Temporal Linear Stability Analysis Research Articles

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.

Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka,G,Pr,L, χ,M,B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ<0. Moreover, the P-mode which is short-wave unstable for χ=0 remains so for χ<0, but becomes long-wave unstable for χ>0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

Linear temporal and spatiotemporal stability analysis of two-layer falling films with density stratification

A detailed temporal and spatiotemporal stability analysis of two-layer falling films with density and viscosity stratification is performed by using the Chebyshev collocation method to solve the full system of linear stability equations. From the neutral curves Re(k) for the surface mode and the interface mode of instability, obtained for different density ratios gamma of the upper layer to the lower layer, it is found that smaller density ratios make the surface mode and the short-wave interface mode much more stable, and can even make the short-wave interfacial instability disappear. Moreover, through the study of the local growth rates of the spatiotemporal instability as a function of the ray velocity V , it is found that for not too small incline angles like theta=0.2, the two-layer flow is always convectively unstable, and there is a transition between long- and short-wave instabilities which is determined by the Briggs-Bers collision criterion. Due to the existence of the absolute Rayleigh-Taylor instability for gamma>0 and theta=0, a transition from convective to absolute instability can be detected at small incline angles, and the corresponding boundary curves are plotted for different Reynolds numbers, viscosity ratios, and incline angles. It is found that there exists a limit Reynolds number above which the two-layer film flow can only be convectively unstable for a fixed small incline angle. The spatial amplification properties of the convective waves are finally presented for both surface and interface modes.

Instability analysis of an inner-driving coaxial jet inside a coaxial electrode for the non-equipotential case

In this paper, the temporal linear stability analysis of a coaxial jet with two coflowing immiscible liquids inside a coaxial electrode is carried out to investigate the non-equipotential case of the inner-driving coaxial electrospray. Accordingly the outer liquid is assumed to be insulating and the inner liquid as the driving medium with finite conductivity. However, the current due to conduction is assumed to be much smaller than that due to convection so that charge is transported mainly by means of convection. In this case, free charge seems to be “frozen” on the interface between two liquids. The analytical dimensionless dispersion relation is derived straightforwardly through theoretical analysis. In order to study the jet instability, the complex frequencies are solved using numerical approach. According to the calculation results, three different unstable modes, i.e. the para-varicose mode, the para-sinuous mode and the transitional mode, are found in the Rayleigh regime. It is also found that the non-dimensional electrostatic force and the Weber number have the similar effects on the unstable modes. In particular, these two parameters both stabilize the para-sinuous mode for relatively long wavelengths and destabilize it for relatively short wavelengths. Moreover, the unstable regions are extended into the wind-induced regime as the electric field intensity or liquid velocity increases, or surface tension decreases. It is predicted that the non-equipotential case closes to the equipotential one when the inner-driving liquid has sufficiently large electrical permittivity. Conversely, if the permittivity of the inner liquid is considerably small, the non-equipotential case is much less unstable than the equipotential case. In addition, the permittivity of the outer insulating liquid shows an unexplainable non-monotonical effect on the jet instability. As a summary, a general dispersion relation expression is given for all the four involved models, according to equipotential and non-equipotential, as well as inner and outer driving. And the essence of the effect of the electric field on the jet instability is outlined for these cases.

Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: Multiple solutions and stability

The simultaneous effects of transpiration through and tangential movement of a semi-infinite flat plate on the self-similar boundary layer flow driven by uniform shear in the far field is considered. Difficulties with standard shooting techniques are overcome using Crocco variables which also serve to better elucidate the solution structure. The stabilities of dual, triple and even quadruple steady flow solutions encountered in different ranges of plate stretching and wall stress are determined using a linear temporal stability analysis for the self-similar flow.

Inertialess temporal and spatio-temporal stability analysis of the two-layer film flow with density stratification

This paper presents a temporal, spatial, and spatio-temporal linear stability analysis of the two-layer film flow down a plate tilted at an angle θ. It is based on a zero Reynolds number approximation to the Orr-Sommerfeld equations and a zero surface tension approximation to both surface boundary conditions. The combined effects of density and viscosity stratifications are systematically investigated. The subtle influence of density stratification is first put into light by a temporal analysis for θ=0.2; when increasing/decreasing the density ratio (upper fluid/lower fluid), the two-layer film flow becomes much more unstable/stable with respect to the finite wavelength instability. Moreover, below a critical density ratio this finite wavelength instability even disappears, whatever the viscous ratio. Concerning the long wave instability, it becomes dominant when decreasing the density ratio below 1 and is even triggered in a region which was stable for equal density layers. The spatio-temporal analysis shows that the instability is convective for incline angles that are not too small as θ=0.2. The study of the local growth rates of the spatio-temporal instability as a function of the ray velocity V shows that there is a transition between long wave and short wave instabilities which has been determined by using the Briggs-Bers collision criterion. Accordingly, there exists a jump for the local oscillatory frequency, spatial amplification rate, and spatial wave number due to this transition. Due to the existence of the absolute Rayleigh-Taylor instability for θ=0, the transition from convective to absolute instability can be detected for values of θ smaller than 0.2, and absolute/convective instability boundary curves have been obtained for varying characteristic parameters.

Linear instability of a coflowing jet under an axial electric field

A temporal linear stability analysis is carried out for a coflowing jet with two immiscible inviscid liquids under a uniform axial electric field. According to the electrical properties of the inner and outer liquids, four cases, i.e., IDOC (inner: dielectric; outer: conductor), ICOD (inner: conductor; outer: dielectric), ICOC (inner and outer: conductor), and IDOD (inner and outer: dielectric), are considered. The analytical dimensionless dispersion relation is derived for both axisymmetric and nonaxisymmetric perturbations and is solved for axisymmetric ones. Three unstable modes, i.e., the paravaricose, parasinuous and transitional modes, are identified in the Rayleigh regime. The influences of the axial electric field, liquid electrical properties, and Weber number are studied at length. The results show that the axial electric field has a generally stabilizing effect on the unstable modes. The effects of the liquid electrical properties are quite different but all great for each case. The change of dominant mode is detected with the variation of the electric field intensity, electrical properties or Weber number. It is found that the parasinuous instability is the easiest to realize in IDOC. And the comparison with the experiment validates that the parasinuous mode is predominant in coaxial electrospray.

Effect of liquid and air swirl strength and relative rotational direction on the instability of an annular liquid sheet

Instability of a swirling annular liquid sheet in swirling inner and outer air streams has been investigated by a temporal linear stability analysis. The effects of the swirling and axial motion of the liquid and the air streams, as well as the effects of relative inner and outer air swirl orientation with respect to the liquid swirl direction on the instability have been investigated. Results show that for a non-swirling liquid sheet axial inner air stream is more effective than axial outer air stream in enhancing the sheet instability. This is opposite of a swirling liquid sheet where axial outer air is more effective in promoting sheet instability compared to axially moving inner air stream. The liquid swirl has a destabilizing effect at the outer interface but has a stabilizing effect at the inner interface. At high liquid swirl Weber number, the outer air (with axial and swirl velocity components) is more effective in enhancing sheet instability compared to the inner air (with axial and swirl velocity components). To understand the effect of air swirl orientation with respect to liquid swirl direction, four possible combinations with both swirling air streams with respect to the liquid swirl direction have been considered. Results show that at high liquid swirl Weber number a combination of counter-rotating-inner air stream and co-rotating-outer air stream has the largest most unstable wave number. However, at low liquid swirl, co-inner/counter-outer combination has the largest most unstable wave number. The combination of inner and the outer air stream co-rotating with the liquid has the highest growth rate. In many combustion applications, the liquid sheet is injected in high pressure environment where the effect of high ambient pressure results in increased aerodynamic interaction due to high air density. Hence the effect of high ambient pressure is studied in terms of the dimensionless parameter of air-to-liquid density ratio. Results show significantly higher disturbance growth rates at high air pressure. However, the qualitative sheet stability behavior is similar to that at atmospheric pressure.

Instability analysis of a coaxial jet under a radial electric field in the nonequipotential case

The linear temporal stability analysis of a coaxial jet model under the radial electric field is carried out on the assumption that all free charges are “frozen” on the outer interface of the conducting liquid (i.e., the nonequipotential case) in the present paper. The analytical dimensionless dispersion relation is derived and solved numerically. Three unstable modes, namely the paravaricose, parasinuous, and transitional modes, are found in the Rayleigh regime and they change into the other two unstable modes, namely mode 1 and mode 2, in the wind induced and atomization regimes if the dimensionless electrostatic force or Weber number is sufficiently amplified. The numerical results show that the dimensionless electrostatic force as well as the Weber number inhibits the jet instability in the long wavelength region while enhances it in the short wavelength region. The relative dielectric constant of the outer liquid alters the behaviors of the nonequipotential case significantly, whereas the electrical permittivity of the inner liquid does not affect the jet instability much. Besides, the relatively small diameter ratio may make the inner and outer interfaces decouple from each other. And the critical diameter ratio is calculated crudely. The differences between the equipotential and nonequipotential cases are clarified step by step.

Temporal linear instability analysis of an electrified coaxial jet with inner driving liquid inside a coaxial electrode

In this paper, a temporal linear stability analysis is performed of a coaxial jet composed of two immiscible liquids inside a coaxial electrode. This analysis is carried out to investigate the case of an inner driving coaxial electrospray system. The assumption is made that the inner liquid has high electric conductivity, and the outer liquid is an insulating dielectric. The dimensionless dispersion equation for both the axisymmetric and non-axisymmetric modes is derived and solved numerically for the axisymmetric case. The effects of the relevant dimensionless parameters on the instability of the jet are discussed in detail. These parameters include the dimensionless electrostatic force E, the dielectric constant ratio ε, the diameter ratios a and b, the velocity ratio Λ, the density ratio S, the Weber number, and the interface tension ratio ζ. Two independent unstable modes, modes 1 and 2, are found and analyzed. Among the various parameters, the dimensionless electrostatic force and the dielectric constant have a similar and remarkable influence on modes 1 and 2, altering drastically the regime of the jet as they vary. The interface tension on the outer interface promotes the instability of both modes 1 and 2 in the region of long wavelengths while suppressing the growth rate in the region characterized by short wavelengths. The interface tension on the inner interface, however, promotes instability of only mode 2 in the same way. The diameter ratio a has a great effect on mode 2 while a negligible influence on mode 1. And the diameter ratio b has a slight effect on both the unstable modes.

On the Rayleigh-Bénard problem in the continuum limit: Effects of temperature differences and model of interaction

The transition to convection in the Rayleigh-Bénard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. Considering a power-law (variable hard-sphere) model of interaction our analysis indicates that for sufficiently large Froude numbers “softer” potentials result in less unstable systems. At small Froude numbers this trend is reversed, i.e., the “softer” interaction potentials correspond to a more unstable response. These results are discussed in terms of the opposing mechanisms of thermal expansion and compressibility. We carry out an asymptotic expansion for small temperature differences and establish the principle of exchange of stabilities for this limit. A singularity appears in this limit when compressibility effects are dominant.

Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets

A discussion is presented on the role of limited conductivity and permittivity on the behavior of electrified jets. Under certain conditions, significant departures with respect to the perfect-conductor limit are to be expected. In addition, an exploration is undertaken concerning the validity of one-dimensional average models in the description of charged jets. To that end, a temporal linear modal stability analysis is carried out of poor-conductor viscous liquid jets flowing relatively to a steady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, are considered. A grounded coaxial electrode is located at variable distance. Most available studies in the literature are restricted to the perfect-conductor limit, while the present contribution is an extension to moderate and low electrical conductivity and permittivity jets, in an effort to describe a situation increasingly prevalent in the sector of small-scale free-surface flows. The influence of the electrode distance b, a parameter α defined as the ratio of the electric relaxation time scale to the capillary time scale, and the relative permittivity β on the growth rate has been explored yielding results on the stability spectrum. In addition, arbitrary viscosity and electrification parameters are contemplated. In a wide variety of situations, the perfect-conductor limit provides a good approximation; however, the influence of α and β on the growth rate and most unstable wavelength cannot be neglected in the general case. An interfacial boundary layer in the axial velocity profile occurs in the low-viscosity limit, but this boundary layer tends to disappear when α or β are large enough. The use of a one-dimensional (1D) averaged model as an alternative to the 3D approach provides a helpful shortcut and a complementary insight on the nature of the jet’s perturbative behavior. Lowest-order 1D approximations (average model), of widespread application in the literature of electrified jets, are shown to be inaccurate in low-viscosity imperfect-conductor jets.

On the Rayleigh–Bénard problem in the continuum limit

The transition to convection in the Rayleigh–Bénard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. No restrictions are imposed on the magnitudes of temperature difference and compressibility-induced density variations. The dispersion relation is calculated by means of a Chebyshev collocation method. The results indicate that occurrence of instability is limited to small Knudsen numbers (Kn≲0.03) as a result of the combination of the variation with temperature of fluid properties and compressibility effects. Comparison with existing direct simulation Monte Carlo and continuum nonlinear simulations of the corresponding initial-value problem demonstrates that the present results correctly predict the boundaries of the convection domain. The linear analysis thus presents a useful alternative in studying the effects of various parameters on the onset of convection, particularly in the limit of arbitrarily small Knudsen numbers.

Temporal Linear Stability Analysis of Three-Dimensional Compressible Binary Shear Layers

Temporal linear stability analysis is performed on the three-dimensional compressible binary shear layer for density ratios ranging from 1 to 32 and convective Mach numbers up to 2. Comparison is made with the results of spatial theory. Some stability properties are shared by both approaches, for example, the independence of the direction of propagation of the most unstable modes from the freestream density ratio, but other features are different: a single secondary mode appears instead of two at high compressibility. An empirical model is proposed for the wave length and phase speed of the most amplified waves. Direct numerical simulations show good agreement with linear inviscid theory for initial Reynolds numbers greater than 400.

Stabilité linéaire d'écoulements symétriques presque parallèles en canal

In this Note, we present a temporal linear stability analysis of symmetric developing flows slightly perturbed from Poiseuille flow. The Chebyshev spectral collocation method is used to resolve the Orr–Sommerfeld equation. For the main flow, the solution considered is analytic. The results of the stability study depend essentially on the shape and amplitude of the velocity profiles imposed at the channel entry. To cite this article: A. Hifdi et al., C. R. Mecanique 332 (2004).

Channel entrance flow and its linear stability

In this work, we present a temporal linear stability analysis of developing channel flow. Forthe main flow, the considered solution is analytic. It is based on the hypothesis of smalldisturbances from fully developed flow and it is valid for intermediate Reynolds numbers.The disturbances are separated into symmetric and anti-symmetric eigenmodes of thevelocity. We deal subsequently with the linear stability of this main flow, taking intoaccount the nearly parallel flow assumption. The stability problem formulationleads to the Orr–Sommerfeld equation. This equation is then resolved using theChebyshev spectral collocation method. The stability results depend essentially onthe shape and amplitude of the velocity profiles imposed at the channel entry.

Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer

The stability of two-layer plane Couette flow of upper-convected Maxwell (UCM) fluids of thicknesses (1− β) R and βR, with matched viscosities η, and relaxation times τ 1 and τ 2 past a soft, deformable solid layer (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η w is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the two fluids and the solid layer is negligible. Interfaces with and without the interfacial tension between the two fluids are considered, while the interfacial tension between the UCM fluid and the deformable solid is neglected. It is well known that the two-layer Couette flow of UCM fluids with different relaxation times undergoes a purely elastic interfacial instability (referred here as mode 1) in rigid-walled channels, even in the creeping flow limit. It was recently shown [J. Non-Newtonian Fluid Mech 116 (2004) 371] that the plane Couette flow of a single UCM fluid past a deformable solid layer also undergoes an instability (referred here as mode 2) in the creeping flow limit when the nondimensional solid elasticity parameter Γ= Vη/( GR) exceeds a certain critical value, for a given Weissenberg number W= τV/ R. In this study, the respective effects of solid layer deformability and fluid elasticity stratification on mode 1 and mode 2, and the concomitant interaction of these two qualitatively different interfacial modes are analyzed in detail. It is shown that the deformability of the solid layer has a dramatic effect on the purely elastic (mode 1) interfacial instability between the two UCM fluids: When the more elastic UCM fluid is present in between the less elastic UCM fluid and the solid layer, the layer deformability has a stabilizing effect. In this configuration, if the thickness of the more elastic fluid is smaller than the less elastic fluid, it is shown that it is possible to stabilize the mode 1 purely elastic instability (except for very long and very short waves, the latter being stabilized usually by the nonzero interfacial tension between the two fluids) by making the solid layer sufficiently deformable (i.e., Γ exceeding a critical value), while such a configuration is unstable in the case of two-layer flow in rigid channels (i.e., in the absence of the deformable solid layer). Increase in the solid layer deformability, on the other hand, destabilizes the interfacial mode between the UCM fluid and the deformable solid (mode 2). It is demonstrated that it is possible to choose the solid elasticity parameter Γ and viscosity ratio η w/ η such that both mode 1 and mode 2 are completely stable in finite experimental geometries. In marked contrast, when the less elastic fluid is present in between the more elastic fluid and the solid layer, it is shown that solid layer deformability has a destabilizing effect on mode 1. For this configuration, if the thickness of the less elastic fluid is smaller, it is shown that mode 1 can be rendered unstable by increasing solid layer deformability, while it is stable in rigid channels (in the absence of the solid layer). In both of the above configurations, the nondimensional solid layer elasticity required respectively to stabilize or destabilize mode 1 varies as Γ∝( k * R) −1 for ( k * R)≪1, where k * is the wavenumber of perturbations. However, for finite experimental geometries the minimum allowed k * is dictated by the system length, and in such cases, it is argued that complete stabilization or destabilization can be achieved for realistic values of Γ. In both the configurations, our results show that deformable solid layer coatings, by design or default, can completely suppress or induce purely elastic interfacial instabilities in two-layer flow of viscoelastic fluids past a deformable solid surface.

Instability of viscoelastic plane Couette flow past a deformable wall

The stability of plane Couette flow of an upper-convected Maxwell (UCM) fluid of thickness R, viscosity η and relaxation time τ R past a deformable wall (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η w is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the fluid and the wall is negligible. The effect of wall elasticity on the stable modes of Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310] for Couette flow of a UCM fluid past a rigid wall, and the effect of fluid elasticity on the unstable modes of Kumaran et al. [J. Phys. II (Fr.) 4 (1994) 893] for Couette flow of a Newtonian fluid past a deformable wall are analyzed. Results of our analysis show that there is only one unstable mode at finite values of the Weissenberg number, W= τ R V/ R (where V is the velocity of the top plate) and nondimensional wall elasticity, Γ= Vη/( GR). In the rigid wall limit, Γ≪1 and at finite W this mode becomes stable and reduces to the stable mode of Gorodtsov and Leonov. In the Newtonian fluid limit, W→0 and at finite Γ this mode reduces to the unstable mode of Kumaran et al. The variation of the critical velocity, Γ c, required for this instability as a function of W ̄ =τ R G/η (a modified Weissenberg number) shows that the instability exists in a finite region in the Γ c − W ̄ plane when Γ c> Γ c,Newt and W ̄ < W ̄ max , where Γ c,Newt is the value of the critical velocity for a Newtonian fluid. The variation of Γ c with W ̄ for various values of H are shown to collapse onto a single master curve when plotted as Γ c H versus W ̄ /H , for H≫1. The effect of wall viscosity is analyzed and is shown to have a stabilizing effect.

Instability of a rotating thread in a second immiscible liquid

We consider the surface-tension-driven instability of a cylindrical liquid column surrounded by a second liquid when the entire system is rotating. Our calculations are in the limit that the flows in both the liquid thread and the outer fluid are viscously dominated, and include the centripetal and Coriolis forces; the effect of the Coriolis force has not previously been studied in the case that the flows in both liquids are viscous. We present numerical results of a linear temporal stability analysis, and an analytical result valid in the large-Taylor-number limit. We also use the boundary-integral method to consider the evolution and instability of a finite cylindrical thread, which then relaxes when the rotation rate is reduced. These results are discussed in connection with recent experimental observations.

Observations d'une instabilité convective apparaissant sous la forme de rouleaux sinueux dans un écoulement de Poiseuille–Rayleigh–Bénard

A both experimental and numerical study of a mixed convective flow in a rectangular channel heated at the bottom and cooled at the top is reported. In such configuration, it is well known that for a Rayleigh number above a critical value and for sufficiently high Reynolds number, longitudinal counter-rotating rolls develop. A linear temporal stability analysis conducted in 1991 detected the onset of a wavy instability of the longitudinal rolls. In our study, experimental results and 3D direct simulations clearly show the development of the wavy pattern and point out the convective behaviour of this instability. To the knowledge of the authors, it is the first time that this instability is observed.