Squire's Theorem Research Articles

Stability of mixed-convection flow in a tall vertical channel under non-boussinesq conditions

We have examined the linear stability of the fully developed mixed-convection flow in a differentially heated tall vertical channel under non-Boussinesq conditions. The Three-dimensional analysis of the stability problem was reduced to an equivalent two-dimensional one by the use of Squire's transformation. The resulting eigenvalue problem was solved using an integral Chebyshev pseudo-spectral method. Although Squire's theorem cannot be proved analytically, two-dimensional disturbances are found to be the most unstable in all cases. The influence of the non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Grashof and Reynolds numbers on the temperature difference. The results show that four different modes of instability are possible, two of which are new and due entirely to non-Boussinesq effects.

Stability of natural convection flow in a tall vertical enclosure under non-Boussinesq conditions

We have examined the linear stability of the fully developed natural convection flow in a differentially heated tall vertical enclosure under non-Boussinesq conditions. The three-dimensional analysis of the stability problem was reduced to a two-dimensional one by the use of Squire's theorem. The resulting eigenvalue problem was solved using an integral Chebyshev collocation method. The influence of non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Rayleigh number on the temperature difference. The results show that two different modes of instability are possible, one of which is new and due entirely to non-Boussinesq effects. Both types of instability are oscillatory, and the critical disturbance wave speed is zero only in the Boussinesq limit.

Moist Eady Waves in a Quasigeostrophic Three-Dimensional Model

Abstract A three-dimensional quasigeostrophic model in which the base state is neutral to moist symmetric stability, following Emanuel et al., is used to obtain the experimental result that the two-dimensional moist Eady waves are the most unstable ones, as was known a priori only for the dry case, from Squire's theorem. The horizontal structure of moist baroclinic waves of nonzero meridional wavenumber presents deviations from the high-low symmetry that it is usually seen in dry models. The evolution of the normal modes of the moist Eady model is also followed in the nonlinear regime, where they display updraft contraction in the meridional direction and a fast meridional drift.

A short note on Squire's theorem for interfacial instabilities in a stratified flow of two superposed fluids

In a recent paper Schaflinger (1994) investigated interfacial instabilities in a stratified flow of two superposed fluids. The paper focused on a viscous resuspension flow (Schaflinger et al., 1990) in which a sediment of non-buoyant, solid particles is resuspended and driven by a clear fluid flow. At the suggestion of one reviewer, Schaflinger (1994) mentions that "[a]ccording to Squire (1933) three-dimensional disturbances in the classical, inviscid problem can be transformed into an equivalent two-dimensional problem, i.e. to each unstable three-dimensional disturbance there corresponds a more unstable two-dimensional one (Drazin and Reid, 1981, p. 129). Even though it seems to be imaginative to carry over Squire's theorem to the current study, the foundation supporting Squire's statement may not hold in the present problem. No studies dealing with this specific issue have been found in the literature. Therefore, this matter is subject to further, thorough investigations." Here we show that Squire's transformation indeed reduces the equations and the boundary conditions for a stratified flow to an equivalent two-dimensional problem.

Early-period dynamics of an incompressible mixing layer

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

Multiple linear instability of layered stratified shear flow

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.

On the analogues and generalizations of squire's theorem

Assertions consistent with Squire's theorem /1/ based on the simultaneous or separate consideration of the factors responsible for the non-linearity of the motion and the non-stationarity of the fluid flow whose stability is investigated, are studied. The results may be of interest from the point of view of the practical applications, and can also be used to solve the problem of the degree of generality of the reasons governing the existence of assertions of the type of Squire's theorem.

The instability of barotropic circular vortices

Abstract The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.

On the stability of a thermally stratified conducting fluid under the action of aligned magnetic field

The stability of a thermally stable stratified viscous electrically conducting shear flow is investigated in the presence of an impressed uniform aligned magnetic field. Only two-dimensional disturbances are studied in this paper because Squire's theorem does not apply in general, owing to the presence of the aligned magnetic field. The analysis is partly analytical and partly numerical. The asymptotic solutions for non-viscous fluid are first obtained analytically and they are then improved by introducing viscous and thermal diffusion terms (but only for σ=1) to get a uniformly valid solution. The neutral stability curves are numerically computed for a range of values of Richardson and Stuart numbers, which show that the flow is completely stabilized when a Stuart number exceeds a certain value for a given R i>0. It is shown that the combined effects of magnetic field and stratification is to make the system stable to two-dimensional disturbances at lower Stuart number than the one given by Stuart (1954) in the absence of thermal stratification.

The stability chart of parallel shear flows with double-diffusive processes ? general properties

The stability of an infinite fluid layer is considered, subject to horizontal flow and arbitrary initial vertical temperature and salinity distributions. Linear stability analysis is applied, using three-dimensional perturbations. A method is outlined for the construction of a general stability chart; parts of the chart are marginal-stability lines obtained by using the Squire theorem, where disturbances in the flow direction are the most unstable; necessary conditions for the theorem to be valid are developed. These conditions are shown to be sufficient in certain cases. A technique is presented to obtain the stability boundary even when these conditions are not satisfied. The results show that an increase in the Reynolds number cannot increase the stability domain.

Stability of flow of a generalized second order fluid down an inclined plane

This investigation concerns the stability of flow of a generalized second-order fluid down an inclined plane with respect to three-dimensional disturbances. The critical Reynolds number is given as a function of dimensionless steady flow velocityU(y), material parameters and the slope of the plane. In this case, for long wave disturbances, Squire's theorem is valid. This result contradicts that of Gupta.

On the nonlinear stability of rotating Newtonian and non-Newtonian fluids

The effect of superimposed rigid body rotations on the nonlinear stability of an arbitrary base flow is examined from a theoretical standpoint for incompressible Newtonian and non-Newtonian fluids. It is proven mathematically, from an analysis of the full nonlinear equations of motion, that instabilities which depend on the state of rotation of the fluid must arise from variations in the velocity disturbances along the axis of rotation of the fluid. Consequently, it follows that Squire's theorem cannot, under any circumstances, apply to rotationally dependent instabilities. The consistency of these results with previous work on Couette flow and rotating plane Poiseuille flow is discussed briefly.

On the interaction between Tollmien-Schlichting and Rayleigh-Bénard instabilities in the presence of a longitudinal magnetic field

AbstractThe method used by Gage and Reid(10) to investigate hydrodynamic stability of thermally stratified fluid is extended to hydromagnetic stability to study the effect of aligned magnetic field on the stability of unstable thermal stratification under the assumption of small magnetic Reynolds number. The interaction between the Tollmien–Schlichting–Stuart mechanism of instability due to shear and magnetic field and Rayleigh–Bénard–Thompson mechanism of instability due to thermally unstable stratification and magnetic field is brought out in detail. It is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, Squire's theorem is not valid. This conclusion follows from the fact that in our analysis the Richardson number Ri ( < 0) will not be greater than the value −0·92 × 10−6. In particular, it is shown that for the values of stratification parameter n ≤ 0·6 the effect of magnetic field for small values of Stuart number N is to augment instability and impose the restriction on the validity of our numerical procedure. However, for η = 0·8 a sharp transition from unstable to stable flow takes place at N = 0·3. A physical explanation for this based on eddies is given.

On the stability of thermally radiative magnetofluiddynamic channel flow

The stability of three dimensional infinitesimal disturbances is examined for the laminar flow of a thermally radiating, viscous, electrically and heat-conducting fluid between parallel walls with transverse magnetic field. In addition to the classical criteria for the non-radiative case a further system of homogeneous differential equations with mixed boundary conditions arises yielding possible further eigenvalues and requirements for stable flows. For the overall configuration the analogue of Squire's theorem is shown to fail. The eigenvalue problem is of unusual type with the eigenvalue parameter appearing non linearly in both the differential equation and boundary conditions. A study of two-dimensional disturbances is carried out in detail for non-thermally conducting fluids and for a particular wave number the further eigenvalues are shown by exact analysis to be always stable. In channels with black walls at the same temperature extensive numerical calculations for a range of wave-numbers and a variety of other parameters all yield the same result.

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.

Stability analysis of stratified sheared flows as an initial-value problem

A mathematical analysis of the stability problem of stratified fluids in horizontal sheared motion is given as an initial-value problem in order to avoid the difficulties encountered in applying classical hydrodynamic stability methods. Good agreement is obtained with results given by other authors in the two-dimensional case. An extension of the stability analysis to the three-dimensional case is given. Such extension is discussed and it is shown how the results obtained are in quite good agreement with the well-known Squire theorem.

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.

Convective instabilities in concurrent two phase flow: Part I. Linear stability

AbstractThe linear stability of thermally stratified horizontal two‐phase Couette flow is analyzed for the case of a constant vertical temperature gradient. Instabilities driven by buoyancy, surface tension gradients, or shear are allowed for. It is shown that the instability can take three possible forms: streamwise oriented roll vortices, long interfacial waves, and short Tollmien‐Schlichting waves. It is shown that the stability limits for rolls are identical to those for plane, stagnant layers. A long wave expansion is presented and the stability limits for this mode are given algebraically. The nonexistence of a Squire's Theorem is demonstrated and some numerical experiments at moderate Reynolds numbers are described. Detailed comparisons with previous work are possible for only one fluid pair, but it is shown that reasonably accurate statements may be made to determine which mode may manifest itself in any given experimental situation.

Stability of plane Poiseuille flow of slightly viscoelastic fluids

The title subject has been examined by the author in a series of papers (Cousins, 1970, 1972a, b), and the assumptions and principal results of those papers are discussed here. The work is motivated by the phenomenon evinced in fluid flow situations, of turbulent drag reduction by certain polymer additives. From a survey of experimental work it is clear that molecular elongation plays an important role in reducing drag by suppressing transverse motions. This effect may be interpreted as a normal stress effect in a continuum theory. A second-order fluid, which is a simple model exhibiting such a property, is used in a linear analysis of disturbances to planePoiseuille flow. Unlike theNewtoniăn case Squire's theorem is not valid (Lockett, 1969a) and a three-dimensional analysis is required. The viscoelastic terms are in general destabilising. Under certain conditions the first growing disturbance will propagate at an angle to the basic flow, giving a longitudinal vortex structure close to the channel boundaries not present at the onset of instability in aNewtonian fluid. The analysis is extended to finite-amplitude disturbances by introducing a time-dependent amplitude, but calculations are here confined to the simpler two-dimensional case. Disturbances which would decay under linear theory may in fact grow provided the initial amplitude is sufficiently large. A threshold amplitude for instability is found as a function ofReynolds number. The viscoelastic terms are again found to be destabilising. Finally, a further viscoelastic property, that of stress relaxation, is introduced through an integral representation of the stress. A linear analysis is developed and stress relaxation is also shown to be a destabilising influence.

Thermal stability of radiating fluids: Asymmetric slot problem

The buoyancy driven thermal stability of a radiating nongray gas between two infinitely long vertical plates is studied analytically. The Squire theorem is shown to hold, and two-dimensional disturbances are more dangerous than the corresponding three-dimensional ones. A combination of the Galerkin and Chandrasekhar methods are used to solve the eigenvalue problem associated with two-dimensional disturbances. In contrast to the Benard problem, the effect of radiation on the onset of stationary cells diminishes as the optical thickness of the gas increases indefinitely.