Wave formation in the gravity-driven low-Reynolds number flow of two liquid films down an inclined plane

Chen, Kangping

doi:10.1063/1.858714

Cited by 55 publications

(55 citation statements)

References 20 publications

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“…This result was extended to the case of arbitrary wavenumber in the Stokes flow limit, and in the absence of interfacial and surface tension, by Loewenherz and Lawrence [14]. Further analysis by Chen [15] allowed for surface tension and inertia and concluded that the clean two-layer flow is always unstable at any Reynolds number if the less viscous fluid is next to the wall. Jiang and Lin [16] showed that the unstable two-layered flow on a vertically inclined plate can be stabilised by oscillating the plate in its own plane.…”

Section: Introductionmentioning

confidence: 87%

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Thompson

Blyth

2016

Phys. Rev. Fluids

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Multilayer film flow down an inclined plane in the presence of an insoluble surfactant is investigated with particular emphasis on determining flow stability and investigating the possibility of travelling wave solutions. The investigation is conducted for two or three layers under conditions of Stokes flow and, separately, on the basis of a long-wave assumption. A normal mode linear stability analysis for Stokes flow shows that adding surfactant to one of the film surfaces can destabilise an otherwise stable flow configuration. For the long-wave system, periodic travelling-wave branches are detected and traced, revealing solutions with pulse-like solitary waves on each film surface travelling in phase with each other, travelling waves with capillary ridge structures, and solutions with two of the film surfaces almost in contact. Time-periodic travelling wave solutions are also found. The stability of the travelling waves is determined by solving initial value problems and by computing eigenvalue spectra. Boundary element simulations for Stokes flow confirm the existence of travelling waves outside of the longwave regime.

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Section: Introductionmentioning

confidence: 87%

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Thompson

Blyth

2016

Phys. Rev. Fluids

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“…In view of its existence in Stokes flows (in the absence of inertia), this is referred to as interfacial inertialess instability. Following Kao [7], stability of multilayer free surface flows has been considered in a number of investigations [11,[13][14][15][16][17][18][19][20][21][22][23]. Loewenherz and Lawrence [11] have examined the effects of viscosity stratification on the linear stability of a two-layer flow with equal densities down an inclined substrate within the framework of Orr-Sommerfeld analysis, in the limit of Stokes flow and negligible surface tension both at the liquid-liquid interface and at the free surface.…”

Section: Introductionmentioning

confidence: 99%

“…Further, the neutral stability is independent of the angle of inclination of the inclined substrate but the growth rate of instability is not. The study by Chen [14] has incorporated the inertial effects in the linear stability analysis and has explored the influence of surface tension at the free surface and the interfacial tension at the liquid-liquid interface. A configuration with less viscous fluid adjacent to the wall has been observed to be unstable for any Reynolds number and any finite surface tension and interfacial tension.…”

Section: Introductionmentioning

confidence: 99%

Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow

Anjalaiah

Usha

2015

Acta Mech

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This manuscript provides a theoretical stability analysis for the configuration of two-layer immiscible flow down an inclined plane with velocity slip along the incline in the limit of zero Reynolds number. Surfactants may be present at the air-liquid interface, liquid-liquid interface, or both. In addition to an Orr-Sommerfeld analysis (again at zero Reynolds number), a long wavelength stability analysis is performed and the results are shown to be consistent. The interface mode, namely the mode of instability that arises because of viscosity stratification, is examined. Stability results (growth rates as a function of slip parameter and neutral stability boundaries) for various configurations of viscosity, surfactant placement, and layer thickness are compared with those of the previous literature and found to agree. It is found that velocity slip along the inclined plane reduces the maximum growth rate of instabilities in configurations where they occur, and the range of unstable wave numbers shrinks as well, indicating that slip has a promise for stabilization. This suggests that there is a possibility of using this favourably as a control option for two-layer flows in the absence or presence of surfactants, in relevant applications by designing the substrate to be a porous substrate with small permeability or a slippery substrate or a rough substrate or a hydrophobic substrate which can be modelled as substrates with velocity slip.

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“…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. Chen (1993) argues that the instability arises from an interaction between the free surface and the interface, while an interpretation of the underlying mechanism has been given recently by Gao & Lu (2008).…”

Section: Introductionmentioning

confidence: 99%

“…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. Chen (1993) argues that the instability arises from an interaction between the free surface and the interface, while an interpretation of the underlying mechanism has been given recently by Gao & Lu (2008). An analogous linear stability study was undertaken by Li (1969), for Couette flow of three superposed fluids of different viscosities; it was shown that the flow can become unstable in the long-wavelength limit for certain values of the depth and viscosity ratio due to resonance between the interfaces, something that does not happen if the additional interface is not present.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear interfacial dynamics in stratified multilayer channel flows

2013

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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. ...

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Wave formation in the gravity-driven low-Reynolds number flow of two liquid films down an inclined plane

Cited by 55 publications

References 20 publications

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow

Nonlinear interfacial dynamics in stratified multilayer channel flows

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