Negligible Reynolds Number Research Articles

Inertial aftermaths on peristaltic drive of MHD micropolar fluid in a two-dimensional asymmetric inclined porous soaked channel independent of wavelength: Galerkin finite element simulation

This manuscript presents a detailed investigation of the peristaltic propulsion of a micropolar fluid in an inclined asymmetric channel, which is also subjected to a magnetic field applied in the normal direction. The medium is considered to be a porous, saturated environment. Unlike traditional lubrication theory, which often assumes long wavelengths and negligible Reynolds numbers, our analysis does not adhere to these constraints. This approach introduces nonlinearity into the modeled equations and allows for significant Reynolds numbers, thereby enhancing our understanding of the peristaltic phenomenon. Numerical solution of coupled partial differential equations is gained by employing a novel Galerkin built finite element method and is presented through graphs of velocity and pressure distributions in accordance with variation in several flow parameters. Streamlines’ gyration, microrotation, and vorticity for different configurations that emerged with varying phase transitions are displayed in this regard as well. It is confessed that peristaltic mixing diminishes for all values of phase differences as the permeability parameter increases, while a rising Hartmann number significantly exacerbates this effect. In addition, the microrotation of micropolar particles is observed to become increasingly distorted with higher Reynolds numbers. Furthermore, the pressure rise throughout both pumping and co-pumping regions is enhanced when the inclined channel is subjected to a greater angle of inclination.

Thermocapillary migration of a compound drop in an arbitrary viscous flow

The thermocapillary migration of a concentric compound drop in an arbitrary viscous flow under the consideration of negligible Reynolds number is investigated. The thermocapillary effect refers to the migration of a drop under the influence of a temperature gradient. The thermal and hydrodynamic problems are examined. The thermal field is governed by the heat conduction equation whereas the hydrodynamic fluid velocities are governed by the linearized Navier–Stokes equations. Presence of temperature gradient results in variation of the interfacial tension which is assumed to depend on temperature linearly. Variation of interfacial gradient leads to the coupling of the hydrodynamic problem with the thermal problem through the boundary condition. A complete general solution of Stokes equations is utilized to obtain closed-form expressions for the velocity vector and pressure. The hydrodynamic forces acting on the compound drop are obtained and expressed in terms of Fax́en’s law. Some important asymptotic limiting cases of hydrodynamic drag are also derived. The hydrodynamic drag for cases of uniform flow, shear flow, and heat source with the known ambient flow are derived and it is found that in the case of shear flow, the hydrodynamic drag is contributed only by the thermal component and the shear flow has no effect on it. The obtained results for drag and torque in the limiting cases are in agreement with the existing results in the literature. Furthermore, the migration velocity of the compound drop is obtained by equating the hydrodynamic drag force to zero. The results obtained for migration velocity are explained with the aid of graphs. The migration velocity is found to be a monotonic function of the Marangoni number and the radius of the innermost drop.

Peristaltic transport of hybrid nanofluid under the effects of thermal radiation through asymmetric curved geometry: a numerical approach

ABSTRACT Magnetohydrodynamics (MHD) has various applications in the field of medicine, particularly in drug delivery and hyperthermia treatment for cancer therapy to the affected area via peristaltic phenomena. Further elaboration is required to optimize the delivery method of hybrid nanoparticles with aluminum oxide and zinc oxide to the target area via the esophagus to get the best-desired results. Keeping the importance of MHD in mind, the present problem highlights the characteristics of Magnetohydrodynamics (MHD) peristaltic transport by incorporating the properties of nanoparticles of aluminum and zinc oxides when suspended in water through an asymmetric curved conduit. The governing equations are mathematically modeled under consideration of Hall current, viscous dissipation, ohmic heating, thermal radiation, and heat sink/source effects. The influence of magnetic field, velocity, and thermal slips is also considered. Negligible Reynolds number and long-wavelength approximations are used to overcome the complexity of the system. To compute the numerical solutions of the simplified nonlinear system, two different techniques are utilized via MATLAB and Mathematica. The outcomes of the different flow parameters on the hybrid nanofluid’s velocity, trapping phenomena, temperature distribution, heat transfer rates, and pressure gradient are analyzed through tables and graphs.

EFFECT OF A VARIABLE MAGNETIC FIELD ON PERISTALTIC SLIP FLOW OF BLOOD-BASED HYBRID NANOFLUID THROUGH A NONUNIFORM ANNULAR CHANNEL

This paper analyzes the impact of hybrid nanoparticles (Cu–TiO2) on a two-dimensional peristaltic blood flow pattern in a nonuniform cylindrical annulus in the presence of an external induced magnetic field with wall slip. Further, this study focuses on the flow dynamics of single and hybrid nanofluids through endoscopic or catheterized effects. The mathematical model consisting of continuity, linear momentum, thermal energy, and Maxwell’s equations is simplified under the assumptions of long wavelength and negligible Reynolds number. The Homotopy perturbation method (HPM) is employed to get an approximate analytical solution of nonlinear dimensionless momentum equations. Based on the mathematical relationships and graphic visualization, the influence of the pertinent parameters described the velocity profile, temperature distribution, induced magnetic field, current density distribution, wall shear stress, and heat transfer coefficient. With the help of contours, the trapping phenomenon is also presented. The results reveal that the Lorentz force significantly reduces the Cu–TiO2/blood nanofluid velocity, whereas the elevating Grashof number does the opposite. Compared with copper nanoparticles, hybrid nanoparticles have a higher wall shear stress. The increasing values of Reynolds numbers amplify the induced magnetic field on annular surfaces. In the axial direction, Lorentz force significantly decreases the current density distribution for hybrid nanofluid. Moreover, hybrid nanoparticles (Cu–TiO2) exhibit superior heat transfer than Copper (Cu) nanoparticles in the blood-based fluid. According to the graphical outcomes, hybrid nanoparticles are comparatively more effective than unitary nanoparticles in the blood.

Second order slip flow of a conducting Jeffrey nanofluid in an inclined asymmetric porous conduit with heat and mass transfer

PurposeThe present work explores the influence of Hall and Ohmic heating effects on the convective peristaltic flow of a conducting Jeffrey nanofluid in an inclined porous asymmetric channel with slip. Also, the authors investigated the impact of viscous dissipation, thermal radiation, heat generation/absorption and cross diffusion effects on the flow. Peristaltic flow has many industrial and physiological applications and most of the biofluids show the non-Newtonian fluid behaviour. Further, in a living body, several biofluids flow through different kinds of systems that are not symmetric, horizontal or vertical. The purpose of this paper is to address these issues.Design/methodology/approachThe authors considered the flow of Jeffrey fluid which is generated by a sinusoidal wave propagating on the walls of an inclined asymmetric channel. The flow model is developed from the fixed frame to the wave frame. Finally, yield the nonlinear governing equations by applying the non-dimensional quantities with the assumptions of lengthy wave and negligible Reynolds number. The exact solution has been computed for the velocity and pressure gradient. The solutions for temperature and concentration are obtained by the regular perturbation technique.FindingsGraphical analysis is made for the present results for different values of emerging parameters and explained clearly. It is noticed that the magnetic field enriches the temperature where it drops the fluid velocity. This work describes that the temperature field is decreasing due to the radiation but it is a rising function of temperature slip parameter. The temperature profile declines for growing values of the Hall parameter. The flow velocity diminishes for boosting values of the Darcy parameter. Further, the authors perceived that the concentration field reduces for large values of the chemical reaction parameter.Originality/valueThe authors validated and compared the results with the existing literature. This investigation will help to study some physiological systems, and heat transfer in peristaltic transport plays key role in medical treatments, so we ensure that these results are applicable in medical treatments like cancer therapy, drug delivery, etc.

Effect of blockage ratio on flow of a viscoelastic wormlike micellar solution past a cylinder in a microchannel.

We present experiments on the flow of a viscoelastic wormlike micellar solution around cylinders (radius R) confined in straight microchannels (width W). Thirteen flow geometries are tested where the blockage ratio is varied over a wide range 0.055 ≤ BR = 2R/W ≤ 0.63. Experiments are performed at negligible Reynolds number, and for Weissenberg numbers Wi = λU/R up to 1000, where U is the average flow speed and λ is the relaxation time of the fluid. Micro-particle image velocimetry is used to characterise the flow state at each BR and Wi. In all of the geometries, a first critical Weissenberg number marks a transition from symmetric flow to an asymmetric but time-steady flow state, while a second higher critical Weissenberg number marks the onset of time-dependent flows. However, we report a clear shift in behaviour over a narrow intermediate range of 0.33 ≲ BR ≲ 0.41. Channels with BR ≤ 0.33 fall in a 'low' BR regime, with instabilities that originate from the downstream stagnation point, while those with BR ≥ 0.44 fall in a 'high' BR regime, with instabilities developing at the upstream stagnation point. Behaviour within the newly-identified intermediate BR regime is complex due to the competing influence of the two stagnation points. We summarise all our results in a flow state diagram covering Wi-BR parameter space, clearly defining the different regimes of blockage ratio for the first time. Our results contribute to the understanding of the complexities of viscoelastic flow in this benchmark geometry.

On the method of determining instantaneous wall shear stress from near-wall velocity measurements in wall turbulence

The determination of the instantaneous wall shear stress (WSS) from near-wall velocity measurements has received considerable attention. However, the most appropriate procedure and the achievable accuracy remain open topics. The present work uses direct numerical simulation datasets of channel flow to investigate the influences of the wall-normal distribution of instantaneous velocity, the method for estimating the velocity gradient, and the wall-normal position of velocity vectors on the accuracy of the instantaneous WSS determined from near-wall velocity measurements. In general, the method of dividing instantaneous velocity vectors by their wall-normal positions performs better than the method of linearly fitting instantaneous velocity profiles for estimating the wall velocity gradients when the wall position is correctly determined. However, the nonlinear instantaneous velocity distribution within the viscous layer means that all methods introduce a negative mean bias error and non-negligible root mean square error for the instantaneous WSS and its statistics. The magnitudes of these errors increase with the wall-normal position of the velocity vectors. An empirical method for correcting the instantaneous WSS statistics is proposed based on the negligible Reynolds number dependence of the bias error of all statistics. The influence of the wall-normal position of the velocity vectors on the WSS statistics and the correction method are verified using experimental data from open channel flows. The verification results show that the correction method significantly improves the accuracy of the statistics of instantaneous WSS determined from near-wall velocity measurements under canonical wall turbulence.

Kepler Orbits in Pairs of Disks Settling in a Viscous Fluid.

We show experimentally that a pair of disks settling at negligible Reynolds number (∼10^{-4}) displays two classes of bound periodic orbits, each with transitions to scattering states. We account for these dynamics, at leading far-field order, through an effective Hamiltonian in which gravitational driving endows orientation with the properties of momentum. This treatment is successfully compared against the measured properties of orbits and critical parameters of transitions between types of orbits. We demonstrate a precise correspondence with the Kepler problem of planetary motion for a wide range of initial conditions, find and account for a family of orbits with no Keplerian analog, and highlight the role of orientation as momentum in the many-disk problem.

Brownian motion and thermophoresis effects on Peristaltic slip flow of a MHD nanofluid in a symmetric/asymmetric channel

The slip and heat transfer effects on MHD peristaltic transport of a nanofluid in a non-uniform symmetric/asymmetric channel have studied under the assumptions of elongated wave length and negligible Reynolds number. From the simplified governing equations, the closed form solutions for velocity, stream function, temperature and concentrations are obtained. Also dual solutions are discussed for symmetric and asymmetric channel cases. The effects of important physical parameters are explained graphically. The slip parameter decreases the fluid velocity in middle of the channel whereas it increases the velocity at the channel walls. Temperature and concentration are decreasing and increasing functions of radiation parameter respectively. Moreover, velocity, temperature and concentrations are high in symmetric channel when compared with asymmetric channel.

From active stresses and forces to self-propulsion of droplets

We study the self-propulsion of spherical droplets as simplified hydrodynamic models of swimming micro-organisms or artificial micro-swimmers. In contrast to approaches that start from active velocity fields produced by the system, we consider active interface tractions, body force densities and active stresses as the origin of autonomous swimming. For negligible Reynolds number and given activity, we compute the external and internal flow fields as well as the centre of mass velocity and angular velocity of the droplet at fixed time. To construct trajectories from single time snapshots, the evolution of active forces or stresses must be determined in the laboratory frame. Here, we consider the case of active matter, which is carried by a continuously distributed rigid but sparse (cyto)-skeleton that is immersed in the droplet interior. We calculate examples of trajectories of a droplet and its skeleton from force densities or stresses, which may be explicitly time-dependent in a frame fixed within the skeleton.

Effects Of Peristaltic Flow Of A Couple Stress Fluids Through A Porous Medium In A Channel At Low Reynolds Number

The present paper discuss the effects of the peristaltic motion of a couple stress fluid through a porous medium in a two dimensional flexible channel under long wave length approximation and negligible Reynolds number. The effects of various physical parameters on axial velocity, transverse velocity, pressure gradient and friction force have been computed numerically by perturbation method. A perturbation method of solution is obtained in terms of wall slope parameter and closed form of expressions has been derived for axial velocity, transverse velocity and pressure also. It is noted that pressure rise increases with increase in couple stress parameter and porous medium. The friction force has an opposite behavior compared with pressure rise.

Full-scale freight train underbody aerodynamics with application to track spraying

ABSTRACTIn this article, measurements of full-scale freight train underbody aerodynamics relevant to top-of-rail spraying are presented. The velocity near the wheel was measured using hot wire anemometers mounted on the train. The pressure was measured using transducers mounted adjacent to the track. Velocity scaled linearly with locomotive speed and pressure scaled linearly with dynamic pressure, implying negligible Reynolds number effects. The mean velocity near the wheels was less than half of the locomotive speed when the vehicle moved at or greater than the ambient wind speed. The velocity upwind of the wheel was 30% greater than downwind of the wheel. These mean velocities are consistent with computational fluid dynamics simulations of the flow field in the vicinity of the wheel. Turbulence intensity levels were measured to be 0.08–0.16. The pressure at the track depends on the configuration of the leading car; the maximum pressure drop was 65% greater when the hopper car preceded the locomotive compared to when the locomotive preceded the hopper car.

Non-colloidal suspensions: Relations between theory and experiment in shearing flows

This paper considers the relation between theory, computation and new experiments in noncolloidal suspensions with Newtonian and viscoelastic matrices in shearing flow.One expects from creeping flow theory that a suspension of rigid spheres in a Newtonian fluid matrix at negligible Reynolds number and large Péclet number would exhibit a constant shear viscosity. However, one usually sees shear-thinning behaviour in experiments for larger volume fractions (φ>0.3). Experiments with a Boger fluid matrix of constant shear viscosity on the other hand, show mild shear thickening at higher shear rates. Explanations for this behaviour are unsatisfying.There are also puzzles for the normal stresses in sheared suspensions. For a suspension with a Newtonian matrix in a simple shear flow various workers have found experimentally that the bulk stress field is no longer a shear plus an isotropic pressure, and that both first and second normal stress differences (N1 and N2) occur and are negative. Our own experiments using an open semi-circular trough to find N2 agree reasonably well with previous experimental results and the Brady–Morris theory of 1997. However, this theory predicts that N1 is zero, whereas it is measured to be negative but smaller in magnitude than N2.The viscometric functions (η, N1 and N2) for non-colloidal suspensions of spheres in a Boger fluid matrix were also measured. Volume fractions (ϕ) from 5% up to 40% were investigated. The relative viscosity (ηr=η/η0) and the (positive) first normal stress difference N1 showed increases with ϕ which were larger than the dilute suspension theory predictions of 1+2.5ϕ, indicating semi-dilute suspension behaviour, even down to 5% concentration.A major concern however centres on the second normal stress difference N2. The Boger matrix fluid showed a zero second normal stress difference, and the measurements showed that N2 was always negative for the suspensions. This agrees with the dilute suspension theoretical prediction found using the Landau–Lifshitz averaging procedure, but not with the result from the ensemble averaging method, which predicts a positive N2. The reason for this predictive failure is suggested to lie in the treatment of the stresses induced by the far distant spheres. It is suggested that using a Brinkman approach to model the effect of the far spheres would be useful.For larger concentrations (up to 0.4) the second normal stress difference was always negative. However, we saw a switch from a positive N1 to a negative value when the volume fraction exceeded 0.3, similar to the results of Aral and Kalyon (1997). This can be understood as a contest between the component of (positive) N1 due to the matrix properties and the influence of the component of (negative) N1 due to the proximity of the spheres. A suggestion for a constitutive model based on this idea is given.

Creeping-flow rotation of a slip spheroid about its axis of revolution

The problem of rotation of a rigid spheroidal particle about its axis of revolution in a viscous fluid is studied analytically and numerically in the steady limit of negligible Reynolds number. The fluid is allowed to slip at the surface of the particle. The general solution for the fluid velocity in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of separation of variables. The slip boundary condition on the surface of the rotating particle is applied to this general solution to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. The torque exerted on the spheroidal particle by the fluid is evaluated for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic torque results and the available analytical solutions in the limiting cases is good. It is found that the torque exerted on the rotating spheroid normalized by that on a sphere with radius equal to the equatorial radius of the spheroid increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip or finite-slip spheroid and vanishes for a perfectly slip spheroid. For a spheroid with a specified aspect ratio, the torque is a monotonically decreasing function of the slip capability of the particle.

Boundary-element method simulation of the impact of bounding walls on the dynamics of a particle group freely moving in a wide-gap Couette flow

In this study, the impact of the bounding walls on the dynamics of a group of neutrally-buoyant identical rigid spheres freely moving at negligible Reynolds numbers in a wide-gap Couette flow, which is important for understanding the particle migrations presented in concentrated suspensions subjected to inhomogeneous shear flows, is simulated by a three-dimensional boundary-element method (BEM) code. The results show that the particle interactions very close to the bounding walls cause the particle group to migrate away from the walls. As the distance of the bounding walls from the group increases, the migration changes direction and the group then move towards the walls. As this distance continues to increase, the migration of the group decreases and beyond a specific distance from the bounding walls the migration of the group is negligible. In addition, the BEM simulations show that the extent and rate of the migration of the group increase as the inter-particle distance decreases.

Boundary-element simulation of hydrodynamic interaction of two smooth spheres suspended in an unbounded Couette flow

The hydrodynamic interaction between two particles, suspended in shear flows, is fundamental to the macroscopic characterization of suspension flows. Although the understanding of the hydrodynamic interaction between two particles suspended in a quiescent or linear shear flow is mature, studies of the interaction in a non-linear shear field are rare. The current study calculates such interactions between two neutrally-buoyant smooth spheres moving at negligible Reynolds numbers in an unbounded wide-gap Couette flow by three-dimensional boundary-element method (BEM) simulations. Both the identical sphere-pair and the disparate sphere-pair are considered. The numerical results show that there is a preferential cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of shear in the unbounded wide-gap Couette flow that does not arise in simple shear-flow. This migration is always directed towards low-shear regions when the sphere having the larger translational velocity approaches the other sphere, and reverses towards high-shear regions when the faster sphere leads the other sphere in the plane of shear. There is also a cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. These migrations are symmetric about the point where the spheres are at the minimum separation, and are only significant when the hydrodynamic interaction of the spheres is sufficiently strong. These results show that the migration of the center-of-gravity of the sphere-pair can be attributed to the non-linearity of the shear field. The hydrodynamic interaction between the two spheres has been quantified under various conditions by the BEM simulations for both identical and disparate spheres.

Hydrodynamic interaction of two neutrally-buoyant smooth spheres suspended in plane Poiseuille flow: the BEM simulations versus the MoR approximations

The hydrodynamic interaction between two particles suspended in shear flows is fundamental to the macroscopic characterization of suspension flows. Although such interaction in quiescent or linear shear flow is well understood, studies on that in a nonlinear shear field are rare. In this study, the hydrodynamic interaction between two neutrally-buoyant smooth spheres moving at negligible Reynolds numbers in an unbounded plane Poiseuille flow has been calculated by three-dimensional boundary element method (BEM) simulations. The BEM results have been compared with the analytical results obtained with the method-of-reflection (MoR) approximations. The BEM simulations have been found to provide satisfactory predictions if the number of elements on the spheres are more than 200, whereas the MoR approximations provide satisfactory predictions only when the minimum separation between the spheres is relatively large although this MoR method has the advantage to easily calculate the hydrodynamic interaction between two spheres freely moving at negligible Reynolds numbers in unbounded quadratic flow by solving ordinary differential equations. Furthermore, it is found that there is a preferential cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of shear in plane Poiseuille flow which does not arise in simple shear flow. This migration is always directed towards low shear regions when the sphere having larger translational velocity approaches the other sphere, and reverses towards high shear regions when the faster sphere leads the other sphere in the plane of shear. There is also a cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. These migrations are symmetric about the point where the spheres are at the minimum separation, and are only significant when the hydrodynamic interaction of the spheres is strong. These results show that the migration of the center-of-gravity of the sphere-pair can be attributed to the nonlinearity of the shear field, which agrees with the MoR approximations. The hydrodynamic interaction between the two spheres has been quantified under various conditions by the BEM simulations for both identical and disparate spheres.

BEM simulation of hydrodynamic interaction between two identical neutrally buoyant smooth spheres freely moving in a wide gap Couette flow

Abstract In this paper, the hydrodynamic interaction between two identical neutrally buoyant smooth spheres freely moving at negligible Reynolds numbers in an unbounded wide gap Couette flow is investigated by three dimensional boundary element method (BEM) simulations. Such information is fundamental to the macroscopic characterization of suspension flows. The numerical results show that the hydrodynamic interaction of the sphere-pair causes both spheres to experience repulsive cross-streamline displacements, which are in opposite directions, when the trailing sphere is approaching the leading one. These cross-streamline displacements reach their maxima when the sphere-pair is at their closest center-to-center distance due to the strongest hydrodynamic interaction at this instant. After that, the initially trailing sphere becomes the leading one, and the sphere-pair begins to separate, causing the hydrodynamic interaction to decrease gradually and the spheres to return to their individual initial streamlines. The center of gravity of the sphere-pair in this non-linear shear field experiences a preferential cross-streamline migration in the plane of shear when they meet, firstly moving from regions of higher shear rates towards those of lower ones before the meeting and then experiencing a “reverse migration”, that is, moving from regions of low shear rates to those of higher ones, after the meeting, which is not presented in a linear shear field such as in a simple shear flow. There is also a cross-streamline migration of the center of gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. All these results suggest that the non-linearity of the shear field is responsible for the preferential cross-streamline displacement and the particle migration in concentrated suspensions undergoing inhomogeneous shearing. In addition, the hydrodynamic interaction between the sphere-pair has also been quantified under various conditions by the BEM simulations.

Hydromagnetic flow of fluid with variable viscosity in a uniform tube with peristalsis

We formulate this problem under an infinitely long wavelength approximation, a negligible Reynolds number and a small magnetic Reynolds number. We decide on a perturbation method of solution. The viscosity parameter α ≪ 1 is chosen as a perturbation parameter. The governing equations are developed up to first-order in the viscosity parameter (α). The zero-order system yields the classical Poiseuille flow when the Hartmann number M tends to zero. For the first-order system, we simplify a complicated group of products of Bessel functions by approximating the polynomial. The results show that the increasing magnetic field increases the pressure rise. In addition, the pressure rise increases as the viscosity parameter decreases at zero flow rate. Moreover, it is independent of the Hartmann number and viscosity parameter at certain values of flow rate. We make comparisons with other studies.

Solitary Vortex Pairs in Viscoelastic Couette Flow

We report experimental observation of a localized structure, which is of a new type for dissipative systems. It appears as a solitary vortex couple ("diwhirl") in Couette flow with highly elastic polymer solutions. A unique property of the diwhirls is that they are stationary, in contrast to the usual localized wave structures in both Hamiltonian and dissipative systems which are stabilized by wave dispersion. It is also a new object in fluid dynamics - a couple of vortices that build a single entity somewhat similar to a magnetic dipole. The diwhirls arise as a result of a purely elastic instability through a hysteretic transition at negligible Reynolds numbers. It is suggested that the vortex flow is driven by the same forces that cause the Weissenberg effect. The diwhirls have a striking asymmetry between the inflow and outflow, which is also an essential feature of the suggested elastic instability mechanism.