Solution to the paradox of the linear stability of the Hagen-Poiseuille flow and the viscous dissipative mechanism of the emergence of turbulence in a boundary layer

It has been shown that the conclusion of the linear instability of the Hagen-Poiseuille flow at finite Reynolds numbers requires the refusal of the use of the traditional “normal” form of the representation of disturbances, which implies the possibility of separation of variables describing disturbances as functions of the radial and longitudinal (along the axis of a tube) coordinates. In the absence of such separation of variables in the developed linear theory, it has been proposed to use a modification of the Bubnov-Galerkin theory that makes it possible to take into account the difference between the periods of the longitudinal variability for different radial modes preliminarily determined by the standard application of the Galerkin-Kantorovich method to the evolution equation of extremely small axisymmetric disturbances of the tangential component of the velocity field. It has been shown that the consideration of even two linearly interacting radial modes for the Hagen-Poiseuille flow can provide linear instability only in the presence of the mentioned conditionally periodic longitudinal variability of disturbances along the axis of the tube, when the threshold Reynolds number Re_th(p) is very sensitive to the ratio p of two longitudinal periods each describing longitudinal variability for its radial disturbance mode. In this case, the threshold Reynolds number can tend to infinity, Re_th(p) → ∞, only at p = p _k = k, p = p _1/k = 1/k, and \(p = p_{\sqrt k } = [k + 1 \pm \sqrt {(k + 1)^2 - 4} ]/2\) , where k = 1, 2, 3, …. The minimum Reynolds number Re_th(p) ≈ 448 (at which p ≈ 1.527) for the linear instability of the Hagen-Poiseuille flow quantitatively corresponds to the condition of the excitation of Tollmien-Schlichting waves in the boundary layer, where Re_th = 420. Similarity of the mechanisms of linear viscous dissipative instability for the Hagen-Poiseuille flow and Tollmien-Schlichting waves has been discussed. Good quantitative agreement has been obtained between the phase velocities of the vortex disturbances and the experimental data on the velocities of the leading and trailing edges of turbulent “puffs” propagating along the axis of the tube.     

Thermophoresis in colloidal suspensions driven by Marangoni forces

In a hydrodynamic approach to thermophoretic transport in colloidal suspensions, the solute velocity u and the solvent flow v(r) are derived from Stokes' equation, with slip boundary conditions imposed by thermal Marangoni forces. The resulting fluid velocity field v(r) significantly differs from that induced by an externally driven particle. We find, in particular, that thermophoresis due to surface forces is insensitive to hydrodynamic interactions. As a consequence, the thermal diffusion coefficient D(T) of polymer solutions is independent of molecular weight and concentration.     

Translational oscillations of a microsphere in superfluid helium

The translational motion of a microsphere (radius 100 μm) in liquid helium is investigated. The sphere is levitating inside a superconducting capacitor and oscillates about its equilibrium position. The velocity amplitude and the resonance frequency are measured as a function of driving force and temperature (0.35 K up to 2.2 K). By increasing the driving force we first find a linear regime (laminar flow) which changes abruptly into a nonlinear one (turbulent flow). For temperatures below 0.7 K the linear drag is given by ballistic roton and phonon scattering whereas for temperatures above 1.1 K the hydrodynamic force on the sphere is described by Stoke's solution. In the turbulent regime, above a temperature independent threshold velocity, we find the drag force to be given by turbulence in the superfluid component plus an essentially laminar drag by the normal component.     

Plasma expansion and current flow in a vacuum arc with a smallanode

A low-density plasma flow in a vacuum arc with a small anode, which intercepts only part of the cathodic plasma jet, was studied theoretically using a two-dimensional approximation. The plasma expansion was modeled using the sourceless steady-state hydrodynamic equations, where the free boundary of the plasma was determined by a self-consistent solution of the gasdynamic and electrical current equations. Magnetic forces from the azimuthal self-magnetic field were taken into account. The influence of the ratio of the anode radius to initial plasma jet radius on the plasma density, velocity, current distribution, and anode sheath potential drop is analyzed. It is shown that the mass and current flow in a 500 A arc are compressed near the axis. This leads to an increase in the plasma density by a factor of two and in the axial current density by a factor of 1.5     

Deformation of a tethered polymer in uniform flow

Static properties of a single polymer fixed at one end and subjected to a uniform flow field are investigated for several polymer models: the Gaussian chain, the freely jointed chain, and the FENE (Finitely Extensible Nonlinear Elastic) chain. By taking into account first the excluded-volume interaction and subsequently also the hydrodynamic interaction, the polymer models are gradually completed and the relevance of each effect for the polymer deformation can be identified. Results from computer simulations of these bead spring chains are compared with analytical calculations using either the conformational distribution function or blob models. To this end, in contrast to the blob model with non-draining blobs introduced for a tethered polymer by Brochard-Wyart, we here develop also a model with free-draining blobs. It turns out that a limited extensibility of the polymer – described by nonlinear spring forces in the model – leads to a flow velocity dependence of the end-to-end distance, segment density, etc. which agrees with the power law predictions of the blob model only for very long chains and in a narrow range of flow velocities. This result is important for comparison with recent experiments on DNA molecules which turn out to be still rather short in this respect. The relative importance of finite extensibility, the excluded-volume effect, and hydrodynamic interactions for polymers in flow is not fully understood at present. The simulation of reasonably long chains becomes possible even when fluctuating hydrodynamic interactions are taken into account without employing averaging procedures by introducing efficient numerical approximation schemes. At medium velocity of the uniform flow the polymer is partially uncoiled and simulations show that the effects of excluded-volume and hydrodynamic interactions are position-dependent. Both are stronger near the free end than near the tethered end of the polymer. A crossover from a nearly non-draining polymer at small flow velocities to a free-draining almost uncoiled chain at large velocities is found in the simulations. Accordingly, models assuming the polymer to be composed of either free- or non-draining subunits, like the two blob models, cannot correctly describe the extension and shape of a tethered polymer in flow, and simple power laws for the polymer extension, etc. cannot be expected. Received 21 June 1999     

Density-density correlations on a gelatin films surface

The spatial correlations on the surface of films obtained from aqueous solutions of gelatin during cooling from 320 to 293 K have been studied using electron microscopy. It has been shown that the total density fluctuation correlation function on the scale R > 4 nm can be represented in the form h(R) ～ R ^?n exp(?R/ζ), where the correlation radius ζ coincides with the hydrodynamic radius of a macromolecule. Unfolding of macromolecules in the coil → helix transformation leads to a decrease in the density and the fractal dimension of the physical network of pinnings of macromolecules and to variation in the index n of the power term in the function h(R) from n = 1 to 2, due to the transition from a continual type of disorder to a cellular type in a solid.     

q\bar q pair creation in a flux tube with confinement

We calculate the effect of radial confinement on the Schwinger pair production rate by solving the Dirac equation in a flux-tube cylinder containing a constant chromoelectric field in the longitudinal direction. We show how the Dirac equation separates into radial and longitudinal equations for a mass term which has an arbitrary radial dependence and introduce radial confinement by having a finite mass inside the cylinder and an infinitely large mass outside. The resulting boundary conditions are equivalent to the MIT boundary condition. The equations are solved analytically for a constant quark mass inside the flux-tube, which acts like a waveguide. The discretization of the transverse wave vector which has a continuous spectrum in the non-confined case leads to a large suppression of the Schwinger pair-production rate for small radii. The minimal radius where pairs are created decreases with increasing field strength. The suppression turns out to be larger for heavier quarks than for light quarks.     

Analysis of the formation of ion beams in varying electric fields of a stationary plasma thruster

We propose a technique for analyzing the distribution function of the velocity components (radial V _r and azimuthal V _φ) of ions in a beam. This technique is used for studying the ion beam emerging from a stationary plasma thruster (SPT). It is shown that the beam contains ions with a radial velocity component in the range V _r /V _z = ? 1.2 to +0.74, as well as ions with the azimuthal velocity component in the range V _?/V _z = ±0.9. Numerical calculations lead to the conclusion that ions acquire the azimuthal velocity component in the field of the azimuthal wave of the plasma potential evolving in the SPT channel.     

Hagen-Poiseuille Flow Solutions in Grad-Type Equations

An analysis of the Hagen-Poiseuille flow for a rarefied gas is presented using Grad’s equations and regularized equations in the 13 moment approximation, which provide a correction for the solution in the hydrodynamic regime. Slip boundary conditions are obtained through a simple wall model in which we take into account diffuse and specular wall-particle interactions. From the solution of the regularized equations, we recover as a special case, the velocity profile in Grad’s approximation and also an equivalent expression in the hydrodynamic regime. In addition, other relevant variables are calculated.     

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.     

Superlattice space-charge-wave analyses in a modified drift-diffusion model and in a simplified balance-equation formulation

Unstable space-charge-waves in superlattice miniband transport are investigated using a modified drift-diffusion model and using a simplified one-dimensional hydrodynamic balance-equation formulation with a relaxation-time approximation. We point out that the earlier widely accepted notion, that in the small wavevector limit the space-chargewave (SCW) propagates at a phase velocity equal to the carrier drift velocity and with an amplitude-growth rate equal to the negative value of the mobility frequency, -ω _c, was crucially based on the implicit assumption that the system momentum relaxation time τ_m is extremely small. Taking account of a finite momentum relaxation time, we find that even the drift-diffusion model would yield results significantly different from the above predictions: the phase velocity of a long-wavelength traveling SCW can be much slower than the carrier drift velocity and its amplitude-growth rate much smaller than -ω_c. A hydrodynamic balance-equation formulation, which properly treats energy dissipation and further reduces the amplitude-growth rate, provides a convenient tool for improved qualitative analyses in SCW-related problem in semiconductor superlattices.     

Polymer dynamics in chaotic flows with a strong shear component

We consider the dynamics of a polymer molecule injected into a chaotic flow with a strong mean shear component. The polymer experiences aperiodic tumbling in such flows. We consider a simplified model of the chaotic velocity field given by the superposition of a steady shear flow and a large-scale isotropic short-correlated random component. In the framework of this model, we present a detailed study of the statistical properties of single-polymer dynamics. We obtain the stationary probability distribution function of the polymer orientation, find the distribution of time periods between consequent events of tumbling, and find the tails of the polymer size distribution. The text was submitted by the author in English.     

Contraction and reswelling of a polymer chain near the critical point of a binary liquid mixture

We have studied the conformation change of a flexible linear polymer chain near the critical point of a binary liquid mixture using fluorescence correlation spectroscopy, which measured the hydrodynamic radius of the chains. Our results indicate that as the critical temperature (Tc) is approached, the chain size decreases. The polymer attains its most compact conformation when the correlation length of the critical fluctuations becomes comparable to the coil size. At very close to Tc, the polymer reexpands dramatically. To our knowledge, this is the first experimental evidence supporting the prediction of Brochard and de Gennes that a polymer chain will collapse and subsequently reswell on approaching Tc.     

Influence of hydrodynamics on the dynamics of a homopolymer

We present an analytical approach of the dynamics of a polymer when it is quenched from a solvent into a good or bad solvent. The dynamics is studied by means of a Langevin equation, first in the absence of hydrodynamic effect, then taking into account the hydrodynamic interactions with the solvent. The variation of the radius of gyration is studied as a function of time. In both cases, for the first stage of collapse or swelling, the evolution is described by a power law with a characteristic time proportional to N ^4/3 (N), where N is the number of monomers, without (with) hydrodynamic interactions. At larger times, scaling laws are derived for the diffusive relaxation time. Received: 10 March 1998 / Received in final form: 15 September 1998 / Accepted: 25 September 1998     

Hydrodynamic screening of star polymers in shear flow

The mutual effects of the conformations of a star polymer in simple shear flow and the deformation of the solvent flow field are investigated by a hybrid mesoscale simulation technique. We characterize the flow field near the star polymer as a function of its functionality (arm number) f . A strong screening of the imposed flow is found inside the star polymer, which increases with increasing f . To elucidate the importance of hydrodynamic screening, we compare results for hydrodynamic and random solvents. The dependence of the polymer orientation angle on the Weissenberg number shows a power law behavior with super-universal exponent --independent of hydrodynamic and excluded-volume interactions. In contrast, the polymer rotation frequency changes qualitatively when hydrodynamic interactions are switched on.     

托卡马克真空室内线圈水冷管道接头强化二次流的数值研究

基于计算流体力学(CFD)理论,研究了不同曲率半径的螺旋导流片的托卡马克真空室内线圈水冷管道接头。利用湍流数值模拟方法,分析了线圈管道接头导流片曲率半径比、冷却水入口流速对线圈管道内流体平均雷诺数分布的影响。结果表明,不同导流片曲率半径比的线圈管道内的流体雷诺数分布曲线相似,平均雷诺数随入口流速的增加而增大,管道接头出口雷诺数随导流片曲率半径比的增大而减小,导流片曲率半径比小的管接头更适用于线圈水冷曲线管的二次流强化。此外,还为导流片曲率半径比为0.2的管接头拟合了管接头出口雷诺数与入口流速的关系式,为进一步研究类似于托卡马克真空室内线圈管道的曲线管接头的二次流强化提供理论基础。     

Shear-flow-induced unfolding of polymeric globules

The behavior of a single collapsed polymer under shear flow is examined using hydrodynamic simulations and scaling arguments. Below a threshold shear rate gamma[.]{*}, the chain remains collapsed and only deforms slightly, while above gamma[.]{*} the globule exhibits unfolding/refolding cycles. Hydrodynamics are crucial: In the free draining case, gamma[.]{*} scales with the globule radius R as gamma[.]{*} approximately R{-1}, while in the presence of hydrodynamic interactions gamma[.]{*} approximately R. Experiments on the globular von Willebrand protein confirm the presence of an unfolding transition at a well-defined critical shear rate.     

Effect of longitudinal electric field on capillary instability of a thin axisymmetric layer of liquid dielectric coating a dielectric fiber

The flow of a viscous dielectric liquid surrounded with a gas is investigated in the process of capillary disintegration of a thin axisymmetric liquid layer on an undeformable cylindrical dielectric fiber in a uniform electric field is investigated. An asymptotic analysis of the system of equations and hydrodynamic boundary conditions written with allowance for surface ponderomotive forces is carried out for the case when the average thickness of the layer is much smaller than the radius of the fiber cross section. The problem of the transition of the liquid configuration from the state of a stationary cylindrical layer to the hydrodynamic state in the form of a regular sequence of drops is formulated. In this formulation, a nonlinear parabolic equation that describes the evolution of the local thickness of the layer on the time interval to the instant of drop formation is derived. The effect of the key parameters on the capillary instability is analyzed based on the linearized version of the resultant equation and the linearized electrostatic problem of calculating the field perturbations.     

Numerical description of axisymmetric blue whirls over liquid-fuel pools

This numerical investigation is focused on determining the structures of blue whirls, recently found to occur in laboratory investigations of fire whirls when the circulation becomes sufficiently large to produce a vortex breakdown that drastically shortens the fire whirl and correspondingly reduces residence times, so that the yellow flames turn blue. The computations address axisymmetric configurations for round pools of liquid fuels flush with and at the center of a larger solid horizontal disc, at the outer edge of which vanes of adjustable angles cause the entrained air to enter with a controllable azimuthal component of velocity. The nondimensionlized conservation equations employed include realistic Lewis numbers with temperature-dependent transport coefficients and a one-step chemical-kinetic approximation that correctly reproduces laminar burning velocities. Buoyancy and radiant energy transport from the flames to the liquid surface are both taken into account, the latter being found to be essential for the blue whirl. Along with the vaporization-equilibrium and energy-conservation boundary conditions at the fuel surface, inflow boundary conditions are provided by a recently developed solution for the boundary-layer flow over the solid disc, while zero-gradient outflow conditions are applied above the whirl. Controlling nondimensional parameters, besides Reynolds, Damköhler, and Froude numbers, are a ratio of radiant to convective energy flux and a ratio of azimuthal to inward radial flow velocity in the boundary layer at the edge of the disc. The computed conditions for the onset of the blue whirl, as well as the computed structure of the whirl itself, bear close resemblance to what was found experimentally.