Shock-wave solutions in two-layer channel flow. I. One-dimensional flows

Mavromoustaki, Aliki; Matar, O. K.

doi:10.1063/1.3497032

Cited by 17 publications

(24 citation statements)

References 35 publications

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“…The fact that the interface between two immiscible liquid layers develops into a complex spatiotemporal phenomena downstream on account of an infinitesimal disturbances has already been examined (Tilley et al 1994a,b;Segin et al 2005;Mavromoustaki, Matar & Craster 2010. Based on the slowly evolving interface with respect to space and time, flow variables are expanded as a series relative to the film parameter ∼ (1/λ), where λ is the wavelength which is large compared with the depth of the channel.…”

Section: Long-wave Modelmentioning

confidence: 99%

“…O( ), solutions are not given. The procedure of finding solutions in detail can also be found in the work ofCharru & Fabre (1994),Tilley et al (1994a),Segin et al (2005) andMavromoustaki et al (2010). Integration of the continuity equation (2.1a) over the domain of lower layer [−n, h] and use of the kinematic boundary condition (2.1h) leads to…”

mentioning

confidence: 98%

“…Equation (4.5) without a surfactant concentration represents a one-dimensional interface equation for a two-layer channel flow. The main difference to the interface equation obtained byCharru & Fabre (1994),Tilley et al (1994a),Segin et al (2005) andMavromoustaki et al (2010) lies on the choice of various dimensionless scales.…”

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confidence: 99%

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Effect of surfactant on two-layer channel flow

Samanta

2013

J. Fluid Mech.

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The effect of insoluble surfactant on the interfacial waves in connection with a two-layer channel flow is investigated for low to moderate values of the Reynolds number. Previous studies focusing on Stokes flow (are extended by including the inertial effect and the study of low-Reynolds-number flow (Blyth & Pozrikidis, J. Fluid Mech., vol. 521, 2004b, pp. 241-250) is enlarged up to moderate Reynolds number. Linear stability analysis based on the Orr-Sommerfeld boundary value problem identifies a surfactant mode together with an interface mode. The presence of surfactant on the interfacial mode is stabilizing at high viscosity ratio and destabilizing at low viscosity ratio. The threshold of instability is determined as a function of the Marangoni number. A long-wave model is developed to predict the families of nonlinear waves in the neighbourhood of the threshold of instability. Far from the threshold, wave dynamics is explored under the framework of a three-equation model in terms of lower layer flow rate q 2 (x, t), lower liquid-layer thickness h(x, t) and surfactant concentration Γ (x, t). Primary instability analysis of a three-equation model captures the result of the Orr-Sommerfeld boundary value problem very well for quite large values of wavenumber. In the nonlinear regime, travelling wave solutions demonstrate deceleration of maximum amplitude and acceleration of speed with the Marangoni number at high viscosity ratio m > 1 and show completely the opposite behaviour at low viscosity ratio m < 1. However, both maximum amplitude and speed attain a fixed value with increasing Reynolds number and this leads to saturation of instability.

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Section: Long-wave Modelmentioning

confidence: 99%

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confidence: 98%

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Effect of surfactant on two-layer channel flow

Samanta

2013

J. Fluid Mech.

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“…In particular, the LA was applied to hydraulic jumps (bores) developing in flows down an inclined substrate when the upstream depth h − exceeds the downstream depth h + (Benney 1966;Mei 1966;Homsy 1974;Lin 1974;Bertozzi & Brenner 1997;Bertozzi et al 1998Bertozzi et al , 2001Bertozzi & Shearer 2000;Chang & Demekhin 2002;Mavromoustaki, Matar & Craster 2010). Note that, in all lubrication models, solutions describing steadily propagating bores exist for all values of h ± and the substrate's inclination angle α.…”

Section: Introductionmentioning

confidence: 99%

Hydraulic jumps in a shallow flow down a slightly inclined substrate

Benilov

2015

J. Fluid Mech.

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This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number Re remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as Re → 0 and Re → ∞, respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the (η, s) parameter space, where η is the bore's downstream-to-upstream depth ratio, and s is a non-dimensional parameter characterising the substrate's slope. The criterion reflects two different mechanisms restricting bores. If s is sufficiently large, a 'corner' develops at the foot of the bore's front -which, physically, causes overturning. If, in turn, η is sufficiently small (i.e. the bore's relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.

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“…In the companion Part I article, 44 we used lubrication theory to develop a single, two-dimensional (2-D) evolution equation for the interface separating two immiscible fluids in an inclined channel. Part I was devoted to determining one-dimensional (1-D) solutions of this equation and the dependence of their structure on the difference between the inlet and outlet interface heights, the degree of density and viscosity stratification, the flow configuration (whether co-or counter-current) and channel inclination.…”

Section: Introductionmentioning

confidence: 99%

Shock-wave solutions in two-layer channel flow. II. Linear and nonlinear stability

Mavromoustaki

Matar

2011

Physics of Fluids

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We investigate the flow of two immiscible fluids in an inclined channel, building on the work presented in Part I of this study. In this paper, we examine the stability of the flow to spanwise perturbations in both the linear and nonlinear regimes. The evolution equation governing the interfacial dynamics, derived using lubrication theory in Part I, is linearised to study the effect of system parameters on the linear stability of the interface. A transient growth analysis of the linearised equation is carried out with no-flux conditions in the spanwise direction. The results of this analysis reveal that increasing the density and=or the viscosity of the upper layer, and=or increasing the counter-current nature of the flow configuration exerts a stabilising influence. Inspection of the flow profiles indicates that single Lax-shocks and the trailing Lax-shocks in Laxundercompressive double-shocks are unstable to finger formation; undercompressive shocks and rarefaction waves are stable. In unstably stratified cases, increasing the channel inclination away from verticality, such that a denser upper layer overhangs a less dense lower one, is found to be destabilising. These results are used to guide our transient numerical simulations aimed at studying the nonlinear development of fingering phenomena.

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Shock-wave solutions in two-layer channel flow. I. One-dimensional flows

Cited by 17 publications

References 35 publications

Effect of surfactant on two-layer channel flow

Effect of surfactant on two-layer channel flow

Hydraulic jumps in a shallow flow down a slightly inclined substrate

Shock-wave solutions in two-layer channel flow. II. Linear and nonlinear stability

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