of the loop. The particle bed was contained between porous ceramic membranes and the whole sealed with shrunk-on PTFE to avoid by-passing flows. Platinum wires, inserted into the loop through Conax fittings and insulated from contact with it by coils of oxidized zirconium wire, served as electrodes upstream and downstream of the bed to measure the potential difference The detailed design and testing of the apparatus containing magnetite powder to verify its correct functioning under operating conditions are described. Since measuring streaming potential involved obtaining the voltage difference across the particle bed alone at the same time as the pressure drop through it from the flowing coolant was measured, the preparatory tests first established baseline values, from measurements on an empty test section, which were subtracted from the measurements on the full test section. Also, immediately before each streaming potential determination, a rapid measurement provided the overall electrical resistance of the test section. The initial measurements of the streaming potential of magnetite, between 20 and 240 °C in a solution containing 1800 ppm of boron and 3.5 ppm of lithium, led to corresponding values of zeta potential. The values show that the apparatus provided reproducible results. The results of the complete series of tests under conditions of the primary system of PWRs, involving nickel ferrite as well as magnetite particles, are the subject of a separate paper.

Unified Theory of Unsteady Planar Laminar Flow in the Presence of Arbitrary Pressure Gradients and Boundary Movement

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications.

An Improved One-Dimensional Flow Model for Side Chambers of Centrifugal Pumps Considering the Blade Slip Factor

Abstract The blade slip factor significantly influences the prediction accuracy of the self-closure one-dimensional flow model for side chambers of centrifugal pumps. Wiesner's and Stodola's slip factors, which are used to formulate the blade outlet pressure and served as the boundary condition for the model, are examined, which is an improvement of the previous study (Gu et al., 2020, “A Pressure Model for Open Rotor–Stator Cavities: An Application to an Adjustable-Speed Centrifugal Pump With Experimental Validation,” ASME J. Fluids Eng., 142(10), p. 101301). Both computational fluid dynamics (CFD) simulations and experiments for the centrifugal pump are conducted to support the improvement. A good agreement exists between the performance at the best efficiency points (BEPs) of different rotating speeds obtained by simulations and experiments. Through the CFD analysis, the flow in the impeller remarkably deviates from the blade-congruent flow, especially in the quasi-triangular regions downstream of the throats. Meanwhile, the reason for pressure over-predictions of the side chamber one-dimensional flow model that embeds Wiesner's slip factor (FMW) is that Wiesner's expression underestimates the impeller flow deflection and overestimates pressure boundary. By contrast, the side chamber one-dimensional flow model with Stodola's slip factor (FMS) is closer to CFD in terms of relative flow angle and chamber inlet pressure. Compared with the side chamber pressure measurements, the accuracy of FMS is upgraded approximately by 3.5% than FMW. At the BEPs of different rotating speeds, FMS generates lower shroud thrust coefficients but slightly greater volumetric efficiencies than FMW. This work provides a simple approach to better calculate flow characteristics in the side chambers.

Exploring the role of fluid-solid interactions for modelling volcano deformation

Investigating the temporal development of magma reservoir pressure and associated surface displacements can reveal fundamental aspects of subsurface magmatic processes and aid in eruption forecasting. The limitation with existing volcano deformation models is that they typically ignore magma intrusion dynamics and focus on the response of surrounding rocks to source boundary pressure. Magma fluid dynamics should be incorporated into magmatic modelling to track the temporal development of a system, instead of the widely used kinematic techniques. Here, we compare analytical and numerical solutions for magma intrusion into a shallow reservoir, using two schemes of intrusion boundary condition, inlet pressure and inlet mass flow. Model sensitivity tests are conducted to explore key factors controlling the two-way coupling between solid and fluid components, assuming an incompressible magma for a first-order approach. For intrusions of viscous magma (⪆108 Pa s) or a narrow feeder conduit (5–20 m), applying an inlet pressure causes the resultant pressure and surface deformation to develop at a very slow rate; lower viscosity magmas produce faster deformation rates. The mass flow boundary condition reduces the number of model parameters as it is independent of poorly constrained parameters such as conduit and magma characteristics. For both boundary conditions, reservoir pressurization, and hence spatiotemporal surface deformation, are strongly influenced by reservoir geometry due to geometric compressibility. Our results provide fundamental knowledge to advance to more complex coupled fluid-solid mechanics models in volcano geodesy.

Embedded Computational Heart Model for External Ventricular Assist Device Investigations

PurposeExternal cardiac assist devices are based on a promising and simple concept for treating heart failure, but they are surprisingly difficult to design. Thus, a structured approach combining experiments with computer-based optimization is essential. The latter provides the motivation for the work presented in this paper.MethodsWe present a computational modeling framework for realistic representation of the heart’s tissue structure, electrophysiology and actuation. The passive heart tissue is described by a nonlinear anisotropic material law, considering fiber and sheetlet directions. For muscle contraction, an orthotropic active-strain model is employed, initiated by a periodically propagating electrical potential. The model allows for boundary conditions at the epicardium accounting for external assist devices, and it is coupled to a circulation network providing appropriate pressure boundary conditions inside the ventricles.ResultsSimulated results from an unsupported healthy and a pathological heart model are presented and reproduce accurate deformations compared to phenomenological measurements. Moreover, cardiac output and ventricular pressure signals are in good agreement too. By investigating the impact of applying an exemplary external actuation to the pathological heart model, it shows that cardiac patches can restore a healthy blood flow.ConclusionWe demonstrate that the devised computational modeling framework is capable of predicting characteristic trends (e.g. apex shortening, wall thickening and apex twisting) of a healthy heart, and that it can be used to study pathological hearts and external activation thereof.

Modified Johnson–Bossler–Naumann method to incorporate capillary pressure boundary conditions in drainage relative permeability estimation

A proposed modification of the Johnson–Bossler–Naumann (JBN) method incorporates capillary pressure at the inlet and outlet of a rock sample. The experimental runs are performed on Berea and Obernkriechner sandstone rock samples. Fluid is injected into Berea rock at capillary to viscous ratios 0.05, 0.1, 0.5, and 1 and into Obernkriechner rock at capillary to viscous ratios 0.05 and 0.25. Rock samples are initially saturated with 20 g/l NaCl water. The experimental observations are analyzed using classical JBN and our modified JBN. The latter's relative permeability curves are found to closely match those derived by applying history matching to a numerical model. This indicates that modified JBN can be used to provide an initial guess for history matching. We also show that accurate estimates of threshold capillary pressure are crucial to obtaining accurate oil relative permeability estimates. In contrast to those derived from the JBN method, modified JBN relative permeabilities are independent of capillary to viscous ratio or injection rate. • The proposed model incorporates capillary pressure at the inlet and outlet boundary. • Experiments are performed to validate the model. • Derived relative permeability curves are less sensitive to capillary number compared to classical methods.

Effect of pressure fluctuation in oil-gas multiphase pump on cavitation and performance of sealing liquid film

During the operation of the oil-gas multiphase pump, due to the variable gas content of the incoming flow, the medium pressure at the high-pressure end fluctuates widely, which is easy to cause the stress change at both ends of the mechanical seal compensation ring and “instability” phenomenon. In this paper, a three-dimensional spiral groove seal film model is established. Considering the cavitation effect of liquid film gap, the variation law of end face cavitation and sealing performance of liquid film seal is explored according to the effect of pressure pulsation in the pump on the inner diameter side (Static mechanical seal, PLAN74 flushing scheme in API 682) and outer diameter side (Rotary mechanical seal, PLAN53A flushing scheme in API 682). The results show that the local pressure drop in the sealing liquid film is the main cause result to cavitation of liquid film. The inner diameter side pressure fluctuation has a great impact on the end face cavitation, and the wave peak is about 100 times that of the outer diameter side pressure. The inner diameter side pressure fluctuation should be avoided in engineering application. There is hysteresis between gas volume fraction (GVF) and boundary pressure fluctuation. The synchronization between sealing performance parameters and outlet pressure fluctuation is good, but there is phase difference with inlet pressure fluctuation. The inner diameter side pressure fluctuation has a great change on the sealing performance. The peak is about 5–20 times that of the outer diameter side pressure, which is easy to cause vibration of the mechanical seal. The amplitude and period of pressure fluctuation have a great impact on the GVF and sealing performance. The larger the fluctuation period is, the less obvious the hysteresis of GVF is, and the closer to the variation law of pressure fluctuation. During the actual operation of the oil-gas multiphase pump, the mechanical seal connected with the medium in the pump is rotary, and the flushing scheme ought to be PLAN53A to ensure the safe and stable operation of the oil-gas multiphase pump.

A three-dimensional hybrid immersed smoothed point interpolation method for fluid-structure interactions

A three-dimensional hybrid immersed smoothed point interpolation method (3D HIS-PIM) has been developed to handle fluid-structure interactions (FSI) with simple geometries. Incompressible viscous fluid is solved by semi-implicit characteristic-based split method with standard Galerkin discretization, smoothed point interpolation method is employed to solve the motion and deformation of nonlinear solids based on gradient smoothing technique, and the virtual fluid is introduced into the entire solid domain to calculate the FSI force. The proposed method can avoid the fluid mesh adjustments, and allows the use of four-node tetrahedral elements for both fluid and solid domains to simplify the pre-processing for mesh generations. The FSI force condition is imposed with a hybrid force, which uses a form of body force for pressure term and boundary force for shear force. The boundary force accords with the physical law and the body force can ensure the numerical stability. The proposed method works well for FSI problems with moving or largely deformable solids, and can achieve more accurate results using a hybrid force than the fully body force in the original method. A large range of mesh size ratios between fluid and solid meshes is allowed due to the treatment of FSI boundary conditions.

Finite element simulation of a viscoelastic cell entering a cylindrical channel: Effects of frictional contact

Finite element method has been extensively used to simulate the process of single cells entering a narrow channel under a prescribed pressure. Yet an in-depth understanding on the effects of the frictional boundary condition in such simulations is lacking. The main challenge lies in the continuously evolving contact region between the deformed cell and the channel wall during the entry process, which makes it difficult to apply proper boundary conditions for the friction and pressure. In this work, we present an approach to continuously update the boundary condition according to the contact status during entry. Using this approach and modeling the cell as a visco-hyperelastic sphere, we study how the pressure boundary conditions and the friction coefficient affect the simulated cell entry process. Specifically, we compare two special cases of pressure boundary conditions: positive pressure at the trailing edge of the cell or negative pressure at the leading edge. It is found that the two cases produce the same result in terms of the cell entry length under frictionless contact, but give considerably different results under frictional contact. Effects of whether the pressure boundary is updated are then studied by considering more special cases. All these results are well explained by the pressure superposition principle, and further insights into the effects of pressure boundary condition and boundary update are also acquired. Finally, focusing on the case of positive pressure with boundary updates, we obtain simulation results revealing how the entry length depends on the pressure, time and friction coefficient. In particular, a larger friction coefficient can significantly reduce the entry length under a prescribed pressure or a certain entry time. Our results suggest that it is necessary to implement accurate pressure boundary conditions (e.g., pressure magnitudes at the two edges of cell and continuously updated pressure boundary regions) in order to faithfully capture the frictional contact between the cell and the channel wall. In addition, the simulation approach can be useful in modeling the general process of a viscoelastic sphere entering a cylindrical channel involving frictional contact.

Effect of Combustion Boundary Conditions and n-Butanol on Surrogate Diesel Fuel HCCI Combustion and Emission Based on Two-Stroke Diesel Engine

The combustion and emission characteristics of surrogate diesel fuel homogeneous charge compression ignition (HCCI) with different combustion boundary conditions and n-butanol (NB) mixing ratios are studied in this work. Engine data of a two-stroke low-speed direct-injection marine diesel engine were selected for the reactor. HCCI combustion was achieved by compressing a completely homogeneous mixture of fuel and air. The results show that NO emissions decrease slightly with the increase of initial boundary pressure at a constant equivalence ratio and initial temperature. In addition, the different initial boundary temperature has little effect on NO emission. The results also indicate that the ignition delay time of the mixed fuel rises with the increase of n-butanol mixing ratio. The emissions and reaction rate of NOx reduce significantly with the increase of n-butanol percentage in surrogate diesel fuel and n-butanol mixing combustion at a constant equivalence ratio and total mole fraction. Meanwhile, CO2 emissions also decrease significantly with the increase of n-butanol mixing ratio.

Droplet evaporation in finite-size systems: Theoretical analysis and mesoscopic modeling.

The classical D^{2}-Law states that the square of the droplet diameter decreases linearly with time during its evaporation process, i.e., D^{2}(t)=D_{0}^{2}-Kt, where D_{0} is the droplet initial diameter and K is the evaporation constant. Though the law has been widely verified by experiments, considerable deviations are observed in many cases. In this work, a revised theoretical analysis of the single droplet evaporation in finite-size open systems is presented for both two-dimensional (2D) and 3D cases. Our analysis shows that the classical D^{2}-Law is only applicable for 3D large systems (L≫D_{0}, L is the system size), while significant deviations occur for small (L≤5D_{0}) and/or 2D systems. Theoretical solution for the temperature field is also derived. Moreover, we discuss in detail the proper numerical implementation of droplet evaporation in finite-size open systems by the mesoscopic lattice Boltzmann method (LBM). Taking into consideration shrinkage effects and an adaptive pressure boundary condition, droplet evaporation in finite-size 2D/3D systems with density ratio up to 328 within a wide parameter range (K=[0.003,0.18] in lattice units) is simulated, and remarkable agreement with the theoretical solution is achieved, in contrast to previous simulations. The present work provides insights into realistic droplet evaporation phenomena and their numerical modeling using diffuse-interface methods.

In Vivo Intravascular Optical Coherence Tomography (IVOCT) Structural and Blood Flow Imaging Based Mechanical Simulation Analysis of a Blood Vessel.

Computer modelling of blood vessels based on biomedical imaging provides important hemodynamic and biomechanical information for vascular disease studies and diagnosis. However due to lacking well-defined physiological blood flow profiles, the accuracy of the simulation results is often not guaranteed. Flow velocity profiles of a specific cross section of a blood vessel were obtained by in vivo Doppler intravascular optical coherence tomography (IVOCT) lately. However due to the influence of the catheter, the velocity profile imaged by IVOCT can't be applied to simulation directly. A simulation-experiment combined method to determine the inlet flow boundary based on in vivo porcine carotid Doppler IVOCT imaging is proposed. A single conduit carotid model was created from the 3D IVOCT structural images using an image processing-computer aided design combined method. With both high- resolution arterial model and near physiological blood flow profile, stress analysis by fluid-structure interaction and computational fluid dynamics were performed. The influence of the catheter to the wall shear stress, the hemodynamics and the stresses of the carotid wall under the measured inlet flow and various outlet pressure boundary conditions, are analyzed. This study provides a solution to the difficulty of getting inlet flow boundary for numerical simulation of arteries. It paves the way for developing IVOCT based vascular stress analysis and imaging methods for the studies and diagnosis of vascular diseases.

Stochastic modeling of fracture initiation and propagation during fluid flow in carbonate reservoir

Fracture initiation and propagation in fractured carbonate reservoirs or shale ones during the production phase of the reservoir goes without paying particular attention because of the complex model that represents in itself the propagation of fracture. It goes without saying how important an assessment of this phenomenon, propagation, would be in quantitative terms for reservoir engineering in reservoir simulators. This fact becomes even more interesting when our data are stochastic, such as boundary conditions, pressure, viscosity, and other targeting functions. Exactly in this paper, we will address such a problem, i.e., the configurations of the crack network as propagating them in time when the inputs are random starting directly with stochastic optimal control Brinkman model in fractured rocks, which is known to be a problem that includes areas where the Darcy flow predominates and areas where the Stokes flow predominates. We will then develop our flow model in fractured rocks under the effects of geomechanical processes by undergoing energy minimization processes. Our method is robust, although at the moment, it requires a bit of calculation time. In the future, the “reduction” of this handicap would pave the way for the implementation of this model on a large Darcy scale.

An improved multicomponent pseudopotential lattice Boltzmann method for immiscible fluid displacement in porous media

Immiscible fluid displacement in porous media occurs in several natural and industrial processes. For example, during petroleum extraction from porous rock reservoirs, water is used to displace oil. In this paper, we investigate the primary drainage and imbibition in a heterogeneous porous medium using an improved numerical model based on the multicomponent pseudopotential lattice Boltzmann method. We apply recent developments from the literature and develop new pressure boundary conditions. We show that the proposed model is able to simulate realistic viscosity ratios, and it allows independent tuning of surface tension from viscosity. Moreover, the model suppresses a nonphysical behavior of previous schemes, in which trapped fluid volumes significantly change with time. Furthermore, we show that the developed model correctly captures the underlying physical phenomena of fluid displacements. We simulate oil–water flows and verify that the measured values of irreducible water and residual oil saturations are realistic. Finally, we vary the wetting conditions of the porous medium to represent different wettability states. For the different scenarios, we show that the simulations are in good agreement with the experimental results.

A Head Loss Pressure Boundary Condition for Hydraulic Systems

Despite the increase in computational power of HPC clusters, it is in most cases not possible to include the entire hydraulic system when doing detailed numerical studies of the flow in one of the components in the system. The numerical models are still most often constrained to a small part of the system and the boundary conditions may in many cases be difficult to specify. The headLossPressure boundary condition is developed in the present work for the OpenFOAM open-source CFD code to include the main effects caused by a large hydraulic system onto a component in the system. The main motivation is to provide a boundary condition for hydraulic systems where known properties are specified by the user and unknown properties are calculated. This paper is a guide to the developed headLossPressure boundary condition. It is based on the extended Bernoulli equation to calculate the kinematic pressure on the patch. An arbitrary number of minor and friction losses are considered to describe the system in terms of head losses. The boundary condition also provides the opportunity to specify the head in relation to a reference elevation. System changes during operations are modelled through Function1 variables, which enables time-varying inputs. The developments are validated against experimental test data, where the varying head between two free surfaces and a valve closing and opening sequence are modelled with the boundary condition. The main effects of the system are well captured by the headLossPressure boundary condition. It is thus a useful and trustworthy boundary condition for incompressible hydraulic system simulations.

Transformational Challenge Reactor Safety Design and Radionuclide Retention Strategy

An integrated safety design and radionuclide (RN) retention strategy is developed to support the Transformational Challenge Reactor (TCR) demonstration. This demonstration aims to showcase viability for rapid deployment of a novel reactor by leveraging the advances in materials, manufacturing, and computational sciences through a highly integrated and agile design and development approach. This strategy provides a logical description and understanding of how RNs are contained within the facility. Rather than discussing fission product barriers individually between separate design and safety basis reports, this paper provides a consistent description and narrative to better facilitate regulatory interactions and focus safety design efforts. The principal barriers credited include the various coating layers in the tristructural isotropic (TRISO) fuel particle, the silicon carbide (SiC) matrix hosting the particles within the fuel element, the helium pressure boundary, and the confinement system. The choice and assumed performance of the credited barriers are highly conservative, which is a direct reflection of the low hazard that the TCR demonstration presents and the need to simplify and focus the safety review process accordingly. However, the strategy and the associated framework are generalized and may be adopted and tailored to support other advanced reactor demonstration efforts.

Modeling of possible CO2 leakage with phase transition in wellbore-reservoir system based on the Ordos CCS project, China

The leakage of injected CO2 through wellbores, during which the CO2 phase transition may occur, is a prime concern in CO2 deep saline aquifer storage projects. In this study, a two-dimensional non-isothermal wellbore-reservoir coupled model that can deal with the CO2 phase transition is established according to the Shenhua Ordos Carbon Capture and Storage (CCS) project in China. Different leakage scenarios through the injection well are simulated to explore the leakage characteristics and reveal the mechanism of important phenomena. The simulation results show that the CO2 leakage through the injection well in the Ordos CCS project, which takes thick multi-layered strata with unusual pressure distribution as the target reservoir, is self-limited, and the leakage stops after the reservoir pressure is offset with the pressure provided by the leaked groundwater in the wellbore. During the leakage, the liquid-phase or supercritical CO2 in the wellbore-reservoir system undergoes a certain degree of gasification, which has a strong and positive effect on the leakage. Moreover, although the leakage eventually stops at the wellhead, the crossflow between different injection layers through the wellbore continues. The parameter analysis indicates that the permeability and the initial and boundary pressure distribution of the reservoir play an important role in the CO2 leakage process, especially when they have a significant impact on the CO2 gasification in the reservoir. These results may provide a basis for evaluation, precaution, and management of similar wellbore leakage.

Thermo-mechanical coupling behavior of plate structure under re-entry aerodynamic environment

A coupled thermo-mechanical finite element (TMFE) algorithm with the computable model on damping effects is proposed to simulate the thermo-mechanical coupling behavior and damping characteristics of a plate structure under re-entry aerodynamic environment. Firstly, an unconditionally stable implicit FE algorithm considering damping effects is constructed. Based on the coupled transient heat conduction problem, the proposed algorithm is verified through comparing the analytical and numerical solution. Combined with the gas-kinetic unified algorithm (GKUA), the aerodynamic force and heating are calculated as the temperature and pressure boundary conditions for the thermo-mechanical coupling calculation. The aerodynamic-structure boundary coupling verification is realized by comparing the GKUA calculated value of the flow field boundary with the TMFE results on the temperature and pressure of the plate surface. Then, the influences of element mesh density, inertia term, coupling term and damping effect on the thermo-mechanical coupling response of a plate are studied. The variation laws among temperature, displacement and stress fields are deeply revealed. The proposed algorithm offers a novel pathway to predict thermal flutter induced by thermo-mechanical coupling response of spacecraft structure.

ТОЧНОЕ РЕШЕНИЕ ЗАДАЧИ О РЕЛАКСАЦИИ НАПРЯЖЕНИЙ В ТОЛСТОСТЕННОЙ ТРУБЕ ИЗ НЕЛИНЕЙНО ВЯЗКОУПРУГОГО МАТЕРИАЛА, ПОДЧИНЯЮЩЕГОСЯ СООТНОШЕНИЮ РАБОТНОВА

We constructed and studied analytically the exact solution of the quasi-static boundary value problem for a hollow cylinder (a tube) made of physically nonlinear homogeneous isotropic viscoelastic material obeying the Rabotnov constitutive equation with two arbitrary material functions (a creep compliance and a function which governs physical nonlinearity). A time-dependent displacement and zero shear stresses are given on the inside cylinder surface, a pressure (normal stress) is given on outside surface and zero axial displacements and zero shear stresses are preset on the end cross sections of the tube (thus, cylinder stress and plain strain are realized in a tube). We supposed that a material is incompressible and that given boundary displacement and pressure vary slowly enough with time to neglect inertia terms in the equilibrium equations. We obtained explicit expressions for strains and stresses at any point via the ratio of a tube radii, given external pressure and integral operators involving composition of two material functions of the constitutive relation and displacement preset on the internal cylindrical surface. In particular, assuming given boundary displacement and pressure are constant we considered the stress relaxation problem. For arbitrary material functions, we calculated all the hereditary integrals involved in the general representation for the stress field and reduced it to simple algebraic formulas convenient for analysis and use. We studied analytically general properties of the stress relaxation curves at any point of a tube and features of stress distributions along radius. We proved that the first material function (relaxation modulus) of the constitutive equation governs completely stresses dependence on time and the second one (non-linearity function) governs only stresses dependence on radial coordinate and found out that hoop stress and axial stress can change sign and can increase, decrease and be non-monotone with respect to radial coordinate. In case of zero external pressure, we proved that all stress relaxation curves at any point of a tube are proportional to each other (ratio of stresses at different points doesn't depend on time), decrease with time and have no flexure points. We also calculated integral mean values of axial and hoop stresses and discovered that their ratio depends on thickness/radius ratio only and doesn't depend on time and material functions although mean stresses do.