On the Squire's transformation for stratified two-phase flows in inclined channels

Barmak, Ilya; Gelfgat, Alexander; Ullmann, Amos; Brauner, Neima

doi:10.1016/j.ijmultiphaseflow.2016.09.018

Cited by 12 publications

(7 citation statements)

References 10 publications

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“…So, the complex wavenumber is the unknown eigenvalue, and φ and ϕ are the unknown eigenfunctions of the eigenproblems described by Eqs. (12) and (13). It is obvious from these equations that the problem is a nonlinear eigenvalue problem.…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…The approximated values of unknown eigenfunctions by rational Chebyshev polynomials are then put into Eqs. (12) and (13). For each n at any ŷ i , these equations are converted to where primes denote differentiation with respect to ŷ i .…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…In Eqs. (12) and 13, u(y) and ρ(y) are non-dimensional background flow velocity and density profiles and will be defined in the following section. α = α r + iα i , where α r and α i are the (non-dimensional) wavenumber and the spatial amplification rate, respectively.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Their two-way coupled approach showed that the increasing number of particles makes the current more stable. Barmak et al [12] examined the applicability of the Squire's transformation for stability analysis of stratified two-phase flow in horizontal and inclined channels. They proved that 2D perturbations are the critical ones also for the case of inclined channel.…”

Section: Introductionmentioning

confidence: 99%

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Linear spatial stability analysis of particle-laden stratified shear layers

Khavasi

Firoozabadi

2019

J Braz. Soc. Mech. Sci. Eng.

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Hydrodynamic instabilities at the interface of stratified shear layers could occur in various modes. These instabilities have an important role in the mixing process. In this work, the linear stability analysis in spatial framework is used to study the stability characteristics of a particle-laden stratified two-layer flow. The effect of parameters such as velocity-to-density thickness ratio, bed slope, viscosity as well as particle size on the stability is considered. A simple iterative method applying the pseudospectral collocation method that employed Chebyshev polynomials is used to solve two coupled eigenvalue equations. Based on the results, the flow becomes stable for Richardson number larger than 0.25 (same as the result of temporal stability analysis); the stability is not affected by spatial wavenumber. The increase in bed slope makes the current more unstable as does in temporal framework. For 1% bed slope, the spatial growth rate increases by 70% in J = 0.23. For R = 5 (velocity-to-density thickness ratio) and zero bed slope, there are four zones: (a) two Kelvin-Helmholtz modes (0 < J < 0.09, J is local Richardson number), (b) two Holmboe modes (0.09 < J < 0.65), (c) no unstable mode (0.65 < J < 2.5) and (d) two Holmboe modes (2.5 < J < 4 where the second type of Holmboe modes appears). The second type of Holmboe modes does not appear in temporal framework in this condition. In spatial analysis, for nonzero bed slope there is no stable region. Also, just one type of Holmboe modes and two types of Kelvin-Helmholtz modes appear. Th existence of particles changes the instability characteristics of the flow. Particles increase the spatial growth rate. In temporal analysis, particles larger than a certain size (e.g., kaolin particles larger than 20 micron in Stokes' law settling velocity) make the flow unstable, but in spatial framework particles with any size do this. As expected, the viscosity makes the current more stable like temporal analysis.

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Section: Numerical Solution Methodsmentioning

confidence: 99%

Section: Numerical Solution Methodsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear spatial stability analysis of particle-laden stratified shear layers

Khavasi

Firoozabadi

2019

J Braz. Soc. Mech. Sci. Eng.

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“…Note, however, that for non-linear viscous fluids (e.g., shearthinning liquids) there is no equivalence for the Squire's theorem (named after Squire, 1933), which was formulated for Newtonian fluids and states the sufficiency of consideration of 2D (in the plane of flow) perturbations for stability analysis, since they are the critical perturbations. Only recently, the applicability of the Squire's theorem for 8 inclined two-phase Newtonian systems was provided by Barmak et al (2017). In the presence of a non-Newtonian liquid, this issue was only verified numerically (e.g., see Nouar and Frigaard, 2009;Sahu and Matar, 2010;Allouche et al, 2015).…”

Section: Linear Stabilitymentioning

confidence: 99%

Stability of stratified two-phase channel flows of Newtonian/non-Newtonian shear-thinning fluids

Picchi

Barmak

Ullmann

et al. 2018

International Journal of Multiphase Flow

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Linear stability of horizontal and inclined stratified channel flows of Newtonian/non-Newtonian shearthinning fluids is investigated with respect to all wavelength perturbations. The Carreau model has been chosen for the modeling of the rheology of a shear-thinning fluid, owing to its capability to describe properly the constant viscosity limits (Newtonian behavior) at low and high shear rates. The results are presented in the form of stability boundaries on flow pattern maps (with the phases' superficial velocities as coordinates) for several practically important gas-liquid and liquid-liquid systems. The stability maps are accompanied by spatial profiles of the critical perturbations, along with the distributions of the effective and tangent viscosities in the non-Newtonian layer, to show the influence of the complex rheological behavior of shear-thinning liquids on the mechanisms responsible for triggering instability. Due to the complexity of the considered problem, a working methodology is proposed to alleviate the search for the stability boundary. Implementation of the proposed methodology helps to reveal that in many cases the investigation of the simpler Newtonian problem is sufficient for the prediction of the exact (non-Newtonian) stability boundary of smooth stratified flow (i.e., in case of horizontal gas-liquid flow). Therefore, the knowledge gained from the stability analysis of Newtonian fluids is applicable to those (usually highly viscous) non-Newtonian systems. Since the stability of stratified flow involving highly viscous Newtonian liquids has not been researched in the literature, interesting findings on the viscosity effects are also obtained. The results highlight the limitations of applying the simpler and widely used power-law model for characterizing the shear-thinning behavior of the liquid. That model would predict a rigid layer (infinite viscosity) at the interface, where the shear rates in the viscous liquid are low, and thereby unphysical representation of the interaction between the phases.

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