Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

Papageorgiou, Demetrios T.; Tanveer, S.

doi:10.1098/rspa.2021.0307

Cited by 4 publications

(3 citation statements)

References 31 publications

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“…By using methods similar to [11] that are detailed in the supplementary file, electronic supplementary material, we arrive at the following theorem for uniform asymptotics of Nfalse[kfalse].…”

Section: Rigorous Results For Uniform Asymptotics For Large α=Kνmentioning

confidence: 99%

“…We emphasize that experimental observations of bistable, see [4], are supported generically by the non-local equations as shown in [8]. For extensions to surfactant-laden three-dimensional flows, the reader is referred to [6,11].…”

Section: Introductionmentioning

confidence: 86%

“…for some constant M that depends on the parameters. By using arguments (see supplementary file, electronic supplementary material) similar to [11], we establish the following two theorems for local and global existence of solutions to the initial value problem (5.1) and (5.2). Remark 5.3.…”

Section: Global Existence For Periodic Initial Conditionsmentioning

confidence: 99%

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Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

Papageorgiou

Tanveer

2023

Proc. R. Soc. A.

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We consider instability of the flat interface in a two-layer Couette flow model developed earlier (Kalogirou & Papageorgiou, 2016, J. Fluid Mech. 802 , 5–36; Katsiavria & Papageorgiou, 2022, Wave Motion 114 , 103018. ( doi:10.1016/j.wavemoti.2022.103018 )) for a thin layer near one of the walls. For the case when the less viscous fluid resides next to the moving wall, we find that even a small slip effect at the interface can destabilize an otherwise highly stable flow to the Turing-type instability. The singular effect of small slip in an otherwise very stable configuration may have important ramifications in physical and technological applications. The neutral points of the dispersion relation give rise to travelling wave solutions that are continued to finite amplitude numerically and their linear stability properties identified for a set of parameter values for disturbances that include subharmonic modes with twice the wavelength of the nonlinear travelling wave. We determined Hopf and regular bifurcation points of travelling waves and rigorously justified their existence for some set of parameter values. Weakly nonlinear analysis close to bifurcation from a flat state is also presented for small amplitude waves in general. We also present global existence and regularity results for periodic initial conditions without any restriction on parameters.

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Section: Rigorous Results For Uniform Asymptotics For Large α=Kνmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 86%

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Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

Papageorgiou

Tanveer

2023

Proc. R. Soc. A.

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Optimal analyticity estimates for non-linear active–dissipative evolution equations

Papageorgiou

Smyrlis

Tomlin

2022

IMA Journal of Applied Mathematics

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Active-dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterised by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative), and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth order dissipation, and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalising the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilising a criterion involving the rate of growth of suitable norms of the $n$-th spatial derivative of the solution, with respect to the spatial variable, as $n$ tends to infinity. An estimate of the rate of growth of the $n$-th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if $\gamma $, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e., if $\gamma $ is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active-dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active-dissipative-dispersive pseudo-differential equations.

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Stability analysis of viscous multi-layer shear flows with interfacial slip

Katsiavria,

Papageorgiou

2024

IMA Journal of Applied Mathematics

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One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin–Helmholtz (KH) instability. It predicts short-wave instabilities that, in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity), render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid–liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid–liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid–liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilization of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.

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Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

Cited by 4 publications

References 31 publications

Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

Optimal analyticity estimates for non-linear active–dissipative evolution equations

Stability analysis of viscous multi-layer shear flows with interfacial slip

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