Instability Of Plane Poiseuille Flow Research Articles

Linear stability of Poiseuille flow of viscoelastic fluid in a porous medium

We study the instability of plane Poiseuille flow of the viscoelastic second-order fluid in a homogeneous porous medium. The viscoelastic fluid between two parallel plates is driven by the pressure gradient. The effects of elasticity number E (depends on fluid properties, geometry; E is defined below) and Darcy number Da (gives the permeability of porous medium; Da is defined below) on flow stability are analyzed through the energy method that provides qualitative behavior of flow stability, and the numerical solution of generalized eigenvalue problem that gives the precise upper bound for stability. The plane Poiseuille flow of second-order fluid becomes unstable for increasing elasticity number while preserving Newtonian eigenspectrum up to a certain range of E. For large elasticity number, instability appears as a part of both wall and center modes for all Darcy numbers. We also noticed that along each neutral stability curve, the eigenfunctions are all antisymmetric with a single extremum near the channel walls. When E = 0.0011, we found an additional new elastic mode, which is unstable and also antisymmetric. For E < 0.0011, the neutral curves split into two lobes with different minima. The critical Reynolds number Rec is found to be decreasing (increasing) for higher (lower) values of fluid elasticity (Darcy number). Physical mechanisms are discussed in detail.

Induction Mechanism of Tollmien–Shlichting Waves in Instability of Plane Poiseuille Flow

The induction mechanism of Tollmien–Schlichting waves (T–S waves) in instability of plane Poiseuille flow is investigated by analyzing the solution to the eigenvalue problem for the Orr–Sommerfeld ...

Linear stability of shear-thinning fluids in deformable channels: Effect of inertial terms

In the present work, the effect of inertial terms is numerically investigated on the hydroelastic stability of shear-thinning fluids in pressure-driven flow between two parallel plates lined with a Mooney–Rivlin hyperelastic polymeric gel. Having obtained the basic flow for the fluid and the basic deformation for the solid, the vulnerability of the basic solutions are examined when subjected to infinitesimally-small, normal-mode perturbations. By dropping all nonlinear perturbation terms, an eigenvalue problem is obtained, in the form of two coupled fourth-order ODEs, which are solved using the shooting or spectral methods. Based on the numerical results obtained in this work, it is concluded that the Mooney–Rivlin gels having a negative first-normal-stress-difference are more stable than the neo-Hookean gels as far as hydroelastic instability in plane Poiseuille flow is concerned. While under creeping-flow conditions the effect of shear-thinning is predicted to be stabilizing, in inertial flows its effect is predicted to be stabilizing or destabilizing depending on the Reynolds number.

Instability of Plane Poiseuille Flow in Viscous Compressible Gas

Instability of plane Poiseuille flow in viscous compressible gas is investigated. A condition for the Reynolds and Mach numbers is given in order for plane Poiseuille flow to be unstable. It turns out that plane Poiseuille flow is unstable for Reynolds numbers much less than the critical Reynolds number for the incompressible flow when the Mach number is suitably large. It is proved by the analytic perturbation theory that the linearized operator around plane Poiseuille flow has eigenvalues with positive real part when the instability condition for the Reynolds and Mach numbers is satisfied.

Instability of plane Poiseuille flow in a fluid-porous system

The instability of Poiseuille flow in a fluid-porous system is investigated. The system consists of a fluid layer overlying porous media and is subjected to a horizontal plane Poiseuille flow. We use Brinkman’s model instead of Darcy’s law to describe the porous layer. The eigenvalue problem is solved by means of a Chebyshev collocation method. We study the influence of the depth ratio d̂ and the Darcy number δ on the instability of the system. We compare systematically the instability of Brinkman’s model with the results of Darcy’s model. Our results show that no satisfactory agreement between Brinkman’s model and Darcy’s model is obtained for the instability of a fluid-porous system. We also examine the instability of Darcy’s model. A particular comparison with early work is made. We find that a multivalued region may present in the (k,Re) plane, which was neglected in previous work. Here k is the dimensionless wavenumber and Re is the Reynolds number.

Hydromagnetic linear instability analysis of Giesekus fluids in plane Poiseuille flow

The effects of a fluid’s elasticity are investigated on the instability of plane Poiseuille flow on the presence of a transverse magnetic field. To determine the critical Reynolds number as a function of the Weissenberg number, a two-dimensional linear temporal stability analysis will be used assuming that the viscoelastic fluid obeys Giesekus model as its constitutive equation. Neglecting terms nonlinear in the perturbation quantities, an eigenvalue problem is obtained which is solved numerically by using the Chebyshev collocation method. Based on the results obtained in this work, fluid’s elasticity is predicted to have a stabilizing or destabilizing effect depending on the Weissenberg number being smaller or larger than one. Similarly, solvent viscosity and also the mobility factor are both found to have a stabilizing or destabilizing effect depending on their magnitude being smaller or larger than a critical value. In contrast, the effect of the magnetic field is predicted to be always stabilizing.

Improving spatial resolution characteristics of finite difference and finite volume schemes by approximate deconvolution pre-processing

A pre-processing step is proposed as a general method to enhance resolution properties of low order numerical differentiation and interpolation. Pre-processing operators are designed by taking two or more terms in the approximate deconvolution formula and using a local filter whose response characteristics are close to those of the numerical operation considered; operators for second order central differencing for first and second derivatives and also for a finite volume method are determined. In addition to the higher resolution the effective order of the truncation error can also be increased. The repeated filtering operations couple the operation over a wider stencil without using direct formulas. The effect of improving the resolution properties is illustrated first by computing the propagation of a square wave by a 1D, linear convection equation. Next, pre-processing is implemented in a standard finite volume code for solving Navier–Stokes equations for incompressible flow. In the Taylor–Green vortex flow, the improvements in order behaviour are demonstrated. The instability of plane Poiseuille flow, which is a sensitive test of resolution ability, shows that the predicted growth rates with the pre-processing scheme are more accurate than those obtained with a second order scheme. Direct numerical simulations of turbulent channel flow show that the improvement allows accurate solutions to be found on smaller grids.

Eigenvalue Bounds for the Orr-Sommerfeld Equation and Their Relevance to the Existence of Backward Wave Motion

Theoretical estimates of the phase velocity cr of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr–Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow or nearly parallel viscous shear flows in straight channels with velocity U(z) (=1−z2, z∈[−1, +1] for plane Poiseuille flow), leave open the possibility that these phase velocities lie outside the range Umin<cr<Umax but not a single experimental or numerical investigation, concerned with unstable waves in the context of flows with (d2U/dz2)max≤0, has supported such a possibility as yet. Umin, Umax and (d2U/dz2)max are, respectively, the minimum value of U(z), the maximum value of U(z), and the maximum value of (d2U/dz2) for z∈[−1, +1]. This gap between the theory on one hand and experiment and computation on the other has remained unexplained ever since Joseph [3] derived these estimates, first in 1968, and has even led to the speculation of a negative phase velocity in plane Poiseuille flow (i.e., cr<Umin=0) and hence the possibility of a “backward” wave as in Jeffrey-Hamel flow in a diverging channel with backflow [1]. A simple mathematical proof of the nonexistence of such a possibility is given herein by showing that if (d2U/dz2)max≤0 and (d4U/dz4)min≥0 for z∈[−1, +1], then the phase velocity cr of an arbitrary unstable wave must satisfy the inequality Umin<cr<Umax, (d4U/dz4)min is the minimum value of (d4U/dz4) for z∈[−1, +1], and therefore cr cannot be negative when Umin=0. Another result that provides valuable insight into the general modal structure of the problem of instability of the above class of flows with Umin≥0 (e.g., plane Poiseuille flow) is that all standing waves, that is, modes for which cr=0, are stable.

Instabilities in plane Poiseuille flow due to the combined effects of stratification and viscosity

The stability of stably stratified plane Poiseueille flow is considered in the case when the buoyancy frequency is given by N2(z)=z4. It is well known that the Tollmien–Schlichting mode of instability for this flow becomes completely stable as the Richardson number Ri↑0.222. For larger values of Ri, however, new modes of instability appear which are of the “center” or “fast” type. These new modes are due to the combined effects of stratification and viscosity which are strongly destabilizing and lead to a minimum critical Reynolds number of about 124. Detailed stability characteristics in the form of curves of marginal stability are presented for various values of the Richardson and Prandtl numbers.

Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

Theoretical estimates of the phase velocityC r of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr-Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow (U(z)=1−z 2,−1≤z≤+1), leave open the possibility of these phase velocities lying outside the rangeU min<C r <U max, but not a single experimental or numerical investigation in this regard, which are concerned with unstable or marginally stable waves has supported such a possibility as yet,U min andU max being respectively the minimum and the maximum value ofU(z) forz∈[−1, +1]. This gap between the theory on one side and the experiment and computation on the other has remained unexplained ever since Joseph derived these estimates, first, in 1968, and has even led to the speculation of a negative phase velocity (or rather,C r <U min=0) and hence the possibility of a ‘backward’ wave as in the case of the Jeffery-Hamel flow in a diverging channel with back flow ([1]). A simple mathematical proof of the non-existence of such a possibility is given herein by showing that the phase velocityC r of an arbitrary unstable or marginally stable wave must satisfy the inequalityU min<C r <U max. It follows as a consequence stated here in this explicit form for the first time to the best of our knowledge, that ‘overstability’ and not the ‘principle of exchange of stabilities’ is valid for the problem of plane Poiseuille flow.

Interfacial instabilities in plane Poiseuille flow of two stratified viscoelastic fluids with heat transfer. Part 1. Evolution equation and stability analysis

An evolution equation is derived that describes the nonlinear development of the interface between two viscoelastic fluids flowing, under the action of imposed pressure gradient and gravity, in a vertical channel. The channel walls are kept at different temperatures, resulting in heat transfer across the layers. The equation, based on the lubrication approximation, models the effects of stratifications in density, viscosity, elasticity, shear thinning, and thermal conductivity. It also describes the capillary and thermocapillary effects, as well as the sensitivity of viscosities to temperature. Linear-stability analysis is performed based on the evolution equation to understand the competing effects of viscous, elastic, and Marangoni instabilities. Particular attention is paid to the active control of the interfacial instabilities through the thermocapillarity.

Molecular orientation and instability in plane Poiseuille flow of a liquid-crystalline polymer

A common rheological hypothesis, that the stress in a fluid element is only a function of its own deformation history, is rendered questionable in liquid-crystalline polymers (LCPs) due to the presence of distortional elasticity, through which neighboring fluid elements may directly influence one another. However, the fine defect texture in LCPs has led to the suggestion that fluid properties may be averaged over a mesoscopic length scale, intermediate between the molecular and macroscopic, so that averaged measures of fluid structure and stress at this scale depend only on their own deformation history [R. G. Larson and M. Doi, J. Rheol. 35, 539 (1991)]. We describe an experimental test of this hypothesis. If true, it should be possible to use rheological and rheo-optical data obtained in simple shear flow to predict the velocity and molecular orientation fields in a nonhomogeneous shear flow. Quantitative flow birefringence experiments are conducted on a liquid-crystalline solution of poly(benzyl glutamate), in plane Poiseuille flow. At low flow rates, the data support the local response hypothesis. As flow rate is increased, however, a profound instability occurs that is unanticipated based on behavior reported in homogeneous simple shear flow. The instability is characterized by large wavelength disturbances in structure oriented perpendicular to the flow direction that are clearly visible to the naked eye. With increasing flow rate, these structures decrease in size and become increasingly chaotic. Despite the onset of the instability, time-averaged measurements of average orientation may be qualitatively predicted based on simple shear flow data.

Numerical simulation of transient turbulent flows by the vorticity-vector potential formulation

Abstract A new eighth-order accurate method is presented for simulating directly turbulent shear flows. In the present method the vorticity-vector potential formulation is adopted in order to satisfy the equation of continuity automatically. The Poisson equations for the vector potentials are discretized also with eighth-order accurate method and the higher-order multi-grid method is employed. The accuracy and efficiency of the present Poisson solver are inspected in detail. The present method is, at the outset, applied to computations of cubic cavity flow and the result shows a striking agreement with that using the pseudo-spectral method. After having shown that the present method predicts accurately the three-dimensional instability of plane Poiseuille flow, the numerical simulation of the transient turbulent flow in a plane channel is dealt with in the 653 grid. The result shows an excellent agreement with that using the pseudo-spectral method; so the present computational method is suitable for the numerical simulation of shear flow turbulences.

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

AbstractThe 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.

On the instability of the flow in a squeeze lubrication film

When two parallel plates move normal to each other with a slow time-dependent speed, the velocity field developed in the intervening film of fluid is approximately that of plane Poiseuille flow, except that the magnitude of the velocity is dependent on time and on the coordinate parallel to the planes. This fact is intrinsic to Reynolds’ lubrication theory, and can be shown to follow from the Navier-Stokes equations when both the modified Reynolds number ( Re M ) and an aspect ratio ( δ ) are small. The modified Reynolds number is the product of δ and an actual Reynolds number ( Re ), which is based on the gap between the planes and on a characteristic velocity. The occurrence of flow instability and of turbulence in the film depend on Re . Typical values of Re , which are known to be required for the linear instability of plane Poiseuille flow, are of order 6000. This condition can be achieved, even if Re M is of order 1, provided that δ is of order 10 -4 . Such parameter values are typical of lubrication problems. The Orr-Sommerfeld equation governing flow instability is derived in this paper by use of the WKBJ technique, δ being the approximate small parameter to represent the small length-scale of the disturbance oscillations compared with the larger scale of the basic laminar flow. However, the coefficients in the Orr-Sommerfeld equation depend on slow space and time variables. Consequently the eigenrelation, derivable from the Orr-Sommerfeld equation and the associated boundary conditions, constitutes a nonlinear first-order partial differential equation for a phase function. This equation is solved by use of Charpit’s method for certain special forms of the time-dependent gap between the planes, followed by detailed numerical calculations. The relation between time-dependence and flow instability is delineated by the calculated results. In detail the nature of the instability can be described as follows. We consider a disturbance wave at or near a particular station, the initial distribution of amplitude being gaussian in the slow coordinate parallel to the planes. In the context of the Orr-Sommerfeld equation and its eigenrelation, the particular station implies an equivalent Reynolds number, while the initial distribution of the disturbance wave implies an equivalent wavenumber. As time increases, the disturbance wave can be considered to move in the instability diagram of equivalent wavenumber against Reynolds number, in the sense that these parameters are time- and space-dependent for the evolution of the disturbance-wave system. For our detailed calculations we use a quadratic approximation to the eigenrelation, an approximation which is quite accurate. If the initial distribution implies a point within the neutral curve, when the plates are squeezed together the equivalent wavenumber falls while the equivalent Reynolds number rises, and amplification takes place until the lower branch of the neutral curve is nearly crossed. If the plates are pulled apart (dilatation) the equivalent wavenumber rises, while the Reynolds number drops, and amplification takes place until the upper branch of the neutral curve has been just crossed. In the case of dilatation the transition from amplification to damping takes place more quickly than for the case of squeezing, in part due to the geometry of the neutral curve.

Numerical simulation of three-dimensional unsteady viscous flow using higher order accurate difference method.

A new direct simulation scheme using the eighth order accurate difference method is presented for three-dimensional instability of plane Poiseuille flow. The vorticity transport equation is transformed into a set of ordinary differential equations in time by using modified differential quadra-ture (MDQ) method for spatial discretization. The time integration is carried out by Runge-Kutta-Gill (RKG) scheme. The Poisson equations with respect to the vector potential are also discretized by MDQ method and solved by the iteration method. The multiple grid method, in addition, is employed to accelerate its convergence. The present method is compared with other method based on the primitive variables for certification. And further, our results of the amplitude decay and oscillation frequency of the disturbance, agree with those of pseudo-spectral method by Orszag, and the accuracy and efficiency of the present method is shown.

On the secondary instability in plane Poiseuille flow

Numerical experiments were performed to clarify apparent differences between experimental observations and a theoretical prediction of the secondary instability in plane Poiseuille flow. It is shown that subharmonic breakdown is unlikely in natural transition as a result of the initial growth of what we call the ‘‘minus’’ modes and consequent forcing of Orr–Sommerfeld modes present in the background noise. Subharmonic breakdown was achieved only when these minus modes were continuously suppressed.

Orientational instabilities in plane poiseuille flow of certain nematic liquid crystals

This paper obtains numerical solutions of the continuum equations for nematic liquid crystals for flow in a channel under a pressure gradient, and examines their stability with respect to certain disturbances. The analysis includes both planar and homeotropic initial alignments, and also cases in which the viscous coefficient α 3 is positive as well as negative. Our main objective is the calculation of instability thresholds for particular perturbations both in and out of the shear plane, and our predictions are compared with relevant experimental observations.

Thermal instability of plane poiseuille flow in the thermal entrance region

The onset of thermal convection in plane Poiseuille flow is investigated theoretically. New stability equations are derived by using the propagation theory considering the variations of disturbance amplitudes in the main flow direction. In the thermal entrance region an analytical procedure to predict the critical conditions for extremely small Prandtl-number fluids is described, based on the local similarity. For xc≤0.01 the critical Rayleigh numbers are well represented in the whole domain of the Prandtl number by Rac = 200(1 + 0.123Pr-1)RaC=200(1+0.123Pr−1)xC−1 under the conventional boundary layer theory. It is of much interest that the time-independent, three dimensional disturbances become more stable with a decrease in the Prandil number.

The convective nature of instability in plane Poiseuille flow

By numerical solution of the Orr–Sommerfeld equation for complex frequency and complex wavenumber for a wide range of Reynolds numbers R and by asymptotic analysis for large R, it is shown that there is no absolute instability in a two-dimensional plane Poiseuille flow for any R and that the flow is convectively unstable for Rc &amp;lt;R&amp;lt;∞, where Rc is the critical Reynolds number.