Linear stability of a plane poiseuille flow of oldroyd fluid

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 5–10, January–February, 1988.     

Stability of plane poiseuille flow of a viscous anisotropic fluid

Experimental investigations show that the presence in a fluid of fibers and rigid asymmetric particles leads to a greater stability of flow in tubes and lowers the turbulent frictional resistance in a certain range of Reynolds numbers [1]. In the present paper, the anisotropic structure of a fluid with additives is described by Ericksen's rheological model [2]. The parameters of the model are particularized in accordance with the paper [3] of Pilipenko, Kalinichenko, and Lemak, and in the limiting case of weak Brownian motion allowance is made for the effect of the predominant orientation of the particles and the influence of additives on the longitudinal and shear viscosity. The stability of the Poiseuille flow is considered in the linear formulation. In an anisotropic viscous fluid, an equation of Orr-Sommerfeld type has a singular point. A rule for choosing the path of integration avoiding the singular point is obtained on the basis of a generalization of the method of Dikii [4] proposed in an investigation of the stability of the flow of an ideal fluid. The results of numerical calculations of the neutral stability curve for two-dimensional perturbations are given.     

Stability of plane poiseuille flow of a highly elastic liquid

The stability of fully developed pressure driven plane laminar flow of a Maxwell fluid has been studied using linear hydrodynamic stability theory. Elasticity is destabilizing in the inertial regime, but the flow is found to be stable to infinitesimal disturbances at low Reynolds numbers. This result contradicts previous calculations, which predicted a low Reynolds number flow instability at a critical recoverable shear of order unity. The previous calculations were carried out using less accurate numerical methods; the eigenvalue problem which must be solved is a delicate one, requiring sophisticated umerical techniques in order to avoid the calculation of spurious unstable modes.This work has direct bearing on the question of the mechanism of a low Reynolds number extrusion instability known as “melt fracture”. It is observed that the intensity of melt fracture increases with increasing die length for high density polyethylene, and it is therfore believed by some experimentalists that fully-developed die flow is unstable for this polymer above a critical recoverable shear. The analysis appears to be at variance with this interpretation of the experimental results.     

Effect of particles suspended in a fluid flow on the decay of isotropic turbulence

Dynamic equations have been obtained for the two-point double correlations of the fluctuation velocities of a fluid and the particles suspended in it at low volume concentrations of the solid phase. In the case of uniform isotropic turbulence these equations can be considerably simplified. The final period of decay of isotropic turbulence has been studied in detail. At this stage in the case of high-inertia particles the inhomogeneous-fluid turbulence is similar to the turbulence of a homogeneous fluid (without particles) in the sense that the presence of the particles affects only the fluctuation energy but leaves unchanged the spatial scales of turbulence and the spatial energy spectrum function. The suspended particles lead to exponential damping of the turbulent pulsations.Little theoretical information is available on the hydrodynamics of a suspension of fine particles in a turbulent liquid or gas. Research has been mainly confined to the behavior of the individual particles in a given turbulence field [1]. The problem of the turbulent motion of the mixture as a whole has been examined by Barenblatt [2], who derived the equations of motion of the mixture, using Kolmogorov's hypothesis to close them. Hinze [3] has also attempted to derive equations for turbulent pulsations of the mixture. However, as Murray showed [4], Hinze' s equations contradict Newton' s third law.The effect of suspended particles on the turbulence of a two-phase flow is governed by the noncorrespondence of the local velocities of the particles and the medium. The forces of resistance to the motion of the particles relative to the fluid lead to additional dissipation of fluctuation energy and decay of turbulence [2]. On the other hand, if the averaged velocities of particles and medium do not correspond, the suspended particles may also have a destabilizing effect [5, 6], causing energy transfer from the averaged to the pulsating motion. Below we shall consider the case where the averaged velocities of the two phases coincide, i.e., we shall deal only with the first of the two above-mentioned effects.The authors thank G.I. Barenblatt for his useful advice.     

Effect of heat transfer on the plane-channel poiseuille flow of a thermo-viscous fluid

A steady-state plane channel flow of viscous incompressible fluid with no-slip and heat transfer boundary conditions is considered. The flow is induced by a fixed pressure difference and the fluid viscosity depends on the temperature in accordance with a power law. It is shown numerically that the dependence of the Peclet number on the nondimensional pressure difference is not single-valued. An investigation of the solution’s dependence on the Biot number shows that for Biot numbers greater than unity the velocity profile has a point of inflection. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–80, March–April, 2000. The work received financial support from the Russian Foundation for Basic Research (project N97-01-00063).     

Non linear transverse disturbances in plane poiseuille flow at low Reynolds numbers — Part I

Summary The behaviour and sign of the Reynolds stress for periodic perturbation of finite amplitude is studied in this paper for the plane Poiseuille flow. The case considered has Reynolds number 250 and wave number =1. The Reynolds stress that in the linear case was of opposite sign with respect to the viscosity, in the case considered becomes such for perturbation amplitudes which are still significant with respect to the dynamics of the mean flow. Some numerical results are given to characterize the phenomenon.

---

Sommario In questo lavoro si studia nel moto piano di Poiseuille il comportamento ed il segno dello stress di Reynolds per perturbazioni periodiche di ampiezza finita. E' trattato il caso del numero di Reynolds 250 e del numero d'onda =1. Si osserva che lo stress di Reynolds, che nel caso lineare era di segno opposto alla viscosità, diventa tale per ampiezza della perturbazione rilevante rispetto alla dinamica del moto medio. Sono dati alcuni risultati numerici che caratterizzano il fenomeno.

---

---

This work is part of a research program on hydrodynamics at the Istituto di Elaborazione dell'Informazione of the C.N.R., Pisa. The first Author has suggested and precisely formulated the problem, whereas the numerical work was carried out by the second Author.     

Influence of suspended particles on the structure of a turbulent flow in a tube

The results of an experimental investigation into the dispersed flow of a system subject to negative pressure gradients are presented. The measurements were based on an optical time-of-flight method in a water channel, using polystyrene spheres as the solid phase. The average and pulsational characteristics of the dispersed flow were obtained in the boundary (wall) region and also in the center (core) of the flow. For zero pressure gradient the influence of the solid phase expressed itself as a reduction in the level of turbulence and an increase in the extent of the viscous sublayer, leading to a fall in the coefficient of friction. For a negative pressure gradient the pressure of the solid phase generated small-scale vortices, reduced the extent of the viscous sublayer, and hence increased the coefficient of surface friction.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 111–118, March–April, 1976.The author wishes to thank Yu. A. Buevich for interest in this work and V. L. Zalukaev for participation in the experiments.     

Development of finite-amplitude perturbations in poiseuille flow

Three-dimensional distortion of initially two-dimensional perturbations has been investigated in a planar channel by means of visualization. The shapes of three-dimensional formations during the stage of breakup of a two-dimensional wave have been obtained for the first time. The formations have the same shape as the Λ-shaped vortices in the boundary layer on a flat plate, and their transverse dimension does not depend on the flow velocity or the frequency of the disturbances that are introduced.     

Heat transfer effects on the stability of low speed plane Couette-Poiseuille flow

The stability problem of low-speed plane Couette-Poiseuille flow of air under heat transfer effects is solved numerically using the linear stability theory. Stability equations obtained from two-dimensional equations of motion and their boundary conditions result in an eigenvalue problem that is solved using an efficient shoot-search technique. Variable fluid properties are accounted for both in the basic flow and the perturbation (stability) equations. A parametric study is performed in order to assess the roles of moving wall velocity and heat transfer. It is found that the moving wall velocity and the location of the critical layers play decisive roles in the instability mechanism. The flow becomes unconditionally stable whenever the moving wall velocity exceeds half of the maximum velocity in the channel. With wall heating and Mach number effects included, the flow is stabilized.     

Longwave stability of two-layer fluid flow in the inclined plane

The exact invariantOstroumov–Birikh solution of the Oberbeck–Bussinesq equationswhich describes two-layer advective thermocapillary flows in the inclined plane is analyzed. The spectrum of the characteristic perturbations of all classes of the flows is investigated and analytical representations of the eigennumbers and eigenfunctions of the corresponding spectral problem are obtained in the zeroth approximation. Stability of the flows with respect to longwave perturbations and the possibility of existence of oscillatory regimes are proved.     

Numerical experiments on stability and transition in plane channel flow

Various secondary and tertiary instabilities in plane channel flow are explored via time-dependent numerical simulations using the incompressible Navier-Stokes equations. Comparisons are made between transitional flows at Reynolds numbers 1500, 5000, and 8000. The lambda vortex, detached shear layer, and inverted vortex regions are identified and the origin of the latter is explained. The laminar breakdown of the Re=1500 flow is computed with high resolution and the nature of its ensuing hairpin eddies is clarified by numerical particle paths. The potential of center-mode rather than wall-mode transitions is proposed and the resulting flow structure is described.     

The motion of rigid rod-like particles suspended in non-homogeneous flow fields

Equations are written for the velocities of rotation and translation of rigid rod-like particles suspended in arbitrary Stokes flows. These make use of the first approximation from slender body theory for the evaluation of drag forces parallel and transverse to the particle axis, and neglect couples induced by shear stress at the particle surface. They are therefore asymptotically valid as the particle axis ratio becomes large. Simple forms of the equations, applying in constant viscosity flows, are solved, where possible analytically and otherwise numerically, and results obtained for particle motion in planar Poiseuille and sink flows. These are discussed and displayed in terms of appropriate dimensionless groups in a comprehensive set of plots, that can conveniently be used to provide information on translational and rotational velocities, and orientation and displacement as a function of time, including particle slip along and across streamlines, for a wide range of cases. In this way the effects of non-homogeneity in the flow fields are quantified.     

Effect of acoustic oscillations on the stability of a plane jet

The problem of the effect of acoustic oscillations on the stability of a compressible ideal-fluid jet flow is examined in the case of a plane jet with standing acoustic waves superimposed across it. The method of dividing the motion into fast and slow with allowance for nonlinear acoustic effects is employed. The acoustic oscillations are found to affect the growth rate of unstable hydrodynamic disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 54–60, July–August, 1991.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

The thermoconvective instability of plane poiseuille flow heated from below: A proposed benchmark solution for open boundary flows

The transient two-dimensional Navier-Stokes and energy equations have been solved numerically for flow in a horizontal channel heated from below in the Boussinesq limit. For the set of dimensionless parameters chosen, the flow consists of periodic transverse travelling waves resulting from a convective instability. The solution is proposed as a benchmark for the application of outflow boundary conditions (OBC) in time-dependent flows with strong buoyancy effects. Richardson extrapolation in both time and space is used in obtaining the solution. Field plots and profiles of velocity, temperature, vorticity and streamfunction at selected axial positions and times are also presented from the finest grid and smallest time step calculation. The calculations have been made on an extended domain so that the effects of OBC used in the present study would be negligible in the test region.