Stability of Liquid Flow down an Inclined Plane

Yih, Chia‐Shun

doi:10.1063/1.1706737

Cited by 940 publications

(511 citation statements)

References 4 publications

Supporting

Mentioning

473

Contrasting

Unclassified

Order By: Relevance

“…The Benney equation asymptotically predicts the linear stability threshold for the longwave hydrodynamic instability in agreement with the Orr-Sommerfeld equation, which corresponds to the exact linear stability problem of the Navier-Stokes equations (Yih 1963). It also allows for bounded nonlinear travelling-wave solutions, i.e.…”

Section: Finite-time Blow-upmentioning

confidence: 66%

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Scheid¹,

Ruyer-Quil²,

Thiele³

et al. 2005

J. Fluid Mech.

View full text Add to dashboard Cite

The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between finite-time blow-up of the Benney equation and absence of one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. Especially, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in any case, even with the thermocapillary effect, in contrast with horizontally heated films. The accuracy of bounded solutions of the Benney equation is also analysed by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponding to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

show abstract

Section: Finite-time Blow-upmentioning

confidence: 66%

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Scheid¹,

Ruyer-Quil²,

Thiele³

et al. 2005

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…The stability of a single-phase falling film was first examined by Binnie (1957), Benjamin (1957) and Yih (1963). Binnie performed experiments on the onset of wave formation on a film of water flowing down a vertical wall, while Benjamin and Yih showed that the critical liquid Reynolds number beyond which instability occurs is proportional to the cotangent of the angle of inclination ft.…”

Section: Gravity-induced Instabilitymentioning

confidence: 99%

Classification of instabilities in parallel two-phase flow

Boomkamp

1996

International Journal of Multiphase Flow

View full text Add to dashboard Cite

Abstract--There is extensive literature on the stability of parallel two-phase flow, both in the context of liquid-liquid as well as gas-liquid flow. Aimed at making this literature more transparent, this paper presents a classification scheme for the various instabilities arising in parallel two-phase flow..To achieve such a classification, the equation governing the rate of change of the kinetic energy of the disturbances is evaluated for relevant values of the physical parameters. This shows the existence of five different ways of energy transfer from the primary to the disturbed flow, which have their origin in density stratification, velocity profile curvature, viscosity stratification or shear effects. Each class is discussed on the basis of references covering the developments over the last 35 years. © 1997 Elsevier Science Ltd.

show abstract

“…As a result, such systems (both closed and open) support instabilities that are not seen in two-layer flows. It has been established that two-layer flows in inclined or pressure-driven channels and single-layer free-surface flows down inclined planes, require fluid inertia for destabilisation, at least when the inclination to the horizontal is less than ninety degrees (see Chen 1995;Benjamin 1957;Yih 1963). However, in the case of two-layer free-surface flows, Kao (1968), Loewenherz & Lawrence (1989) and Chen (1993) showed that when the less viscous fluid is adjacent to the wall, then a long-wave instability can appear in the absence of inertia (zero Reynolds number); this instability has been termed inertialess instability.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear interfacial dynamics in stratified multilayer channel flows

2013

View full text Add to dashboard Cite

The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically the particular system of three stratified layers with two internal fluid-fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared to the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney-type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilisation of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearised advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic-elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto-Sivashinsky type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularise the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. ...

show abstract

Stability of Liquid Flow down an Inclined Plane

Cited by 940 publications

References 4 publications

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Classification of instabilities in parallel two-phase flow

Nonlinear interfacial dynamics in stratified multilayer channel flows

Contact Info

Product

Resources

About