Stability Of Plane-parallel Flow Research Articles

Stability of plane parallel flow revisited for particle-fluid suspensions

Abstract An alternative model is proposed for hydrodynamic stability of plane parallel flow of an incompressible Newtonian fluids with suspended solid particles. For heavy particle-laden dusty gases with negligible particle volume fraction, the effective complex-form mean velocity in the modified Orr-Sommerfeld equation derived by the present model is showed to be essentially identical to the well-known Saffman's classical results. In the limit cases of small or large Stokes number of particles, simple formulas are derived for the effective Reynolds number ratio of the particle-laden suspension to the clear fluid without particles under otherwise identical conditions. The derived formula for particles of finite particle-to-fluid density ratio and small Stokes number is verified by comparing predicted results with known data, although a comparison of the derived formula with known results for particles of finite density ratio and large Stokes number cannot be made here due to the lack of available data. It is hoped that the present work could offer a conceptually novel and relatively simplified model for hydrodynamics of solid particle-fluid suspensions.

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

A conjecture on the least stable mode for the energy stability of plane parallel flows

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

Instability of plane-parallel flow of incompressible liquid over a saturated porous medium.

The linear stability of plane-parallel flow of an incompressible viscous fluid over a saturated porous layer is studied to model the instability of water flow in a river over aquatic plants. The saturated porous layer is bounded from below by a rigid plate and the pure fluid layer has a free, undeformable upper boundary. A small inclination of the layers is imposed to simulate the riverbed slope. The layers are inclined at a small angle to the horizon. The problem is studied within two models: the Brinkman model with the boundary conditions by Ochoa-Tapia and Whitaker at the interface, and the Darcy-Forchheimer model with the conditions by Beavers and Joseph. The neutral curves and critical Reynolds numbers are calculated for various porous layer permeabilities and relative thicknesses of the porous layer. The results obtained within the two models are compared and analyzed.

Rayleigh Instability of Plane-parallel Liquid Flows

Studying in this paper the stability of plane-parallel flows of an ordinary liquid can be naturally translated into the language of the theory of hydrodynamic resonances. Thus, resonant absorption of oscillations induces stability of the flows of an ideal liquid having a velocity profile without inflection points (Rayleigh theorem), while resonant emission leads to Rayleigh instability in the presence of an inflection point. The flow velocity profile has an inflection point. Thus, the presence of inflection points is a necessary condition for instability. If, however, the velocity profile has inflection points, the flow is stable (Rayleigh's theorem). Note that the sign of the jump depends on whether the neutral oscillations are regarded as the limiting cases of growing or damped oscillations.

On the problem of the stability of convective two-phase medium flows under high-frequency vibration

The principles and methods of constructing a model of vibrational convection in a medium consisting of a liquid (gas) and a solid admixture are discussed. A closed system of averaged equations is first obtained. The system admits passage to the limits of the equations of both vibrational convection in a homogeneous fluid and convection in a dusty medium in the static case. As a measure of the difference with respect to the homogeneous fluid, in addition to the sedimentation parameter, which also manifests itself in the absence of vibrational accelerations, it is possible to take the inhomogeneity parameter introduced in this study and responsible for the pulsatory transport of the average fields. The problem of the stability of plane parallel flow in a vertical layer of a two-phase medium under horizontal longitudinal vibration with respect to infinitesimal perturbations is considered. It is shown that the introduction of particles into the flow leads to qualitatively novel effects which cannot be predicted within the framework of the homogeneous fluid model.

Instability of combined convective flow in a vertical layer

In the present paper, a study is made of the stability of plane-parallel flow induced by a transverse temperature difference between the boundaries of a layer and a longitudinal pressure gradient. This problem was solved earlier by the author [3] in a purely hydrodynamic formulation without allowance for thermal factors; the results then obtained correspond to the limiting case of small Prandtl numbers. In the paper, a numerical solution to the problem with the complete formulation is given.

Stability of plane parallel flows in a magnetized plasma

We present an investigation of the problem of stability of plane parallel flows in a magnetized plasma, based on magnetohydrodynamics equations. The corresponding eigenvalue problem, for incompressible velocity perturbations, is solved numerically, by using continuous profiles for both plasma density and velocity at equilibrium.

Branching of solutions in the problem of the wavy flow of a viscous liquid with a free boundary

The linear problem of the stability of plane-parallel flow with a free boundary was studied on the basis of the Navier--Stokes equations in [5-9]. The nonlinear problem was investigated in [10-14] within the framework of the equations of P. L. Kapitsa; the Korteweg--de Vries equation and equations similar to it were used in [15, 16] to describe the flow in a liquid film. Nonlinear flow conditions in a liquid film and nonlinear stability were studied in [17-20] on the basis of the Navier--Stokes equations in a long-wave approximation. In the two latter works the Landau constant was calculated using a modified Reynolds--Potter method [21] and a conclusion was drawn with respect to the absence of long-wave subcritical motions.

On the Stability of Plane Parallel Flow Between Differentially Heated, Tilted Planes

The linear and energy theories of stability of a plane parallel, antisymmetric flow between differentially heated, tilted planes are examined. The linear and energy results do not coincide indicating that subcritical motion may be possible. Also, it is seen that the energy stability limit corresponds to solutions independent of the coordinate in the direction of the flow. An interesting application of Busse’s theorem in this context is demonstrated.

Stability of plane-parallel MHD flows in transverse magnetic field

We study within the framework of linear theory the stability of plane-parallel flows of a viscous, electrically conducting fluid in a transverse magnetic field. The magnetic Reynolds numbers are assumed small. The critical Reynolds number as a function of the Hartmann number is obtained over the entire range of variation of the latter. The small perturbation spectrum is studied in detail on the example of Hartmann flow. Neutral curves are constructed for symmetric and antisymmetric disturbances. The destablizing effect of a magnetic field is studied in the case of modified Couette flow. The results obtained agree with the calculations of Lock and Kakutani (where they meet) and are at variance with the results of Pavlov.

Unsteady viscous flow over a wavy wall

The method of conformal transformation is used to investigate the steady streaming generated by an oscillatory viscous flow over a wavy wall. By assuming that the amplitude of the wall is much smaller than the Stokes layer thickness, the equations are linearized and solved for large and small values of the parameterkR. This parameter is the ratio of the amplitude of oscillation of a fluid particle to the wavelength of the wall. WhenkR[Lt ] 1, the results due to Schlichting (1932) are recovered, and whenkR[Gt ] 1 the equations resemble closely those derived in the theory of stability of plane parallel flows. With the aid of this theory the first-order steady streaming is found.

Non-linear wave-number interaction in near-critical two-dimensional flows

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Stability of a laminar boundary layer in an incompressible fluid with variable physical properties

The stability of plane-parallel flows of an incompressible fluid with variable kinematic viscosity in the presence of solid walls has been discussed in [1–5]. The stability of Couette flow is considered in [1]. The method of solution, which is the same as that used in [2], differs from the Tollmien-Schlichting method, since the expansion of the solutions in powers of αR which is used assumes the smallness of this quantity. A general formulation of the problem of stability of a nonuniform fluid is presented in [3]. Di Prima and Dunn [4] used the Galerkin method to study the stability of the boundary layer relative to vortex-like disturbances in the case of variable kinematic viscosity. Since the development of this sort of disturbance depends only weakly on the form of the velocity profile in the boundary layer, a marked change of the viscosity had little effect on the critical Reynolds number. This same problem is considered in [5], The present author was not able to find in the literature any references to study of the stability of the laminar boundary layer in an incompressible fluid with variable kinematic viscosity relative to disturbances of the Tollmien-Schlichting type, with the exception of mention in [4] of an unpublished work of MacIntosh which showed the essential dependence of the critical Reynolds number on the viscosity gradient. In Section 1 an analysis is made of the effect of variability of the kinematic viscosity on the stability of the boundary layer relative to Tollmien-Schlichting waves under the condition of constant fluid density. Two approaches are possible to the study of the development of disturbances in a heterogeneous fluid. On the one hand, we can assume that the displacement of the fluid particles does not cause changes in the distribution of p(y) and υ*(y), i.e., the velocity pulsations are not accompanied by pulsations of ρ and υ*. This will be the case if a particle which is characterized by the quantities ρ1,υ 1 entering a layer with the different values ρ2 υ 2, instantaneously alters its properties so that its density becomes equal to ρ2, and its kinematic viscosity becomes equal to υ2. On the other hand, we can consider that a fluid particle moving from layer 1 into layer 2 fully retains the properties which it had in layer 1. In this case the velocity pulsations naturally lead to pulsations of the quantities ρ and υ*. In actuality, the phenomenon develops along some intermediate scheme, since the particle alters its properties as it moves in a heterogeneous fluid. The degree of approximation of the process to the first or second scheme depends on the rate of these changes. The analyses below are based on the first scheme.

Shear flow instabilities in rotating systems

A theoretical description is given for infinitesimal non-axisymmetric disturbances of a shear flow caused by differential rotation. It is assumed that the deviations from the state of rigid rotation are small corresponding to the case of small Rossby number. The shear flow becomes unstable at a finite critical value of the Rossby number, at which the inertial forces overcome the friction in the Ekman boundary layers and the constraint imposed by the variation in depth of the container. The governing equation is closely related to the Rayleigh stability equation in the theory of hydrodynamic stability of plane parallel flow. Experimental observations by R. Hide show reasonable agreement with the theoretical predictions.

On the stability of parallel flows with parallel magnetic fields

The first part of the paper is a physical discussion of the way in which a magnetic field affects the stability of a fluid in motion. Particular emphasis is given to how the magnetic field affects the interaction of the disturbance with the mean motion. The second part is an analysis of the stability of plane parallel flows of fluids with finite viscosity and conductivity under the action of uniform parallel magnetic fields. We show that, in general, three-dimensional disturbances are the most unstable, thus disagreeing with the conclusion of Michael (1953) and Stuart (1954). We show how results obtained for two-dimensional disturbances can be used to calculate the most unstable three-dimensional disturbances and thence we prove that a parallel magnetic field can never completely stabilize a parallel flow.

Kelvin-Helmholtz instability of a dusty gas

In a recent paper ((2)) Saffman gave a representation of the flow of a dusty gas in which the dust was described in terms of a number density of small particles with very small volume concentration but appreciable mass concentration. In the same paper the problem of the stability of plane parallel flows to small disturbances was formulated, and subsequently the author ((1)) gave some approximate results for the stability of plane Poiseuille flow.

Note on the stability of plane parallel flows

The subject of this note is the behaviour of three-dimensional small disturbances to plane parallel flows, which have a variation in the direction normal to the plane of mean flow, in relation to two-dimensional disturbances which vary in the plane of mean flow only. It was pointed out by Squire (1933) that, in linearized theory, the disturbancewhichisneutrallystable at the criticalReynolds number R, is two-dimensional in form. More recently interest has turned to the question as to which kind of disturbance is most rapidly amplified at a given Reynolds number above the critical. Jungclaus (1957) pointed out that for certain values of R and of the resolved wavelength in the plane of mean flow, three-dimensional disturbances may be more unstable than plane ones. Recently, Watson (1960) has shown further that a two-dimensional disturbance is the one most rapidly amplified in a certain range of R starting from the critical. In thi8 note we take a slightly different view of the problem which enables us to define specifically the upper end of this range of R, when it exists.

On the stability of plane parallel flows of an inhomogeneous fluid

On the stability of plane parallel flows of an inhomogeneous fluid

On the stability of plane-parallel flow of a viscous fluid over an inclined bottom

On the stability of plane-parallel flow of a viscous fluid over an inclined bottom