Asymptotic analysis of wall modes in a flexible tube revisited

The stability of wall modes in fluid flow through a flexible tube of radius R surrounded by a viscoelastic material in the region R < r < HR is analysed using a combination of asymptotic and numerical methods. The fluid is Newtonian, while the flexible wall is modelled as an incompressible viscoelastic solid. In the limit of high Reynolds number (Re), the vorticity of the wall modes is confined to a region of thickness O(Re ^-1/3) in the fluid near the wall of the tube. Previous numerical studies on the stability of Hagen-Poiseuille flow in a flexible tube to axisymmetric disturbances have shown that the flow could be unstable in the limit of high Re, while previous high Reynolds number asymptotic analyses have revealed only stable modes. To resolve this discrepancy, the present work re-examines the asymptotic analysis of wall modes in a flexible tube using a new set of scaling assumptions. It is shown that wall modes in Hagen-Poiseuille flow in a flexible tube are indeed unstable in the limit of high Re in the scaling regime Re∼Σ^3/4. Here Σ is a nondimensional parameter characterising the elasticity of the wall, and Σ≡ρGR ²/η², where ρ and η are the density and viscosity of the fluid, and G is the shear modulus of the wall medium. The results from the present asymptotic analysis are in excellent agreement with the previous numerical results. Importantly, the present work shows that the different types of unstable modes at high Reynolds number reported in previous numerical studies are qualitatively the same: they all belong to the class of unstable wall modes predicted in this paper. Received 12 June 2000 and Received in final form 8 November 2000     

Stability of fluid flow past a membrane

The stability of the flow of a fluid past a solid membrane of infinitesimal thickness is investigated using a linear stability analysis. The system consists of two fluids of thicknesses R and H R and bounded by rigid walls moving with velocities and , and separated by a membrane of infinitesimal thickness which is flat in the unperturbed state. The fluids are described by the Navier-Stokes equations, while the constitutive equation for the membrane incorporates the surface tension, and the effect of curvature elasticity is also examined for a membrane with no surface tension. The stability of the system depends on the dimensionless strain rates and in the two fluids, which are defined as and for a membrane with surface tension , and and for a membrane with zero surface tension and curvature elasticity K. In the absence of fluid inertia, the perturbations are always stable. In the limit , the decay rate of the perturbations is O(k ³ ) smaller than the frequency of the fluctuations. The effect of fluid inertia in this limit is incorporated using a small wave number asymptotic analysis, and it is found that there is a correction of smaller than the leading order frequency due to inertial effects. This correction causes long wave fluctuations to be unstable for certain values of the ratio of strain rates and ratio of thicknesses H. The stability of the system at finite Reynolds number was calculated using numerical techniques for the case where the strain rate in one of the fluids is zero. The stability depends on the Reynolds number for the fluid with the non-zero strain rate, and the parameter , where is the surface tension of the membrane. It is found that the Reynolds number for the transition from stable to unstable modes, , first increases with , undergoes a turning point and a further increase in the results in a decrease in . This indicates that there are unstable perturbations only in a finite domain in the plane, and perturbations are always stable outside this domain. Received: 29 May 1997 / Revised: 9 October 1997 / Accepted: 26 November 1997     

Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces

The stability of linear shear flow of a Newtonian fluid past a flexible membrane is analysed in the limit of low Reynolds number as well as in the intermediate Reynolds number regime for two different membrane models. The objective of this paper is to demonstrate the importance of tangential motion in the membrane on the stability characteristics of the shear flow. The first model assumes the wall to be a “spring-backed” plate membrane, and the displacement of the wall is phenomenologically related in a linear manner to the change in the fluid stresses at the wall. In the second model, the membrane is assumed to be a two-dimensional compressible viscoelastic sheet of infinitesimal thickness, in which the constitutive relation for the shear stress contains an elastic part that depends on the local displacement field and a viscous component that depends on the local velocity in the membrane. The stability characteristics of the laminar flow in the limit of low are crucially dependent on the tangential motion in the membrane wall. In both cases, the flow is stable in the low Reynolds number limit in the absence of tangential motion in the membrane. However, the presence of tangential motion in the membrane destabilises the shear flow even in the absence of fluid inertia. In this case, the non-dimensional velocity (Λ_t) required for unstable fluctuations is proportional to the wavenumber k ( Λ _t∼k) in the plate membrane type of wall while it scales as k² in the viscoelastic membrane type of wall ( Λ _t∼k ²) in the limit k→ 0. The results of the low Reynolds number analysis are extended numerically to the intermediate Reynolds number regime for the case of a viscoelastic membrane. The numerical results show that for a given set of wall parameters, the flow is unstable only in a finite range of Reynolds number, and it is stable in the limit of large Reynolds number. Received 8 November 2000 and Received in final form 20 March 2001     

Velocity measurement of a settling sphere

We study experimentally the motion of a solid sphere settling under gravity in a fluid at rest. The particle velocity is measured with a new acoustic method. Variations of the sphere size and density allow measurements at Reynolds numbers, based on limit velocity, between 40 and 7 000. At all Reynolds numbers, our observations are consistent with the presence of a memory-dependent force acting on the particle. At short times it has a t ^-1/2 behaviour as predicted by the unsteady Stokes equations and as observed in numerical simulations. At long times, the decay of the memory (Basset) force is better fitted by an exponential behaviour. Comparison of the dynamics of spheres of different densities for the same Reynolds number show that the density is an important control parameter. Light spheres show transitory oscillations at Re∼ 400, but reach a constant limit speed. Received 12 April 2000 and Received in final form 13 July 2000     

Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection

We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow vs. convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque G = T/ν² L, where L is the cylinder length, scales with Reynolds number R and gap width η, G = 1.46η^3/2(1 - η)^-7/4 R ^3/2. At larger Reynolds number, velocity fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the torque versus Reynolds. Instead, we obtain logarithmic corrections to the classical ultra-hard (exponent 2) regimes: G = 0.50 . These predictions are found to be in excellent agreement with avail-able experimental data. Predictions for scaling of velocity fluctuations are also provided. Received 7 June 2001 and Received in final form 7 December 2001     

Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

The fluctuation-induced attractive interaction of rigid flat objects embedded in a fluid membrane is calculated for a pair of parallel strips and a pair of equal circular disks. Assuming flat boundary conditions, we derive the interaction from the entropy of the suppressed boundary angle fluctuation modes. Each mode entropy is computed in two ways: from the boundary angles themselves and from the mean-curvature mode functions. A formula for the entropy loss of suppressing one or more mean-curvature modes is developed and applied. For the pair of disks we recover the result of Goulian et al. and Golestanian et al. in a direct manner, avoiding any mappings by Hubbard-Stratonovitch transformations. The mode-by-mode agreement of the two computed entropies in both systems confirms an earlier claim that mean curvature is the natural measure of integration for fluid membranes. Received 15 December 2000     

Effects of upstream roughness and Reynolds number on separated and reattached turbulent flow

This paper presents experimental investigation of upstream roughness and Reynolds number effects on the recirculation region over a smooth forward facing step. The upstream rough wall was produced from 1.5 mm sand grains and the Reynolds number based on step height, Re_h, was varied from 2040 to 9130 for both the upstream smooth and rough walls. For the smooth wall, the reattachment length increased monotonically with Re_h to an asymptotic value of 2.2 step heights for Re_h ≥ 6380. Upstream roughness reduced the reattachment length by 44% because of larger momentum deficit and higher turbulence level in the rough wall boundary layer. The mean velocities and Reynolds stresses were also reduced by roughness. The Reynolds shear stress and production of turbulent kinetic energy showed high negative values at the leading edge of the step indicating counter-gradient diffusion. The implications of these results for standard eddy viscosity models are discussed.     

Simulation of homogeneous turbulent shear flows at higher Reynolds numbers: numerical challenges and a remedy

In a recent study, Isaza and Collins [J. Fluid Mech., 637 (2009), pp. 213–239] found the asymptotic state of homogeneous turbulent shear flows (HTSFs) to be sensitively dependent on the initial shear parameter (), and yet be almost independent of the initial Reynolds number (R _λ≡qλ/ν). The stringent resolution criteria they employed, however, restricted their studies to relatively low Reynolds numbers. In this paper, we present higher resolution direct numerical simulations of HTSFs over a wider range of Reynolds numbers, aided in part by an improved parallelisation scheme that utilises two-dimensional domain decomposition. We maximise the time-window for our simulations by determining appropriate settings for the initial energy spectrum, viscosity and domain configuration, thereby ensuring that we attain the highest possible asymptotic Reynolds number at the chosen grid resolution. In the course of our study, we find that the pseudo-spectral method suffers from Gibbs oscillations while resolving the thin vortical structures that tend to form in HTSFs. The nonlinear growth of these oscillations leads to spurious energy buildup in the high-wavenumber region of the spectrum, and contaminates the flow field. Consequently, the growth of the integral length scale is found to be numerically stunted, well before the intended final Reynolds number is attained. The issue is rectified by the application of exponential-type spectral filters, which stabilise the simulations and extend the runtime window, permitting attainment of larger asymptotic Reynolds numbers. Various large-scale flow statistics are then studied, and their dependence on the initial value of the shear parameter and Reynolds number corroborates the findings of Isaza and Collins.     

The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: Application of an endoscope

This Letter discusses the influence of heat transfer and magnetic field on the peristaltic flow of Newtonian fluid in a vertical annulus under a zero Reynolds number and long wavelength approximation. The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. The flow is investigated in a wave frame of reference moving with velocity of the wave. Numerical calculations are carried out for the pressure rise and frictional forces. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail.     

On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

Understanding of the structure of turbulent flows at extreme Reynolds numbers (Re) is relevant because of several reasons: almost all turbulence theories are only valid in the high Re limit, and most turbulent flows of practical relevance are characterized by very high Re. Specific questions about wall-bounded turbulent flows at extreme Re concern the asymptotic laws of the mean velocity and turbulence statistics, their universality, the convergence of statistics towards their asymptotic profiles, and the overall physical flow organization. In extension of recent studies focusing on the mean flow at moderate and relatively high Re, the latter questions are addressed with respect to three canonical wall-bounded flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Main results reported here are the asymptotic logarithmic law for the mean velocity and corresponding scale-separation laws for bulk flow properties, the Reynolds shear stress, the turbulence production and turbulent viscosity. A scaling analysis indicates that the establishment of a self-similar turbulence state is the condition for the development of a strict logarithmic velocity profile. The resulting overall physical flow structure at extreme Re is discussed.     

圆管内潜热型功能流体对流换热的实验研究

本文实验研究了由正十四烷和尿素甲醛树脂制成的相变微胶囊和水混合组成的潜热型功能流体流过等热流圆管时的对流换热特性。相变微胶囊的加入可以显著增强流体与壁面间的对流换热,显著降低壁面温度和流体温度;在融化段对流换热系数呈增加分布,流体和壁面温度各自基本稳定在相应的温度值。强化对流换热的效果主要在融化段,并随流体中相变微胶囊浓度的增大而增强,也随R-eynolds数的增大而增强。     

Heat transfer by laminar flow of an elastico-viscous liquid along a plane wall with periodic suction

The problem of heat transfer by the laminar flow of an elastico-viscous liquid along a plane wall with periodic suction has been considered. A perturbation technique has been used to obtain an approximate solution of the differential equations. The flow phenomenon has been characterized by the non-dimensional parameters like the elastic number (S), the Reynolds number (R), the Prandtl number (P) and the Eckert number (E). The effects of these parameters on the temperature distributions and the rate of heat transfer at the wall have been studied.     

Flow of Newtonian Fluid in Non-Uniform Tubes with Application to Renal Flow: A Numerical Study

In this paper, a numerical method employing a finite difference technique is used for an investigation of viscous, incompressible fluid flow in a tube with absorbing wall and slowly varying cross-section. The effect of fluid absorption through permeable wall is accounted by prescribing flux as a function of axial distance. The method is not restricted by the parameters in the problem such as wave number, permeability parameter, amplitude ratio and Reynolds number. The effects of these parameters on the radial velocity and mean pressure drop are studied and the results are presented graphically. Comparison is also made between the results obtained by perturbation method of solution and present approach.     

Experimental Investigation of Mixed Convection in a Channel With an Open Cavity

Mixed convection in an open cavity with a heated wall bounded by a horizontally unheated plate is investigated experimentally. The cavity has the heated wall on the inflow side. Mixed convection fluid flow and heat transfer within the cavity is governed by the buoyancy parameter, Richardson number (Ri), and Reynolds number (Re). The results are reported in terms of wall temperature profiles of the heated wall and flow visualization for Re = 100 and 1000, Ri in the range 30–110 (for Re = 1000) and 2800–8700 (for Re = 100), the ratio of the length to the height of cavity (L/D) is in the range 0.5–1.5, and the ratio of the channel height to cavity height (H/D) is in the range of 0.5 and 1.0. The present results show that the maximum dimensional temperature rise values decrease as the Reynolds and the Richardson numbers decrease. The flow visualization points out that for Re = 1000 there are two nearly distinct fluid motions: a parallel forced flow in the channel and a recirculation flow inside the cavity. For Re = 100 the effect of a stronger buoyancy determines a penetration of thermal plume from the heated plate wall into the upper channel. Nusselt numbers increase when L/D increase in the considered range of Richardson numbers.     

Aging kinetics of porous media due to freezing-thawing cycles

The two-dimensional Ausloos et al. model of fluid invasion, freezing and thawing in a porous medium is elaborated upon and investigated in order to take into account the pore volume redistribution and conservation during freezing. The results are qualitatively different from previous work, since the damaged pore sizes are found to be much less than the possible maximum value and is reached after a large number of invasion-freezing-thawing cycles, e.g. the material is “slowly damaged”. The pore size distribution is thus found in better agreement with expected practical findings. The successive invasion percolation clusters are still found to be self-avoiding with aging. The cluster size decreases with a power law as a function of invasion-frost-thaw iterations. The aging kinetics is also discussed through the normalized totally invaded pore volume. Received 24 September 1999 and Received in final form 5 January 2000     

Asymptotic Analysis of a Bingham Fluid in a Thin Domain with Fourier and Tresca Boundary Conditions

In this paper we prove first the existence and uniqueness results for the weak solution, to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition; then we study the asymptotic analysis when one dimension of the fluid domain tends to zero. The strong convergence of the velocity is proved, and a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.     

Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope

This paper discusses the influence of an endoscope on the peristaltic flow of a couple stress fluid in an annulus under a zero Reynolds number and long wavelength approximation. The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. Analytical expressions for the axial velocity, stream function and axial pressure gradient are established. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Numerical calculations are carried out for the pressure rise, frictional forces and trapping. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail.     

Three-sphere low-Reynolds-number swimmer with a cargo container

A recently introduced model for an autonomous swimmer at low Reynolds number that is comprised of three spheres connected by two arms is considered when one of the spheres has a large radius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of this system can be studied analytically using the Stokes Green's function of a point force in front of a sphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shown to scale as 1/R ³ with the radius of the sphere.