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Review Article

Stability of Two-Immiscible-Fluid Systems: A Review of Canonical Plane Parallel Flows

[+] Author and Article Information
Alireza Mohammadi

Department of Mechanical and
Aerospace Engineering,
Princeton University,
Princeton, NJ 08544
e-mail: alirezam@princeton.edu

Alexander J. Smits

Department of Mechanical and
Aerospace Engineering,
Princeton University,
Princeton, NJ 08544;
Department of Mechanical and
Aerospace Engineering,
Monash University,
Victoria 3800, Australia
e-mail: asmits@princeton.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 5, 2016; final manuscript received May 26, 2016; published online July 29, 2016. Assoc. Editor: Francine Battaglia.

J. Fluids Eng 138(10), 100803 (Jul 29, 2016) (17 pages) Paper No: FE-16-1139; doi: 10.1115/1.4033969 History: Received March 05, 2016; Revised May 26, 2016

A brief review is given on the stability of two-fluid systems. Our interest is primarily driven by drag reduction using superhydrophobic surfaces (SHS) or liquid-infused surfaces (LIS) where the longevity and performance strongly depends on the flow stability. Although the review is limited to immiscible, incompressible, Newtonian fluids with constant properties, the subject is rich in complexity. We focus on three canonical plane parallel flows as part of the general problem: pressure-driven flow, shear-driven flow, and flow down an inclined plane. Based on the linear stability, the flow may become unstable to three modes of instabilities: a Tollmein–Schlichting wave in either the upper fluid layer or the lower fluid layer, and an interfacial mode. These instabilities may be further categorized according to the physical mechanisms that drive them. Particular aspects of weakly nonlinear analyses are also discussed, and some directions for future research are suggested.

Copyright © 2016 by ASME
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Figures

Grahic Jump Location
Fig. 1

Sketch of the flow configuration for Poiseuille flow. Parameters dj, ρj, and μj denote the thickness, density, and dynamic viscosity of each fluid, where subscript j=1, 2 correspond to the upper and lower fluids, respectively, and U0 defines the interface velocity.

Grahic Jump Location
Fig. 2

Neutral stability diagram in (n, m)-plane for long-wavelength disturbances for r=1, F=0, and S=0. The zones of stable and unstable regions are marked by S and U (reproduced based on asymptotic results of Ref. [43]).

Grahic Jump Location
Fig. 3

Eigenvalue spectrum for two-fluid flow with Re = 6000, α = 1, β = 0, n = 1, r = 1, and F = S = 0. (a) m = 1, the only unstable mode is σ=0.259 815 871+0.000 323 089i, a shear mode belonging to branch A, (b) m = 0.5, the two unstable modes are σ=1.006 908 625+0.006 689 680i an interfacial mode belonging to branch P, and σ=0.248 975 403+0.003 531 211i a shear mode belonging to branch A. ○ and + denotes OS and Sq modes, respectively.

Grahic Jump Location
Fig. 4

Neutral stability diagrams for β=0, r=1, F=0, and S=0. (a) m=20, Re = 10 and (b) m=0.05. The vertical line is n=m, which is a neutral stability line. Labels S and U denote stable and unstable zones, respectively. With a proper change of variables (see Eqs.(13a)(13d)) points (a′,b′) are mapped onto points (a,b). (Reprinted with the permission from Yiantsios and Higgins [43]. Copyright 1988 of AIP Publishing.)

Grahic Jump Location
Fig. 5

Neutral stability diagram for β=0, m=20, r=1, F=0, Re=10, and different S. The dashed curves are for S = 0. Labels S and U denote stable and unstable zones, respectively. (Reprinted with the permission from Yiantsios and Higgins [43]. Copyright 1988 of AIP Publishing.)

Grahic Jump Location
Fig. 6

Neutral stability diagram for β=0, m = 0.05, S = 0, and Re = 10. (a) Stabilizing density stratification (F>0) with r=1.5. (b) Destabilizing density stratification (F<0) with r=0.5. The dashed curves are for F = 0. Labels S and U denote stable and unstable zones, respectively. (Reprinted with the permission from Yiantsios and Higgins [43]. Copyright 1988 of AIP Publishing.)

Grahic Jump Location
Fig. 7

Neutral stability diagram for the shear mode for flow parameters n=1, r=1, F=0, and S=0. Re* is Reynolds number defined by maximum velocity across the channel and is related to Re by Re*=Re{1+(m−1)2/[8(m+1)]}. Labels S and U denote stable and unstable zones, respectively. (Reprinted with the permission from Yiantsios and Higgins [43]. Copyright 1988 of AIP Publishing.)

Grahic Jump Location
Fig. 8

Variations of the imaginary part of the phase speed ci=σi/α (a) and the real part of the phase speed cr=σr/α (b) as a function of wavenumber α for flow in a channel with an air layer overlying a water layer with m = 64, n = 1, ReST = 10,000. Reynolds number ReST is defined based on properties of air layer and its thickness and the average velocity across the channel. TSA-mode and I-mode refer to the TS wave in the air layer and the interfacial mode, respectively. (Reprinted with the permission from Shapiro and Timoshin [62]. Copyright 2005 of AIP Publishing.)

Grahic Jump Location
Fig. 9

Evolution of the energy growth obtained by all modes G(t) and the first three adjoint modes, Ĝ1(t), Ĝ2(t), and Ĝ3(t), respectively, for three-dimensional two-fluid flow when ReMH=3000, α=0, β=2.41, m=0.5, n=1, and r=1. Reynolds number ReMH is defined by half-channel height, maximum velocity across channel Umax, and average viscosity across the channel. Solid, dashed, dashed–dotted, and thick lines correspond to G(t), Ĝ2(t), Ĝ1(t), and Ĝ3(t), respectively. (Reprinted with the permission from Malik and Hooper [45]. Copyright 2007 of AIP Publishing.)

Grahic Jump Location
Fig. 10

Energy growth G(t) at Reynolds number ReY=ρ2U0d2/μ2=900, r=1.11, α=0, and β=1 and different values of Weber number defined by We=ρ2U02L2/γ. (a) m=0.5, inset shows early time and (b) m=0.05. (Reprinted with the permission from Yecko [65]. Copyright 2008 of AIP Publishing.)

Grahic Jump Location
Fig. 11

Stability diagram showing the regions of convective (σo,i<0) and absolute instability (σo,i>0) in (Re, m)-plane for different values of β. Curves separating the absolute instability regions from the convective instability regions correspond to σo,i=0. Here ReSM denotes Reynolds number and is defined as ReSM=ρ1Um(d1+d2)/μ1, where Um is the average velocity of the base flow across the channel. The parameters used are ReSM=500, n=3/7, r=1.2, G=(ρ2−ρ1)g(d1+d2)2/μ1Um=10 is a dimensionless gravitational parameter, and Γ=γ/μ1Um=0.01 is an inverse capillary number. (Reprinted with the permission from Sahu and Matar [66]. Copyright 2011 of AIP Publishing.)

Grahic Jump Location
Fig. 12

Flow regimes and interfacial structure. ○: Laminar– turbulent transition in water phase, •: laminar–turbulent transition in oil phase, —: indicates transition in interfacial wave structure. The superficial Reynolds number for each phase is defined based on hydraulic diameter of the conduit, superficial velocity of that phase defined as the volumetric flow rate of thephase divided by the conduit cross-sectional area and kinematic viscosity of that phase. (Reprinted with the permission from Charles and Lilleleht [68]. Copyright 1965 of AIP Publishing.)

Grahic Jump Location
Fig. 13

Sketch of the flow configuration for unbounded two-layer Couette flow

Grahic Jump Location
Fig. 14

Sketch of the flow configuration for semibounded two-layer Couette flow

Grahic Jump Location
Fig. 15

Sketch of the experimental device. (Reprinted with the permission from Charru and Barthelet [74]. Copyright 1999 of AIP Publishing.)

Grahic Jump Location
Fig. 16

The “double-exponent” velocity profile defined by Eq.(14) and used in Ref. [55]

Grahic Jump Location
Fig. 17

Sketch of the flow configuration for free surface flow down an inclined plane

Grahic Jump Location
Fig. 18

Topological features of ci curves. (a) γ=0 and θ=π/2. Both axes are neutral stability curves (ci=0). (b) γ=0 and θ<π/2. At the bifurcation point B, the flow Reynolds number Re is Re=5cot(θ)/4. The line α=0 is a neutral stability line (ci=0). (c) γ≠0 and θ=π/2. ci=0 on α=0. (d) γ≠0 and θ<π/2. ci=0 on α=0. ci→−S′/2 as α→∞, where S′=S  Re. Re=Rec=5cot(θ)/4 at the bifurcation point B. (Reprinted with the permission from Yih [39]. Copyright 1963 of AIP Publishing.)

Grahic Jump Location
Fig. 19

Neutral curves for θ=0.5′ (a) and θ=4′ (b) for various ζ (nondimensional surface tension number defined by ζ=(3ργ3/gμ4)1/3) for the shear and surface modes. The values of the critical Reynolds number for the surface mode are Rec=5cot(θ)/4=8594.4 in Fig. 15(a) and Rec=1074.3 in Fig.15(b). Symbol ′ denotes minute of arc, one minute of arc equals to 1/60 of one degree, i.e., 1′=(1/60) deg. Labels S and U denote stable and unstable zones, respectively. (Reprinted with the permission from Floryan et al. [76]. Copyright 1987 of AIP Publishing.)

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