Steady two-layer flow in a topographically patterned channel

Lenz, Richard D.; Kumar, Satish

doi:10.1063/1.2784809

Cited by 21 publications

(33 citation statements)

References 29 publications

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“…It is also complementary to previous work on two-layer flows, which has no pressure-driven or gravity component but accounts for rupture and dewetting by intermolecular forces and surface heterogeneities. 5,[31][32][33][34] Finally, in part II we will examine the linear and nonlinear stability of the flow to spanwise disturbances, which have not been done previously.…”

Section: Introductionmentioning

confidence: 99%

“…This is of relevance to flow in microfluidic devices where inertia is negligible 4 and coating flow applications featuring bilayers. 5 The present paper corresponds to part I of a two-part study and focuses on onedimensional flows; the companion part II paper explores the stability of these flows to spanwise perturbations. The flows under consideration here are driven by a combination of pressure-and gravity-driven, co-and countercurrent configurations.…”

Section: Introductionmentioning

confidence: 99%

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

Mavromoustaki

Matar

2010

Physics of Fluids

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We study the dynamics of an interface separating two immiscible layers in an inclined channel. Lubrication theory is used to derive an evolution equation for the interface position that models the two-dimensional flow in both co-and countercurrent configurations. This equation is parameterized by viscosity and density ratios, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are treated as additional parameters. In the present work, which corresponds to part I of a two-part paper, we focus on one-dimensional flows. We use an entropy-flux analysis to delineate the existence of various types of shocklike solutions, which include compressive Lax shocks, pairs of Lax and under-compressive shocks, and rarefaction waves. Flows characterized by unstably stratified layers are accompanied by the formation of propagating, large-amplitude interfacial waves, which are not shocklike in nature. The results of our transient numerical simulations accord with our analytical predictions and elucidate the mechanisms underlying spatio-temporal development of the various types of waves; the stability of these waves to spanwise perturbations is investigated in part II of this work.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Mavromoustaki

Matar

2010

Physics of Fluids

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“…The flow of two immiscible fluid layers in channels appears in a range of applications such as lubricated piping, oil recovery, and lithographic printing [7]. Such flows are also common in microfluidic technologies [16].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we propose a simple technique to build solutions for the two-layer flow problem in corrugated channels. It is based on the analysis of Lenz and Kumar [7] who reduced the Navier-Stokes equations to a nonlinear ordinary differential equation for the interface profile in the long-wave approximation limit. This flow configuration is particularly suitable as a benchmark problem as the interaction between the wall topography and the two-fluid layer interface is nontrivial, as recently demonstrated by Luo and Pozrikidis [9] and Luo et al [8].…”

Section: Introductionmentioning

confidence: 99%

A Note on Approximate Benchmark Solutions for Viscous Two-Layer Flows

Sellier¹,

Lenz²

2010

ANZIAM J.

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An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the twolayer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.2000 Mathematics subject classification: primary 76T99; secondary 65L09.

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“…Lenz & Kumar (2007) analysed steady states of the interface between two viscous fluids flowing between one flat boundary and one with periodic steps using lubrication theory. This set-up was later considered by Zhou & Kumar (2012) with full numerical simulations that showed good agreement with those from lubrication theory.…”

Section: Introductionmentioning

confidence: 99%

Two-layer electrified pressure-driven flow in topographically structured channels

Dubrovina

Papageorgiou

2017

J. Fluid Mech.

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The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a stream wise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large amplitude long waves in the Stokes flow regime. For flat walls the electrified flow is long-wave unstable with a critical cut-off wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously -time modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low amplitude sinusoidal boundary, time modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large amplitude waves that span the channel and are subharmonic relative to the wall. A type of "walking" motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Nonuniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

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Steady two-layer flow in a topographically patterned channel

Cited by 21 publications

References 29 publications

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

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

A Note on Approximate Benchmark Solutions for Viscous Two-Layer Flows

Two-layer electrified pressure-driven flow in topographically structured channels

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