Film flow for power-law fluids: Modeling and linear stability

Amaouche, Mustapha; Amar, D.; Abderrahmane, Hamid Ait

doi:10.1016/j.euromechflu.2012.02.001

Cited by 15 publications

(22 citation statements)

References 21 publications

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“…Indeed, a small variation of the regularization parameter can completely change the dynamics of the P. Noble and J.-P. Vila thin film. In Amaouche et al (2012), the second-order terms are divergent if n < 1, and simply cancelled, so that the model is only first-order.…”

Section: Introductionmentioning

confidence: 96%

“…In another recent paper, Amaouche, Djema & Ait Abderrahmane (2012) derived a full second-order model for shear thickening fluids, n > 1, whereas it is only first order in the shear thinning case, n < 1, where the weighted residual method fails to converge. The full second-order model derived in Amaouche et al (2012) is too cumbersome in practice (being a system of seven evolution partial differential equations), so various reduced models have been considered, but they are no longer second-order accurate.…”

Section: Introductionmentioning

confidence: 99%

“…In the shear thickening case, n 1, the apparent viscosity remains bounded all along the fluid height: the solutions of the Cauchy momentum equations can be written in a basis of functions bounded in the cross-stream variable z. In Amaouche et al (2012), these computations were carried out directly for the power law, and in Ruyer-Quil et al (2012) for a regularized version of the power law. In this latter paper, the coefficients of the shallow-water equations converge as the regularization parameter goes to zero.…”

Section: Introductionmentioning

confidence: 99%

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Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

Noble

Vila

2013

J. Fluid Mech.

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The aim of this paper is to provide a better understanding of the derivation of two moments approximate models of shallow water type for thin power law films down an incline. We limit ourselves to the case of laminar flow for which the boundary layer issued from the interaction of the flow with the bottom surface has an influence all over the transverse direction to the flow. In this case the concept itself of thin film and its relation with long wave asymptotic leads naturally to flow conditions around a uniform free surface Poiseuille flow.This Poiseuille flow exhibits a divergence of the apparent viscosity at the free surface which, in turn, introduces a singularity in the formulation of the Orr-Sommerfeld equations and in the derivation of shallow water models. We remove this singularity by inverting the relation between the fluid strain and the deformation tensor and by keeping the fluid strain as a variable. No regularization procedure is needed here, in contrast to RQ [RQ]. In order to derive the shallow water equations, we compute a second order asymptotic expansion of the fluid velocity field and fluid strain in the shallow water regime through a nonlinear iterative scheme. We show that a linearized version of this scheme naturally provides an expansion of the eigenfunctions of the Orr-Sommerfeld equations in the small wavenumber/small frequency regime. The shallow water models account for the streamwise diffusion of momentum which is important to describe accurately the full dynamic of the thin film flows when instabilities like roll-waves arise.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

Noble

Vila

2013

J. Fluid Mech.

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“…Amaouche, Djema & Bourdache (2009) derive a Shkadov-type model for a power-law fluid thin-film flow. Very recently, Amaouche, Djema & Abderrahmane (2012) and Ruyer-Quil, Chakrabortya & Dandapata (2012) have extended the weighted residual approach to thin-film flows of shear-thinning fluids. In both cases the authors propose special treatment of the low-shear-rate limit close to the interface in order to avoid the singularity in the viscosity.…”

Section: Introductionmentioning

confidence: 99%

A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability

2013

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We study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shearthinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr-Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel-Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr-Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.

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“…They presented two versions of lubrication theory for superposed layers of non-Newtonian fluid with powerlaw rheology and carried on both linear analysis and nonlinear numerical simulation. Amaouche et al [18] conducted modeling and linear stability for a power-law fluid film flowing down an inclined plane with the method of lubrication theory and weighted residual approach. Also, they performed an incomplete regularization procedure to cure the rapid divergence of the reduced two-equation model.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Instability Analysis of Power-Law Sheets

Liu¹,

Yang²,

Liu³

et al. 2017

Topical Problems of Fluid Mechanics 2017

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Research on nonlinear instability of Power-law plane sheets has been conducted using Carreau law. Combined with asymptotic expansion and long wave assumption, the governing equations and boundary conditions were performed using integral transform. The first-order dimensionless dispersion relation between unstable growth rate and wavenumber was obtained and the second-order interface disturbance amplitude was calculated. By comparison and analysis of components of the second-order interface disturbance amplitude, the effects of power-law index n (n<1) were investigated and the condition under which the shear-thinning effect can be evident was concluded, thus contributing to theoretical basis and technological means to improve atomization of power-law sheets.

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Film flow for power-law fluids: Modeling and linear stability

Cited by 15 publications

References 21 publications

Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability

Nonlinear Instability Analysis of Power-Law Sheets

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