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SIAM REVIEW c© 2000 Society for Industrial and Applied MathematicsVol. 42, No. 3, pp. 417–439

Numerical Study of Flows ofTwo Immiscible Liquids at LowReynolds Number∗

Jie Li†

Yuriko Renardy†

Abstract. We review our recent results on fingering, bamboo waves, and drop breakup in low Reynoldsnumber flows composed of two viscous liquids under shear.

Key words. volume-of-fluid scheme, interface tracking, semi-implicit Stokes solver, two-layer flows,low Reynolds number flows

AMS subject classifications. 76D05, 65-02, 76E05

PII. S0036144599354604

1. Introduction. Many configurations are possible for the flow of two immisci-ble fluids: layers, fingers, encapsulated regimes, and drops [1, 2, 3]. We describeour numerical simulations of these two-fluid systems with the volume-of-fluid (VOF)scheme. In the case of the layered regime, we describe what happens when the twoliquids are moving past each other and form large amplitude waves, sheets, fingers,and drops as a result of an interfacial instability driven primarily by the difference inviscosities. The jump in the viscosity from one fluid to the other results in the jumpin the tangential velocity gradient across the interface [4] and can be thought of asa viscous counterpart of the Kelvin–Helmholtz instability. In the case of a drop de-forming under shear, we focus on the breakup regime, driven by shear acting againstthe restoring force of interfacial tension.

2. Numerical Method. One of the greatest difficulties in the study of two im-miscible fluid flows is that the domain of interest contains an unknown interface: theinterface moves from one location to another and may sometimes undergo severe de-formations, including even breakup; the interface plays a major role in defining thesystem and must be determined as part of the solution. In the numerical treatmentof the interface, we must answer three questions: (1) How do we represent the in-terface on a mesh? (2) How will the interface evolve in time? and (3) How shouldwe apply boundary conditions on the interface? There are many interface trackingmethods, such as the moving grid method, the front tracking method, the level setmethod, and the VOF method. The VOF method provides a simple way of treatingthe topological changes of the interface, as well as ease of generalization to the three-

∗Received by the editors April 19, 1999; accepted for publication (in revised form) November3, 1999; published electronically July 31, 2000. This work was supported by NSF-CTS 9612308,NSF-INT9815106, and NCSA-CTS grants.

http://www.siam.org/journals/sirev/42-3/35460.html†Department of Mathematics, 460 McBryde Hall, Virginia Polytechnic Institute and State Uni-

versity, Blacksburg, VA 24061-0123 ([email protected], [email protected]).

417

418 JIE LI AND YURIKO RENARDY

dimensional case. This approach was first introduced by DeBar [5] in 1974, followedby significant advance from Youngs’s work [6] eight years later. More recent worksinclude [7, 8, 9, 10], and the review article of [11].

2.1. The Equations of Motion. We suppose that the two fluids are immiscibleand that the density ρ and the viscosity µ are constant in each fluid, but we do allowfor the possibility of a jump across the interface. We use a concentration function Cto represent and track the interface:

C(x) ={

1, fluid 1,0, fluid 2.

(2.1)

The concentration function is transported by the velocity field u:

∂C

∂t+ u · ∇C = 0.(2.2)

The average values of the density and the viscosity are given by

ρ = Cρ1 + (1 − C)ρ2, µ = Cµ1 + (1 − C)µ2,(2.3)

where subscripts refer to fluids 1 and 2.We suppose also that the flow is incompressible,

∇.u = 0,(2.4)

governed by the Navier–Stokes equation

ρ

(∂u∂t

+ u · ∇u)

= −∇p + ∇ · µS + F,(2.5)

where S is the viscous stress tensor

Sij =12

(∂uj

∂xi+

∂ui

∂xj

).

The body force F includes the gravity and interfacial tension force [12, 7, 8, 13, 14, 15,16]. The interfacial tension force is Fs = σκnSδS , where σ is the interfacial tension,κ is the mean curvature, and nS is the normal to the interface.

2.2. Temporal Discretization and Projection Method. The simultaneous solu-tion of the large number of discrete equations arising from (2.4) and (2.5) is verycostly, especially in three spatial dimensions. An efficient approximation can be ob-tained by decoupling the solution of the momentum equations from the solution ofthe continuity equation by a projection method [17].

In the projection method, the momentum equations (2.5) are first solved for anapproximate u∗ without the pressure gradient, assuming that un is known:

u∗ − un

∆t= −un · ∇un +

1ρ

(∇ · (µS) + F)n.(2.6)

In general, the resulting flow field u∗ does not satisfy the continuity equation (2.4).It is corrected by the pressure

un+1 − u∗

∆t= −∇p

ρ(2.7)

TWO-LIQUID FLOW AT LOW REYNOLDS NUMBER 419

x i∆

zk∆

y∆ j

x

y

z

Fig. 2.1 Three-dimensional Cartesian mesh with variable cell sizes.

U

CP

W

i-1/2,j,k

i,j,k-1/2

i,j

i,j Vi,j-1/2,k

Fig. 2.2 Location of variables in a Mac mesh cell.

in order to yield a divergence-free velocity un+1. Pressure is not known at this momentbut can be found from a Poisson equation,

∇ ·(∇p

ρ

)=

∇ · u∗

∆t,(2.8)

which is obtained by taking the divergence of (2.7). In the problems we addressbelow, the boundary conditions for the velocity are periodicity and the Dirichletcondition. Analogously, the boundary conditions for the pressure are periodicity andthe Neumann condition, respectively.

2.3. Spatial Discretization. An Eulerian mesh of rectangular cells is used, withvariable sizes: ∆xi for the size of the ith mesh cell in the x-direction, ∆yj for the sizeof the jth mesh cell in the y-direction, and ∆zk for the kth mesh cell in the z-direction(Figure 2.1).

The momentum equations are finite-differenced. As Figure 2.2 shows, the x-component of velocity ui− 1

2 ,j,k, the y-component of velocity vi,j− 12 ,k, and the z–

component of velocity wi,j,k− 12

are centered at the right face, front face, and top faceof the cell, respectively, whereas the pressure pi,j,k is located at the center. This isthe so-called Mac method [18]. This mesh has the advantage that the pressure field


does not allow spurious checkerboard oscillations. Another advantage of the Macmethod is that the Neumann condition for the pressure is automatically involved inthe numerical solution when the boundary condition for the velocity is the Dirichletcondition [19]. No numerical boundary condition is needed for the pressure.

The convective terms appearing in (2.6) are treated with a nonconservative scheme,

DUL = (ui− 12 ,j,k − ui− 3

2 ,j,k)/∆xi−1,

DUR = (ui+ 12 ,j,k − ui− 1

2 ,j,k)/∆xi,

(u

∂u

∂x

)i,j,k

= ui,j,k∆xiDUL + ∆xi−1DUR

∆xi−1 + ∆xi;(2.9)

the derivatives are weighted by cell size such that the correct order of approximationis maintained on a variable mesh. The pressure and viscous terms in the momentumequation are calculated using second-order central finite differences.

The numerical solution of (2.8) is the most time consuming part of our Navier–Stokes solver and, consequently, an efficient solution is crucial for the performance ofthe whole method. Potentially, the multigrid method is the most efficient method.The basic idea of the multigrid method [20] is to combine two complementary proce-dures: one basic iterative method to reduce the high frequency error and one coarsegrid correction step to eliminate the low frequency error. We choose a two-colorGauss–Seidel iterative method because it breaks the dependence between the variablesand therefore allows for parallelization of the scheme. We use a Galerkin method toprovide a good coarse grid correction.

2.4. Advection of the Interface. At the discrete level, the concentration functionis the volume fraction field Cij : when a cell is filled by the fluid 1, Cij = 1; when acell does not contain any of this fluid, Cij = 0. The interfaces are in the cells with Cij

between 0 and 1. Given an interface, we can calculate a unique volume fraction field.On the other hand, when we represent the interface by a volume fraction field, we loseinterface information and we cannot determine a unique interface; the interface needsto be reconstructed. Piecewise linear interface calculation (PLIC) methods have beendeveloped in [7, 8] for two-dimensional and three-dimensional cases. The gist of thesemethods is to calculate the approximate normal n to the interface in each cell, sincethis determines one unique linear interface with the volume fraction of the cell. Thediscrete gradient of the volume fraction field yields

n =∇hC

|∇hC| .(2.10)

A least-squares method [9] has also been implemented in this work. Numerical testssuggest that this approximate interface is a second-order approximation.

The second step of the VOF method is to evolve the volume fraction field C.Because the interface evolution is governed by a transport equation, (2.2), the La-grangian method is the natural choice [21, 7]. In this scheme, once the interface isreconstructed, the velocity at the interface is interpolated linearly and then the newinterface position is calculated via xn+1 = xn + u(∆t). Figure 2.3 illustrates howthe Lagrangian method performs on an arbitrary two-dimensional mesh. In compari-son with the Eulerian method, the Lagrangian method has the following advantages:when the Courant condition (Max|u|)∆t/h < 1/2 is satisfied, this method is stable,and the volume fraction always satisfies the physical constraint 0 ≤ C ≤ 1.


Fig. 2.3 The Lagrangian method on an arbitrary two-dimensional mesh. The shaded polygon repre-sents the part occupied by the fluid in the central cell. The broken line shows the polygon’sposition after advection in the local velocity field, represented by arrows. The fluid is re-distributed between neighboring cells, which the new polygon partially overlaps.

In the VOF method, the interfacial tension condition across the interface is ap-plied not directly but rather as a body force over the cells that contain the interface.Two such formulations have been implemented in this work. The first is the continu-ous surface force formulation [22], in which fs = σκnS and Fs = fs∇C. The secondis the continuous surface stress formulation [12], in which Fs = ∇ · T = σδSκnS andT = [(1 − nS ⊗ nS)σδS], which leads to a conservative scheme for the momentumequation.

2.5. Semi-Implicit Stokes Solver. The above description completes the charac-terization of our numerical method. However, one weakness is that it is an explicitmethod and not suitable for simulations of low Reynolds number flows. For an explicitmethod, the time step ∆t should be less than the viscous time scale, Tµ = ρh2/µ,where h denotes the mesh size. This stability limit is much more restrictive than theCourant–Friedrichs–Levy (CFL) condition for simulations of low Reynolds numberflows. In order to carry out calculations for times of order 1, the implicit treatmentof the viscous terms is imperative.

Our approach is an original semi-implicit method. The time integration schemeis constructed to be implicit for the Stokes operator and is otherwise explicit. In theu component of the momentum equation (2.6), we treat only the terms related to u(the terms with upper index ∗) implicitly and leave the other terms (the terms withupper index n) in the explicit part, that is,

u∗ − un

∆t= (un · ∇)un +

1ρn

F n1 +

1ρn

∂

∂x(2µnu∗

x) +1ρn

∂

∂y(µnu∗

y + µnvnx )

+1ρn

∂

∂z(µnu∗

z + µnwnx ).(2.11)

This can be expressed as[I − ∆t

ρ

(∂

∂x

(2µ

∂

∂x

)+

∂

∂y

(µ

∂

∂y

)+

∂

∂z

(µ

∂

∂z

))]u∗ = explicit terms.(2.12)

This procedure decouples the u component from the previous parabolic system. Thisis still a first-order method, but the effort needed to solve (2.12) is significantly reduced


in comparison with the full implicit treatment. The same idea applies to the othervelocity components.

As far as the viscous terms are concerned, our semi-implicit scheme is uncondi-tionally stable [23, 24]. In addition, the factorization technique of [25] is applied tothe left-hand side of (2.12):

[I − ∆t

ρ

(∂

∂x

(2µ

∂

∂x

))] [I − ∆t

ρ

(∂

∂y

(µ

∂

∂y

))] [I − ∆t

ρ

(∂

∂z

(µ

∂

∂z

))]

u∗ = explicit terms.

The error of this factorization is O(∆t3). The inversion of the left-hand side of theabove equation requires solving only tridiagonal matrices, which results in a significantreduction in computing and memory. In fact, the solution of these tridiagonal systemscan be done in only O(N) operations (where N is the grid point number). In summary,this scheme is first-order accurate and unconditionally stable. The stability of thisscheme is crucial for simulation of low Reynolds number flow.

The direct simulation of two-fluid flow is often limited by computing cost andmachine memory, especially in the three-dimensional case. Three issues have been ofthe utmost importance in improving performance: accuracy, stability, and efficiency.The entire code (including the semi-implicit Stokes solver) is parallelized: on theOrigin2000 with 16 parallel processors, the efficiency of our code is more than 80%.

3. Simulation of Two-Layer Couette Flow. Flows composed of two immiscibleliquids and undergoing shearing motions can form fingers as a result of an interfacialinstability due primarily to the viscosity jump. We attempt to capture the qualitativefeatures of fingering by simulating two-layer Couette flow, which is one of the simplestof all the shearing flows of two fluids one might consider. A base flow solution is shownin Figure 3.1. The lower fluid is fluid 1 (with viscosity µ1) and occupies 0 < z < l1;fluid 2, with viscosity µ2, occupies l1 < z < 1. There are four parameters: theviscosity ratio m = µ1/µ2; the average depth of the lower liquid l1; the interfacialtension parameter T = σ/(µ2Ui), where σ is the interfacial tension coefficient and Ui

is the dimensional interfacial speed of the base flow; and a Reynolds number based onthe lower fluid R1 = Uil

∗ρ1/µ1, where l∗ is the dimensional plate separation. If thedensities are equal, then gravitational force is balanced by a pressure gradient andcan be neglected. If the densities are not equal, then two additional parameters arethe Froude number F , where F 2 = U2

i /(gl∗), and the density ratio r = ρ1/ρ2.

3.1. Stability of Two-Layer Couette Flow. We explore the evolution of smallamplitude disturbances, the weakly nonlinear regime, saturation to spatially periodictraveling waves, and fingering of large amplitude periodic disturbances. In order toachieve the computational results, we have implemented the semi-implicit scheme ofsection 2.5 to enable faster computations at low Reynolds numbers and a second-ordervelocity interpolation for greater accuracy of the interface advection.

The stability of two-layer Couette flow pictured in Figure 3.1 for small perturba-tions of wavenumber α has been addressed in [26, 27, 1]. When the initial conditionis an eigenmode, the full numerical simulations yield the correct growth rates for theinterface, as well as for the L∞-norm and L2-norm of the velocity field. Mesh conver-gence studies were conducted for small initial amplitudes and also with different timesteps. This comparison with linear theory provides a test for our code [23].

At low speeds, the flow can become unstable due to an interfacial instability thatarises from the jump in shear rates across the interface. This flow instability is a Hopf


Fluid 1

Fluid 2

interface

z=1

z=0

z=l1

z

xy

Fig. 3.1 Flow schematics for two-layer Couette flow.

bifurcation. The weakly nonlinear theory of [27, 13] yields a Stuart–Landau equationfor the amplitude function Z(t) of the primary mode, dZ/dt − ωZ = κ|Z|2Z, whereκ denotes the Landau coefficient, and ω the critical eigenvalue from the linearizedstability analysis. The dynamics just above the onset of instability is determined bythe interaction of the primary mode with the mean flow mode and the second har-monic and by a cubic self-interaction. The traveling wave solution Z(t) = exp(ict)Z0is predicted to saturate when the real part of the Landau coefficient is negative.The saturation amplitude and wave shape at saturation are also predicted by thetheory.

3.2. Long-Time Saturation. In the laboratory experiments [28], the long-timesaturation predicted by Hopf bifurcation theory is reached after a time on the orderof 1000 s. The lower fluid is a water-glycerine (32/68%) mixture with viscosity 0.0191Pa·s, density 1169 kg/m3; the upper fluid is mineral oil with viscosity 0.0297 Pa·s anddensity 846 kg/m3. Interfacial tension is 0.03 Pa m. The onset condition is modeledin [29], with upper plate velocity 0.44 m/s, channel depth 20mm, wavelength 6.8 cm,and depth of lower fluid 12.74 mm. Our parameters for this condition [27] are shownin the caption of Figure 3.2. Qualitative features of this flow are successfully capturedon a 256×256 mesh: Figure 3.2(a) illustrates the evolution of the interface amplitudevs. time, showing saturation after a time of order 1000 s. Figure 3.2(b) shows thesimulated wave shape at 2500 s with flat crests and sharp troughs, in agreement withthe weakly nonlinear theory and the experimental observations.

3.3. Two-Dimensional Fingers or Sheets. Large amplitude perturbations typi-cally lead to the formation of fingers.

3.3.1. Moderate Reynolds Number. Figure 3.3 shows the case of viscosity ratiom = 0.5, equal density, T = 0.01, Reynolds number R1 = 500, and wavenumberα = π/2. The initial amplitude is A(0) = 0.05. Interface profiles are plotted fort = 0, 5, 10, 12, 13, 14, 15, 20. The interface moves vertically under the viscosity-jumpinstability and is elongated horizontally by the base shear flow. The two fluids donot penetrate into each other in the same manner. Note that the upper fluid is moreviscous than the lower one. The qualitative features of this flow are retained if theamplitude is increased, but the sequence occurs faster for larger amplitudes.

3.3.2. Low Reynolds Number. Figure 3.4 shows the case of a low Reynoldsnumber, R1 = 40, with wavenumber α = 6.3. The initial amplitude of the interfaceis A(0) = 0.05. Interface profiles for time t = 0, 2, 5, 8, and 10 are plotted. One


A(t)

(b) Waveform

(a) time

Fig. 3.2 Simulation of saturation for the experiments in [28], R1 = 394, T = 3.14, F 2 = 0.528,density ratio r = 1.4, viscosity ratio m = 0.65, depth ratio l1 = l∗1/l∗ = 0.637, dimension-less wavenumber α = 1.9. (a) Maximum of interface amplitude against time. (b) Waveshape at time t = 2500.

0 1 2 3 40

0.5

1t=0

0 1 2 3 40

0.5

1t=13

0 1 2 3 40

0.5

1t=5

0 1 2 3 40

0.5

1t=14

0 1 2 3 40

0.5

1t=10

0 1 2 3 40

0.5

1t=15

0 1 2 3 40

0.5

1t=12

0 1 2 3 40

0.5

1t=20

Fig. 3.3 Sequence of interface positions for A(0) = 0.05, α = π/2, Re1 = 500, m = 0.5, l1 = 0.372,T = 0.01, equal densities, and zero gravity. The calculation was carried out on a 512×512mesh. The flow domain is 4 × 1 with periodicity in the x-direction.

striking fact compared to the flow with Reynolds number R1 = 500 is that the verticalgrowth of the fingers is small and the horizontal elongation is large. This is becausethere are two competing mechanisms governing the shape of the interface. First, thegrowth rate of the instability originating from the viscosity stratification contributes


t = 10

t = 8

t = 5

t = 2

t = 0

Fig. 3.4 Sequence of interface positions for A(0) = 0.05, α = 6.3, Re1 = 40, m = 0.5, l1 = 0.372,T = 0.01, equal densities, and zero gravity. t = 0, 5, 10, 15, and 20. The calculation wascarried out on a 160 × 160 mesh. The vertical axis is magnified from 0.25 to 0.50 to showthe details; the flow domain is 1 × 1 with periodicity in the x-direction.

to the vertical growth of the interface; this tends to zero as the Reynolds numbertends to zero. Second, the base shear flow of Figure 3.1 convects the large, initiallysinusoidal interface, as illustrated in Figure 3.5; the crest of the wave moves forwardfaster than the trough. At low Reynolds numbers, the second mechanism is dominantand initiates the fingering that is observed in Figure 3.4. Elongated two-dimensionalfingers have also been recorded [30] for creeping flow.

3.3.3. Three-Dimensional Fingers. The response to periodic perturbations inthree dimensions is analyzed by using a horizontal undulation of the phase in theinitial condition. The initial interface is

z = l1 + Ax(0) cos(αxx + φ(y)), φ = Ay(0) cos(αyy),(3.1)

where Ax(0) is the two-dimensional perturbation amplitude, Ay(0) the spanwise per-turbation amplitude, αx the x-direction wavenumber, and αy the y-direction wavenum-ber. The perturbation formulation is inspired by the experiment of [31] on shearlayers.

The two-dimensional fingering study in the previous section provides useful in-formation for the subsequent three-dimensional investigation. As we have seen, small


x

z

moving wall

wall at rest

Fig. 3.5 Large amplitude disturbance in creeping flow leads to fingering because the trough is leftbehind while the crest hurries on.

Fig. 3.6 Simulation of two-layer Couette flow for Reynolds number R1 = 500, α = π/2, m = 0.5,l1 = 0.372, T = 0.01, equal densities, and zero gravity. The initial interface height isz = 0.372 + 0.05 cos{6.3x + 0.1 cos[y/(4π)]}. The interface position is shown at t = 12. Afinger forms in the low viscosity fluid. The three-dimensional finger is in the process ofbreaking up into a series of drops of the low viscosity fluid.

perturbations to low Reynolds number flow involves small interface structures, anda fine mesh is needed to capture them. In addition, a three-dimensional simulationis limited by machine memory and computation time. Figure 3.6 shows the case ofReynolds number R1 = 500. The spanwise wavenumber is selected to be αy = 4π;therefore, the computational domain is a 4×0.5×1 box, and the mesh is 128×32×128.As in the previous section, the initial two-dimensional amplitude is Ax(0) = 0.05 fora flow with wavenumber αx = π, R1 = 500, viscosity ratio m = 0.5, undisturbedinterface height l1 = 0.372, and T = 0.01. We use a spanwise amplitude Ay(0) = 0.1.


Fig. 4.1 (a) Schematic of core-annular flow. (b) Axisymmetric waves.

In Figure 3.6, we have inverted our three-dimensional visualization box vertically toprovide a better view. The finger is in the lower fluid; the case t = 12 shows thisfinger to be breaking up, yielding a series of drops of the low viscosity fluid.

A droplet will pinch off in a VOF scheme when the neck has a thickness on theorder of a grid cell. It is not clear, then, that a numerically observed droplet breakupcorresponds to a physical one, unless a refinement study is conducted. In [32] itwas observed that at low resolution for a two-dimensional computation of two-layerCouette flow, drops were observed to form, while at high resolution, these dropletswere replaced by an elongated finger or filament. The conclusion here is that whendrops break off of filaments and are of the size of the mesh, then a refinement needsto be done in order to ascertain whether the drops are numerical or physical. Thisproblem is inherent in the VOF approach. There is a difference between the breakupinto drops in the two-dimensional case and the three-dimensional Couette flow. Thebreakup recorded in [23] for two-dimensional Couette flow is not physical becausethere is no breakup mechanism in two dimensions. Hence, the numerical refinementin [32] resolved the issue as thin fingers. Drop breakup in three dimensions, however,is physical for two reasons. First, there is clearly an interfacial tension driven breakupmechanism, and the VOF results are similar to the more accurate boundary integralscheme for neck formation [33]. Second, a numerical resolution study for drop breakupis shown in section 5.3 in reference to Figure 5.3, which compares satisfactorily withits refinement in Figure 5.4.

4. Simulation of Core-Annular Flow. Core-annular flow is a pressure-drivenflow through a pipe of one fluid at the core and another fluid in the annulus (Figure4.1). This arrangement arises naturally for fluids with markedly different viscosities,because higher viscosity material tends to become encapsulated by lower viscositymaterial. An industrial application is the lubricated pipelining of crude oil with theaddition of water [2, 3]. The purpose is to efficiently transport a very viscous liquid,which on its own would require costly work. However, when the viscous fluid just alongthe wall is replaced by a much less viscous immiscible one, in this case water, thenthe work required for transportation is significantly lowered. The ideal arrangementhas a perfectly cylindrical interface (Figure 4.1(a)) and is called perfect core-annularflow (PCAF), but a wavy interface is also viable (Figure 4.1(b)).


4.1. Parameters. Consider the PCAF of Figure 4.1(a). The dimensionless piperadius is a = R2/R1, where R2 is the dimensional pipe radius and R1 is the radius ofthe undisturbed interface. The pressure gradient in the axial direction is a constant:dP ∗/dx = −f∗. The viscosity ratio is denoted m = µ2/µ1, the radius ratio is a, thedensity ratio is ζ = ρ2/ρ1, and the ratio of driving forces in the core and annulus isK = (f∗ + ρ1g)/(f∗ + ρ2g). The base velocity field is in the axial direction and isa function of the radial variable. An interfacial tension parameter is J = σR1ρ1/µ2

1,where σ denotes the interfacial tension. Reynolds numbers Rei are defined by Rei =ρiV0R1/µi, i = 1, 2, where V0 denotes the centerline axial velocity. In summary,PCAF is characterized by six dimensionless parameters: m, a, ζ, J , K, and Re1 = Re.

PCAF can lose stability to a variety of regimes. Several qualitatively differentregimes of flow have been documented in the experiments of [34], which were con-ducted for a pipe diameter of 3/8 inch, m = 1/601, ζ = 1.10, R2 = 3/16 inch,σ = 8.54 dyn/cm, and aJ = 0.102. The oil at the core is lighter than the water in theannulus, so that buoyancy and the pressure gradient act in the same sense in up-flow,where the core oil is observed to produce bamboo waves, and in the opposite sense indown-flow, where the core is compressed and typically buckles into corkscrew waves[35]. The volume flow rates of water and oil are denoted by Qw and Qo, respectively.The superficial water velocity is defined by Vw = Qw/A and the superficial oil velocityby Vo = Qo/A. Here, A = πR2

2 is the cross-sectional area of the pipe. Conservationof volume implies that the parameter a will remain constant as the PCAF evolvesinto the nonlinear regime. The evolution is, however, more complicated for superficialvelocities Vo and Vw.

The hold-up ratio h, defined as the ratio of the input oil/water flow rate ratioto the in situ oil/water volume ratio, is an important practical parameter for core-annular flow. Experiments show that h is constant in up-flow and fast flows. Theradius ratio a is related to the flow rates via the hold-up ratio h. If the flow is perfectlycore-annular, then it can be shown that

a =√

1 + hQw/Qo =√

1 + hVw/Vo.(4.1)

4.2. Bamboo Waves. Our motivation for studying bamboo waves in verticalcore-annular flow is that these structures are well documented in the experimentsof [34]. In the bamboo wave regime, trains of Stokes-like waves with sharp crestsare connected by long filaments. The waves are axisymmetric and occur in a veryrobust regime of up-flow. The main difference between up-flow and down-flow is thatin down-flow, the driving pressure gradient and gravity act in the same direction,making the heavier fluid, water, fall while the buoyancy holds the oil back; in up-flow,gravity hinders the water and the oil is encouraged to flow upward. Naturally, ifthe driving pressure gradient is sufficiently strong and dominant, then the differencebetween up-flow and down-flow vanishes.

The flow regime denoted #1 in the flow chart of [34] has the superficial oil velocityVo = 10 cm/s and a = 1.28. This is shown on the right of Figure 4.2. The otherparameters for this flow are given in the figure caption. The experimental snapshotshows the coexistence of waves with wavenumbers roughly between 1.5 and 2.0. Sincethe observed wavespeeds and wavelengths of the bamboo wave regimes are close tothe linear theory for stability of PCAF, the initial condition for our numerical simu-lation is seeded with an eigenmode [34, 36, 2, 35, 24]. When perturbations grow tolarger amplitudes, saturation to the fully nonlinear bamboo waves is achieved. Figure4.2 shows a direct comparison of the saturated waves from our simulations with a


Fig. 4.2 Up-flow for Re = 0.93, a = 1.28, m = 0.00166, ζ = 1.1, J = 0.0795, and K = −0.4552. Thephotograph on the right was experimentally obtained in [34]. Our numerical simulationsare on the left, for dimensionless wavenumbers α = 1.5, 1.75, 2.0.

photograph from experiments. Previously, a steady solution was calculated in [37]under the assumptions of solid core and density matching; their interface shape is toorounded and asymmetric compared to the experimental snapshot, which shows analmost symmetric form of the crest, with a pointed peak. The crest is slightly sharperon the front and less sharp on its back. These details are successfully reproduced inour results.

4.3. The Hold-Up Ratio. The parameters for Figure 4.2 can be questioned foraccuracy regarding the Reynolds number and K. This is because the experimentalvalue for the pressure gradient is not known; rather, the superficial velocities aregiven in [34, 37]. The reconstruction of the rest of the parameters leaves some roomfor guesswork. In fact, our investigation of the hold-up ratio at saturation shows thatthe experiments are probably closer to a larger Reynolds number: Figure 4.3 showsthe base velocity field at Re = 3.0, K = −0.9993. This is fully up-flow, in both fluids.We performed a calculation for this flow from an initial amplitude A(0) = 0.005 ona 256 × 256 mesh. The wavenumber is 2. The evolution of the hold-up ratio againsttime is shown in the lower plot. The hold-up ratio decreases dramatically betweent = 20 and t = 40. This period corresponds to the transition from the linear regime,in which the eigenmode grows according to the growth rate predicted by linearizedstability theory for PCAF, to a nonlinear regime. In the fully nonlinear bamboo waveregime, the hold-up ratio stabilizes around 1.46, which is a reasonable approximationof the experimental value 1.39.

The interface position after saturation is plotted in Figure 4.4(a) and is verysimilar to the one in Figure 4.2 for α = 2, Re = 0.93, with the crest a little more


Fig. 4.3 The upper plot shows the velocity profile of PCAF for Re = 3, a = 1.28, m = 0.00166,ζ = 1.1, J = 0.0795, K = −0.9993, and wavenumber α = 2. The lower plot shows thecalculated hold-up ratio h against time. At t = 0, the PCAF hold-up ratio h = 2.61. AsPCAF evolves, h decreases first and stabilizes around 1.46.

pointed. The streamlines and contours of the pressure field are shown in Figure4.4(b). For an incompressible fluid, the axisymmetric stream function ψ is definedby u = −(1/r)(∂ψ/∂x), and v = (1/r)(∂ψ/∂r). This is calculated numerically by astandard central difference scheme. In the left half of Figure 4.4(b), the contours ofthe pressure field are plotted. In the water, the pressure field reaches its maximumvalue above the crest, at location A, and its minimum value below the crest, at B.The force that arises from this pressure field in the water and the buoyancy in theoil work together in up-flow, where they lead to stretching of the core into bamboowaves (cf. the cartoon of Figure 15.5 in [2]). The pressure field in the oil core is alsoshown in the figure. In the right half of Figure 4.4(b), the streamlines are plotted inthe frame of reference moving with the oil core. C denotes the stagnant line that isat rest in the reference frame. This stagnant line distinguishes the set of streamlinesinto two categories. One category of streamlines is inside the stagnant line and formsa recirculation zone. Another category of streamlines is outside the stagnant line.While some of them are completely in the water, others enter into the oil on theupper side of the crest and return to the water on the down side of the crest. Thebehavior of these streamlines is due to the fact that the waves move slowly comparedto the oil core. The wavespeed is 10% smaller than the core velocity. Although thewaveform is stationary, the velocity and pressure fields are not.

4.4. Effect of Radius Ratio and Temporally Periodic Flow. In the previous ex-ample, the radius ratio is a = 1.28, so that the oil core is relatively close to thepipe wall and the interaction between them is strong, leading to the vortices foundnear the wave crests. For the experimental data point #1 in the flow chart of [34],a = 1.61, J = .063354, K = −2.0303, and Re1 = 3.73754. Linear theory shows that


(b)(a)

A

B

C

Fig. 4.4 Up-flow for Re = 3.0, a = 1.28, m = 0.00166, ζ = 1.1, J = 0.0795, and K = −0.9993. (a)Interface shape at t = 100. (b) Pressure contours on the left half. Pressure field reachesthe maximum value on the upper side of the crest, at location A and the minimum value onthe down-side of the crest, at location B. Streamlines in the frame of reference moving withthe oil core are shown on the right half. The broken line represents the axis of symmetry.

the most dangerous mode for this flow has dimensionless wavenumber α = 2.4. Weset the initial amplitude of perturbation A(0) = 0.01. The evolution of the maximumamplitude A(t) against the time is plotted, on a log10-linear scale, in the upper plot ofFigure 4.5. The calculation is carried out on a 256×256 mesh over one spatial periodon a domain [0, 1.61]× [0, 2.618]. The initial growth of the perturbation compares wellwith the predicted linear theory. After roughly 40 seconds, a fully nonlinear evolutionbegins to take shape. Linear theory yields a wavespeed 0.9542, and the computedwavespeed is 0.8068 at time t = 200 (middle plot of Figure 4.5).

The most surprising feature of this flow is that it demonstrates temporal peri-odicity, as evident from the evolution of the hold-up ratio vs. time (lower plot ofFigure 4.5). As the PCAF evolves into the nonlinear bamboo wave regime, the hold-up ratio first decreases and then oscillates around 2.15, with a well-defined temporalperiod. The period is about 8.

4.5. Effect of Reynolds Number and Temporally Periodic Flow. As the speedis increased, the bamboo waves shorten and peaks become more pointed. This is illus-trated in Figure 4.6. Based on the linear stability theory for PCAF, the wavenumberfor the fastest growing mode decreases as the Reynolds number increases. Figure 4.6shows that the slow and long waves (low Reynolds number flow) have asymmetricalcrest shape, flat on the upper side and steeper on the down side. This asymmetryis due to the effect of the buoyancy. In fast flow (large Reynolds number flow), theasymmetric effect of buoyancy is less important, as evidenced by the almost symmetricshape of wave crests for Re = 3.74.


Fig. 4.5 Up-flow for α = 2.4, Re1 = 3.73754, a = 1.61, m = 0.00166, ζ = 1.1, J = .063354, andK = −2.030303. For the interfacial mode, theoretical linear growth rate is 0.066 and thewave speed is 0.954. The upper plot shows maximum amplitude A(t) vs. time on a log10-linear scale. The middle plot shows wave crest position vs. time. The lower plot showshold-up ratio vs. time. The hold-up ratio oscillates around 2.15, with period approximately8. The solid line represents the theoretical linearized growth rate, and circles represent thecalculation carried out on a 256 × 256 mesh.

At Re = 3, a temporally periodic streamline pattern with period of about 10 isfound. While the periodicity appears already in low Reynolds number flow (Re = 1.0,1.5, and 2.0), it becomes quite marked in the Re = 3 flow. When the Reynolds numberis increased, it is not surprising that a steady solution would lose stability to time-dependent and eventually chaotic solutions as the flow transitions. However, it is sur-prising that the time-dependent bamboo waves would appear to the eye to be steady.

5. Simulation of Drop Breakup in Three Dimensions. The deformation andbreakup of a drop in shear flow lies at the foundation of dispersion science and mul-tiphase flow and has invoked a great deal of interest in the scientific community sincethe time of G. I. Taylor [38, 39, 40, 41, 42, 43, 44, 45, 46]. More recently, experimen-tal observations of the sheared breakup have been recorded in [43]: a strong shearis applied to a single drop, which elongates and undergoes end pinching via a pro-cess termed elongative end pinching as opposed to retractive end pinching, studiedin [47]. These processes, which yield daughter drops, are paradigms of theoreticalinvestigations into emulsification and mixing [46, 42, 48, 49].


Re = 1.5 Re = 3.7Re = 2.0 Re = 3.0Re = 1.0

Fig. 4.6 Up-flow for a = 1.61, m = 0.00166, ζ = 1.1, J = 0.063354, and K = −2.030303.

The experimental work of [43] focuses on a viscous drop suspended in a secondimmiscible liquid (the matrix liquid) in a cylindrical Couette device. The differencein density between the two liquids is a minor effect, and the flow is sufficiently slow,so that centrifugal effects in the cylindrical device are not important. A theoreticalmodel for this is simply three-dimensional Couette flow with zero gravity, as shownin Figure 5.1. The liquid drop has an undeformed radius a and viscosity µd, and thematrix liquid has viscosity µm. The matrix liquid is undergoing a simple shear flowbetween two parallel plates, placed a distance d∗ apart. The undisturbed velocity fieldis u = γzi, where γ is the imposed shear rate. Additional parameters for our numericalsimulations are the interfacial tension σ and the spatial periodicities λ∗

x and λ∗y in the

x- and y-directions, respectively. There are six dimensionless parameters: a capillarynumber Ca = aγµm/σ, where an average shear rate is defined as γ = U∗/d∗; theviscosity ratio λ = µd/µm; the Reynolds number Re = ρmγa2/µm; the dimensionlessplate separation d = d∗/a; and dimensionless spatial periodicities λx = λ∗

x/a andλy = λ∗

y/a. The shear stress induced by the flow competes with the interfacial tensionto deform and rupture the drop. The capillary number denotes the ratio between thesetwo competing effects and provides a useful measure of efficiency of the shear flow todeform the drop.

Computational studies [33] have elucidated regimes where the drop deforms tothe point of breaking, but results on the motion past breakup are limited. We present


plate

plate

z

fluid 2 fluid 1

x

Fig. 5.1 A drop in simple shear flow, bounded at the top and bottom by walls. The y-axis pointsinto the paper.

L - BL + B

D = L

B

a θ x

-z

Fig. 5.2 Scalar measures of deformation and orientation.

a numerical exploration of the stages in the formation of daughter drops under shear.The simulations were conducted as three-dimensional initial value problems [16].

5.1. Drop Breakup in Simple Shear. In the past, all numerical studies of a vis-cous drop in shear flow have been performed with the boundary integral method,combined with a front tracking method. An advantage of the front tracking methodis the use of marker particles to track the interface explicitly. However, the implemen-tation of boundary integral methods poses a major obstacle, because it is difficult tohandle merging and folding interfaces. The boundary integral method incorporates asimple shear flow out to infinity. In [16], the limitation of this assumption has beeninvestigated by changing the plate separation.

5.2. Evolution to Steady Drop Shape. Two dimensionless parameters, the cap-illary number Ca and the viscosity ratio λ, characterize the behavior of the suspendeddrop, provided the Reynolds number Re = ρmγa2/µm is small. Previous studies haveshown that when λ is less than 4, there is a “critical capillary number” Cac, abovewhich the drop disintegrates. Below the critical capillary number, the drop evolves toa steady shape. The critical capillary number in shear flow is lowest for λ ≈ 0.5; forλ = 1, Cac ≈ 0.41 is critical [50]. In the case where the drop evolves to a steady shape,two parameters have been used to measure the deformation attained by the drop inits final stage. The first is the Taylor deformation parameter, D = (L − B)/(L + B),where L and B are the half-length and half-breadth of the drop, respectively. Thesecond parameter is the angle θ of orientation of the drop with the axis of shear strain(Figure 5.2).


Fig. 5.3 Evolution of drop shape for Ca = 0.42 in domain 3 × 1 × 2. Viscosities and densities areequal, Re = 0.0.

The initial condition is shown in Figure 5.1: the drop is a sphere. The no-slipcondition is imposed on the top and bottom plates and periodic conditions in the x-and y-directions. Constant velocities are imposed on the top and bottom plates suchthat the shear rate is constant during the entire computation. The steady parametersD and θ of the previous works [44, 51] were retrieved for the computational domain2 × 1 × 4. When the plate separation is smaller, e.g., 2 × 1 × 1 domain, the drop isfound to break up at lower capillary numbers.

5.3. Rupture of a Drop in Shear. When the shear rate is increased past a criticalvalue, the drop ruptures. The critical capillary number is roughly 0.41. Indeed, ourcomputation predicts an unsteady solution for Ca = 0.42 in the domain 3 × 1 × 2.Figure 5.3 shows the evolution of the drop shape on a 96 × 32 × 64 mesh grid. Thedrop continues to deform and eventually breaks up. The competition between theexternally imposed shear flow and the surface tension driven flow is clearly evidentin the figures. Initially, the most noticeable motion is the elongation of the drop,stretched by the viscous shear stress of the external flow (time T = 0.0, 10.0, and20.0). To time T = 30, we see clearly that a waist is formed near the center of thedrop, and the drop continually thins. The drop is beginning to lengthen slowly, and avisible neck is formed near the bulbous end. This neck will eventually lead to the endspinching off [52], and the remaining liquid thread in the middle will form some smallsatellite droplets. The number of these satellite droplets and their volume cannot becomputed precisely at this time, because the mesh grid is not sufficiently fine, but wesee here that the VOF method provides the possibility for this kind of investigationin the future.


1.4 2 2.70.8

0.9

1

1.1

1.2

1.3

T = 36.0

1.4 2 2.70.8

0.9

1

1.1

1.2

1.3

1.4 2 2.70.8

0.9

1

1.1

1.2

1.3

1.4 2 2.70.8

0.9

1

1.1

1.2

1.3

T = 39.0

T = 38.5

T = 38.0

Fig. 5.4 Velocity fields during the breakup of a drop in the simple shear flow for capillary numberCa = 0.42. This shows a cross-sectional cut.

To examine the breakup procedure more carefully, we have done the calculationon a 192×64×128 mesh grid. We present the velocity field on the cross-sectional cut inthe x-z plane in Figure 5.4; the flow pattern is symmetrical, and we need to show onlythe right half-field. The precise role of the surface tension–driven flow during breakupcan be examined from this figure. At time T = 36, the result of the competitionbetween the external flow and the surface tension force is a vortical motion inside thebulbous end of the drop, except near the neck; the surface tension force drives a fastflow motion toward the bulbous end, while in the waist near the center, the flow ismuch weaker. The consequence is that the neck quickly and continually narrows (theneck has the same size as the mesh grid at this time), while the width of the central


Fig. 5.5 Reproduction from Marks [43], showing a typical breakup. Matrix viscosity 7.0 Pa·s, dropviscosity 4.3 Pa·s, interfacial tension 10.7 mN/m, initial drop radius 0.048 cm, shear rate2.17/s, and equal densities. Reprinted with permission.

Fig. 5.6 Interface evolution as viewed from the top of the computational box 12 × 1 × 1 duringbreakup for Ca = 0.55, λ = 0.77, and equal densities. The calculation was performed on a256 × 64 × 64 grid.

waist remains almost unchanged at time T = 38. At T = 38.5, the drop breaks upat the neck and produces a main drop and a middle liquid thread. More resolution isrequired to investigate the breakup of the middle liquid thread.

The deformation and breakup of a drop in shear flow has recently been investi-gated experimentally by Marks [43]. Figure 5.5 shows one of Marks’s observations,and Figure 5.6 shows our simulation for parameters close to the experimental case.Each breakup experiment of [43] began with the largest daughter drops being formedby the elongative end pinching, which is shown in both of these figures. Subsequently,there is a sequence of small, then large drops, as both these figures show. The readeris referred to [16].

6. Summary. Our investigation of two-layer Couette flow has shown the devel-opment of elongated fingers, which allow the migration of the more viscous into theless viscous liquid, and vice versa, depending on the fluid parameters. Our numericalsimulations of axisymmetric core-annular flow show that bamboo waves arise from lin-early unstable small amplitude perturbations on core-annular flow; the disturbancesgrow and saturate at fully nonlinear waveforms. In order to proceed further with thedrop breakup problem and explore the details of the breakup regime and daughter


drops, we need to enhance the efficiency of our scheme, for example, with an adaptivemesh technique [53].

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