Bubble motion in a converging-diverging channel

Harsha Konda, Manoj Kumar Tripathi and Kirti Chandra Sahu∗

Department of Chemical Engineering,

Indian Institute of Technology Hyderabad,

Yeddumailaram 502 205, Telangana, India

(Dated: October 9, 2015)

Abstract

The migration of a bubble inside a two-dimensional converging-diverging channel is numerically

studied using a finite volume flow approach. A parametric study is conducted to investigate the

effects of the Reynolds and Weber numbers, and the amplitude of the converging-diverging channel.

It is found that increasing the Reynolds number and the amplitude of the converging-diverging

channel increases the oscillation of the bubble, and promotes the migration of the bubble towards

one of the channel wall. While travelling inside the channel from trough to crest regions, the bubble

undergoes oblate-prolate deformation periodically at the early times, which becomes chaotic at the

later times. This phenomenon is a culmination of the bubble path instability as well as the Segre-

Silberberg effect. These oscillations in the shape of the bubble and the complex path travelled by

the bubble can enhance mixing, which is desirable in many applications involving small-scale flows.

∗ ksahu@iith.ac.in

1

I. INTRODUCTION

The bubble dynamics moving through channels and pipes have been the subject of nu-

merous experimental, theoretical and numerical studies due to their relevance to a number of

applications, ranging from microscale to macroscale flows, such as mixing in microfluidics,

biological applications, chemical reactors, etc. [1–3]. In microscale flows, surface-tension

force is dominant over gravitational force, but in large-scale systems, gravity plays a sig-

nificant role. Thus, several researchers studied the motion of bubbles and drops due to

buoyancy in vertical channels and pipes [4] as well as in unbounded domains [5, 6]. At

smaller scales, spatially varying walls are frequently encountered, and play a vital role in the

resultant flow dynamics [7, 8]. First, the studies of flows associated with spatially varying

geometries, which have received considerable attention in the recent past, mainly, in the

context of small-scale flows are briefly discussed below.

Many researchers (see e.g., [9–12]) have investigated flow through converging-diverging

channels by conducting numerical simulations, experiments and linear stability analysis.

The main emphasis of these studies has been on the enhancement of mixing, heat and

mass transfer rates in small-scale devices (which are associated with low flow rates) by

introducing spatially varying geometries. Also, flow through corrugated channels increases

the residence time of the fluid without having to increase the length of the device [13, 14].

This helps in designing compact and efficient devices for thermally sensitive substances, such

as those encountered in food processing and pharmaceutical industries. Stone & Vanka [15]

numerically studied the flow field in a wavy channel and observed enhanced mixing above

a certain Reynolds number (Re) after which the separation bubbles distort/divide to form

multiple roll-up structures near the walls.

A very few studies experimentally investigated the motion of a single bubble or multiple

bubbles inside corrugated channels focusing on the average features of the flow, such as

pressure drop and friction factor. Gardeck & Lebouche [16] experimentally measured wall

shear stress due to gas-liquid flow in a corrugated channel. Vlasogiannis et al. [17] studied

the heat transfer characteristics of a plate heat exchanger, and observed recirculating zones

of the liquid phase in the near-wall regions, and the gaseous phase in the centreline region

of the pipe. For gas-liquid flow in wavy channels having different phase-differences between

the walls (i.e. Φ = 0, π/2 and π), Nilpueng & Wongwises [18] obtained correlations between

2

pressure drop and friction factor. They also found that for Φ = π the recirculation zone

is bigger than that observed for Φ = 0 or π/2. Slug flow was observed only for Φ = π

(named as symmetrical channel in the present study), which was not seen for other Φ values

considered in their study. Although the migration of a single bubble in horizontal channels

and pipes has been well studied [19–21], to the best of our knowledge, none of them have

investigated a single bubble motion in converging-diverging geometries numerically, which

is the focus of the present study.

Another objective of the present work is to investigate the Segre-Silberberg effect [22]

for bubble migration inside a converging-diverging channel. This phenomenon was first

documented by Segre & Silberberg [22], where a macroscopic rigid sphere moving inside a

pipe or a channel, migrates toward the wall. This migration of particle may be attributed

to the nonlinear effects, which arise inside a flow for Re > 0. Although the original study

was conducted for a rigid sphere, this phenomenon is also applicable to liquid-liquid or gas-

liquid systems [23]. Since then the Segre-Silberberg effect has been investigated by several

researchers for various parameter regimes (see e.g. [24, 25]). In the present study, the

dynamics of a bubble in a converging-diverging channel are studied. Here, it is expected

that the wavy motion of the bubble will accentuate due to the Segre-Silberberg effect, which

in turn could enhance mixing.

The rest of the paper is organized as follows. The details of the problem formulation

are given in section II, and the numerical approach used in the present study is provided

in section III. In section IV, a parametric study is conducted, and the effects of various

dimensionless numbers on the flow dynamics are discussed. Concluding remarks are given

in section V.

II. FORMULATION

The dynamics of an air bubble (designated by fluid ‘A’) of initial radius r, moving inside

a converging-diverging two-dimensional channel, consisting of another fluid (designated by

fluid ‘B’) as shown in Fig. 1, are investigated numerically using a Volume-of-Fluid (VoF)

approach. A Cartesian coordinate system (x, y) with its origin at the center of the chan-

nel inlet is used to model the flow, wherein x and y represent the horizontal and vertical

directions, respectively. The positions of the sinusoidal top and bottom walls are described

3

by y = R + hsin(2πxλ

+ Φ), and y = −R + hsin

(2πxλ

), respectively. Here, h, λ and Φ are

the half-amplitude, the wavelength of the converging-diverging portion of the channel, and

the phase difference between the walls, respectively. The distance between the walls in the

straight section and the radius of the bubble are 2R and r, respectively, such that R/r = 4.

It is expected that increasing R/r will decrease the effect of the converging-diverging geome-

try. The length of the horizontal section at the inlet is 12r, and initially the bubble is placed

at (x, y) = (9r, 0). The total length of the channel is considered to be 78r, and λ = 8r.

Both the fluids are considered to be incompressible and Newtonian. The gravitational force

is assumed to be very small as compared to the inertial force, thus the former is neglected

in the present study. This is a valid approximation for the flows inside a channel of height

of the order of a millimetre. The viscosity and density of fluids A and B are µA, ρA and µB,

ρB, respectively.

The governing equations of the problem are:

∇ · u = 0, (1)

ρ

[∂u

∂t+ u · ∇u

]= −∇p+∇ ·

[µ(∇u +∇uT )

]+ δσκn, (2)

∂c

∂t+ u · ∇c = 0, (3)

where u(u, v) represents the velocity field, wherein u and v are the components of velocity

in the horizontal and vertical directions, respectively; p denotes the pressure field; c is the

volume fraction of fluid B, whose values are 0 and 1 in the air and the liquid phases,

respectively; δ is the Dirac delta function, whose value is one at the interface and zero in

the rest of the domain; κ = ∇ · n is the curvature, wherein n is the unit normal to the

interface pointing towards fluid B, and σ is the interfacial tension coefficient of the liquid-

gas interface. The surface tension force is included in Eq. (2) using continuum surface force

formulation [26].

The density, ρ and viscosity, µ are calculated as volume averaged quantities as follows:

ρ = ρA(1− c) + ρBc, (4)

µ = µA(1− c) + µBc. (5)

The following scaling is employed in order to render the governing equations dimensionless:

(x, y) = r (x, y) , t =r

Vt, (u, v) = V (u, v), p = ρBV

2p, µ = µµ0, ρ = ρρB, (6)

4

where tildes designate the dimensionless quantities, and V is the velocity of the fluid injected

at the channel inlet, given by V ≡ Q/2R, where Q is the volume flow rate at the inlet; the

value of Q is taken to be one in the present study. After dropping tildes from all the

non-dimensional terms, the governing dimensionless equations are given by

∇ · u = 0, (7)

ρ

[∂u

∂t+ u · ∇u

]= −∇p+

1

Re∇ ·[µ(∇u +∇uT )

]+

δ

Wen∇ · n, (8)

∂c

∂t+ u · ∇c = 0, (9)

where Re(≡ ρBV R/µB) and We(≡ ρBRV2/σ) denote the Reynolds number and the

Weber number, respectively. The dimensionless viscosity and density are given by:

ρ = ρr(1− c) + c, (10)

µ = µr(1− c) + c, (11)

where ρr (≡ ρA/ρB) and µr (≡ µA/µB) are the density ratio and the viscosity ratio, respec-

tively.

III. NUMERICAL METHOD AND VALIDATION

An open-source finite-volume fluid flow solver, Gerris [27] is used to simulate the dynamics

of bubble motion in a converging-diverging symmetrical channel (Φ = π). In the present

study, the density and the viscosity ratios are fixed at ρr = 10−3 and µr = 10−2, respectively,

which correspond to the air-water system. Numerically it is difficult to handle such high

values of ρr and µr, and is known to create spurious currents at the interface separating

the fluids. Gerris minimizes this problem by using a balanced-force continuum surface force

formulation [26] for the calculation of surface tension. In addition, due to the curvature at

the walls very large number of grids are required to resolve the boundary layer. The adaptive

refinement feature of the Gerris is very useful for this purpose. In the present study, the

near-wall and the central regions are refined separately. For the near-wall regions, stationary

uniform grid sizes are used, whereas adaptive grid refinement based on vorticity magnitude

and position of the interface separating the fluids is implemented for the fluid region.

The present code has been validated extensively by simulating several interfacial flow

problems. The reader is also referred to Tripathi et al. [4–6], wherein extensive validations

5

of the present code have been reported. It is ensured that the grid convergence is indeed

achieved upon mesh refinement (the maximum deviation in bubble shape is within 0.01%

of the radius of bubble), which is shown in Fig. 2 for We = 10 and 25. The rest of the

parameter values are ρr = 10−3, µr = 10−2 and Re = 100. Thus, in order to generate the

rest of the results presented in this paper, the grid − 1 for which the smallest grid sizes at

the near-wall and the fluid regions are 0.031 and 0.015, respectively, is used.

The influence of various dimensionless numbers, such as the Reynolds number (Re), the

Weber number (We), and the half-amplitude of the converging-diverging channel (h) is

discussed below.

IV. RESULTS AND DISCUSSION

The presentation of our results is begun by investigating the effects of Reynolds number

for the symmetrical channel (Φ = π) by plotting the location and the velocity of center-of-

gravity, and the aspect ratio (Ar ≡ w/l) of the bubble in Figs. 3 and 4 for We = 1 and

We = 5, respectively. Here w and l represent the height and length of the bubble, and the

rest of the parameter values are ρr = 10−3, µr = 10−2. It can be seen in Fig. 3(a) that for

We = 1, the bubble migrates towards the bottom wall as it travels inside the channel for all

the values of Reynolds number considered. Inspection of this plot also reveals that increasing

Re takes the bubble away from the centreline, which is consistent with the literature [24],

however the corrugations do not allow for an equilibrium vertical location, as expected by

Segre and Silberberg [22] for straight pipes.

It can be seen that the bubble moves along the centreline of the channel at the early

times and then undergoes an oscillatory motion as it migrates towards the bottom wall.

The amplitude of oscillation increases as the bubble translates in the axial direction. Close

inspection of Fig. 3(a) divulges that the starting time of departure of the bubble from the

centreline reduces with increasing Re. The vertical velocity component of the center-of-

gravity of the bubble shown in Figs. 3(b) also demonstrates this oscillatory behaviour. In

Fig. 3(c), the instantaneous aspect ratio of the bubble is plotted versus time for different

values of the Reynolds number. It can be seen that the bubble undergoes a continuous

oblate-prolate type deformation, which is periodic at early times (t < 70) for this set of

parameter values, but the deformation is more chaotic at later times. Similar dynamics are

6

also observed for We = 5 as shown in Fig. 4. It can be seen in Fig. 4(a) that the bubble

migrates towards the bottom wall for Re = 100, whereas for Re = 120 and 140, it moves

towards the top wall. However, it is to be noted that the path chosen by the bubble is

arbitrary, and may be influenced by the initial conditions. As the value of We considered

in Fig. 4 is higher than that in Fig. 3, the surface tension is less dominant, and the bubble

undergoes larger deformation for the cases considered in Fig. 4 as compared to those in Fig.

3.

In order to get more insight into the dynamics of the bubble, the spatio-temporal evolution

of horizontal and vertical components of velocity are plotted in Figs. 5 and 6 for Re = 70

and We = 1, and Re = 120 and We = 5, respectively. It can be seen in Fig. 5 that

the horizontal velocity is higher near the centre of the channel as compared to the near-

wall regions. Along the centreline, the horizontal velocity is maximum near the troughs

of the converging-diverging channel. When the bubble crosses the trough regions of the

converging-diverging channel, it elongates to become a prolate shaped bubble. It deforms to

an oblate shape as it reaches the crest regions of the channel. This effect is more pronounced

for We = 5 due to the lesser dominance of surface tension force as compared to that for

We = 1. The contours of the vertical velocity component show the appearance of four lobes

of maximum and minimum vertical velocity inside the bubble as it migrates towards the

bottom wall.

Next, the effect of the half-amplitude, h of the converging-diverging symmetrical channel

(Φ = π) is investigated in Fig. 7 for two typical sets of Reynolds and Weber numbers.

It is to be noted that h = 0 case corresponds to bubble motion inside a straight channel.

It can be seen that for h = 0, the bubble migrates towards the top wall (classical Segre–

Silberberg effect). Moreover, it is observed that bubble starts oscillating for h ≥ 0.25, and

the oscillation amplitude magnifies with increasing the value of h. Inspection of aspect ratio

of the bubble in Figs. 7(b) and (d) reveals that the shape of the bubble remains spherical for

h = 0, but undergoes a periodic oblate-prolate deformation for intermediate values of h. For

h ≥ 0.5 the deformation of bubble is periodic till t = 90 (approximately), but experiences a

larger deformation for t > 90.

All the results presented so far are for a symmetrical channel, i.e. when top and bottom

walls are described by y = R+hsin(2πxλ

+ Φ), and y = −R+hsin

(2πxλ

), with Φ = π. Finally,

the bubble migration through asymmetrical channels is investigated by setting Φ = 0 and

7

π/2. The variation of yCG and Ar for asymmetrical channels are compared with those

obtained for the symmetrical channel (Φ = π) in Fig. 8 for Re = 60, We = 1, ρr = 10−3

and µr = 10−2. It can be seen that the amplitude of oscillation of the center-of-gravity of

the bubble is more in case of asymmetrical channel with Φ = 0. The oscillations observed in

these channels are due to the Segre-Silberberg effect as well as the asymmetrical nature of

the channels for Φ = π and π/2. The larger deformation of the bubble shown in Fig. 8(b)

for Φ = π can be attributed to the presence of bigger recirculation zones inside the channel

with Φ = π as compared to those for channels with Φ = 0 and π/2.

V. CONCLUDING REMARKS

The dynamics of bubble motion through converging-diverging channels having different

phase differences between the top and the bottom walls are studied using an open-source

finite volume flow solver, Gerris. A parametric study is conducted, and the effects of various

dimensionless parameters, such as the Reynolds number, Weber number, and amplitude of

the converging-diverging channels are investigated. It is found that increasing Re and h

increases the amplitude of oscillation of the bubble migrating inside the channels. The

oscillation of the shape of the bubble and the complex path followed by the bubble can lead

to more mixing, which is desirable in many applications involving small-scale flows.

[1] Haeberle, S and Zengerle, R, 2007, “Microfluidic Platforms for Lab-on-a-Chip Applications,”

Lab Chip, 7, 1094–1110.

[2] Stone, H. A., Stroock, A. D. and Ajdari, A., 2004, “Engineering Flows in Small Devices:

Microfluidics towards a Lab-on-a-Chip,” Annu. Rev. Fluid Mech., 36, 318–411.

[3] Le, T. -L., Chen J. -C., Shen B. -C., Hwub, F. -S. and Nguyen, H. -B, 2015, “Numerical

Investigation of the Thermocapillary Actuation Behavior of a Droplet in a Microchannel,”

Int. J. Heat Mass Trans., 83, 721–730.

[4] Tripathi, M. K. and Sahu, K. C., Karapetsas, G., Sefiane, K. and Matar, O. K., 2015, “Non-

Isothermal Bubble Rise: Non-Monotonic Dependence of Surface Tension on Temperature,” J.

Fluid Mech., 763, 82–108.

8

[5] Tripathi, M. K., Sahu, K. C. and Govindarajan, R., 2014,“Why a Falling Drop Does Not in

General Behave Like a Rising Bubble,” Nature Scientific Reports, 4, 4771.

[6] Tripathi, M. K., Sahu, K. C. and Govindarajan, R., 2015,“Dynamics of an Initially Spherical

Bubble Rising in Quiescent Liquid,” Nat. Commun., 6, 6268.

[7] Sahu, K. C. and Govindarajan, R. 2005, “Stability of Flow Through a Slowly Diverging Pipe,”

J. Fluid Mech., 531, 325–334.

[8] Jose, B. M. and Cubaud, T., 2012, “Droplet Arrangement and Coalescence in Diverg-

ing/Converging Microchannels,” Microfluid Nanofluid, 12, 687–696.

[9] Blancher, S. and Creff, R., 2004 “Analysis of Convective Hydrodynamics Instabilities in a

Symmetric Wavy Channel,” Phys. Fluids, 16 (10), 3726–3737.

[10] Selvarajan, S., Tulapurkara, E. G. and Ram, V. V., 1999“Stability Characteristics of Wavy

Walled Channel Flows,” Phys. Fluids, 11 (3), 579–589.

[11] Szumbarski, J. and Floryan, J. M., 2006, “Transient Disturbance Growth in a Corrugated

Channel,” J. Fluid Mech., 568, 243–272.

[12] Duryodhan, V., Singh, S. G. and Agarwal, A., 2013, “Liquid Flow Through a Diverging

Microchannel,” Microfluid Nanofluid, 14, 53–67.

[13] Santacesaria, E., Di Serio, M., Tesser, R., Casale, L., Verde, D., Turco, R. and Bertola, A.,

2009, “Use of a Corrugated Plates Heat Exchanger Reactor for Obtaining Biodiesel with Very

High Productivity,” Energy Fuels, 23, 5206–5212.

[14] Theron, F., Anxionnaz-Minvielle, Z. and Cabassud, M., Gourdon, C. and Tochon, P., 2014,

“Characterization of the Performances of an Innovative Heat-Exchanger/Reactor,” Chem.

Eng. Process., 82, 30–41.

[15] Stone, K. and Vanka, S., 1999, “Numerical Study of Developing Flow and Heat Transfer in a

Wavy Passage,” ASME J. Fluids Eng., 121, 713–720.

[16] Gradeck. M. and Lebouche, M., 2000, “Two-Phase Gas-Liquid Flow in Horizontal Corrugated

Channels,” Int. J. Multiphase Flow, 26, 435–443.

[17] Vlasogiannis, P., Karagiannis, G., Argyropoulos, P. and Bontozoglou, V., 2002, “AirWater

Two-Phase Flow and Heat Transfer in a Plate Heat Exchanger,” Int. J. Multiphase Flow, 28,

757–772.

[18] Nilpueng, K. and Wongwises, S., 2006, “Flow Pattern and Pressure Drop of Vertical Upward

GasLiquid Flow in Sinusoidal Wavy Channels,” Exp. Therm. Fluid Sci., 30, 513–534.

9

[19] Andreussi, P., Paglianti, A. and Silva, F. S., 1999, “Dispersed Bubble Flow in Horizontal

Pipes,” Chem. Eng. Sci., 54, 1101–1107.

[20] Murai, Y., Fukuda, H., Oishi, Y., Kodamai, Y. and Yamamoto, F., 2007, “Skin Friction

Reduction by Large Air Bubbles in a Horizontal Channel Flow,” Int. J. Multiphase Flow, 33,

147–163.

[21] Bohm, L., Kurita, T., Kimura, K. and Kraume, M., 2014, “Rising Behaviour of Single Bub-

bles in Narrow Rectangular Channels in Newtonian and Non-Newtonian Liquids,” Int. J.

Multiphase Flow, 65, 11–23.

[22] Segre, G. and Silberberg, A., 1961, “Radial Particle Displacements in Poiseuille Flow of Sus-

pensions,” Nature, 189, 209–210.

[23] Douglas-Hamilton, D. H., Smith, N. G., Kuster, C. E., Vermeiden, J. P. W. and Althouse, G.

C., “Capillary-Loaded Particle Fluid Dynamics: Effect on Estimation of Sperm Concentra-

tion,” J Androl, 26, 115-122.

[24] Pan, T. -W and Glowinski, R., 2002, “Direct Simulation of the Motion of Neutrally Buoyant

Circular Cylinders in Plane Poiseuille Flow,” J. Comput. Phys, 181, 260–279.

[25] Matas, J. -P., Morris, J. F. and Guazzelli, E., 2009, “Lateral Force on a Rigid Sphere in

Large-Inertia Laminar Pipe Flow,” J. Fluid Mech., 621, 59–67.

[26] Brackbill, J., Kothe, D. B. and Zemach, C., 1992 “A Continuum Method for Modeling Surface

Tension,” J. Comput. Phys, 100, 335–354.

[27] Popinet, S., 2003, “Gerris: a Tree-Based Adaptive Solver for the Incompressible Euler Equa-

tions in Complex Geometries,” J. Comput. Phys, 190, 572–600.

10

List of figure captions

Fig. 1: Schematic diagram (not to scale) of the bubble motion inside a two-dimensional

converging-diverging symmetric channel. Fluid ‘B’ constitutes the continuous phase and

fluid ‘A’ is initially confined to a circular region (bubble) of radius, r.

Fig. 2: Comparison of shapes of the bubble (a,c) at x = 9.537 and (b,d) at x = 14.37

for two different gird refinements. The panels (a,b) and (c,d) correspond to We = 10

and We = 25, respectively. Grid − 1 (shown by solid line): minimum grid sizes near the

boundary and the fluid regions are 0.031 and 0.015, respectively. Grid − 2 (shown by

dashed line): minimum grid sizes near the boundary and the fluid regions are 0.015 and

0.008, respectively. The rest of the parameter values are ρr = 10−3, µr = 10−2 and Re = 100.

Fig. 3: The variation of (a) yCG, (b) vCG, and (c) Ar in the axial direction for dif-

ferent values of Re. The rest of the parameter values are ρr = 10−3, µr = 10−2 and We = 1.

Fig. 4: The variation of (a) yCG, (b) vCG, and (c) Ar in the axial direction for dif-

ferent values of Re. The rest of the parameter values are ρr = 10−3, µr = 10−2 and We = 5.

Fig. 5: The spatio-temporal evolution of horizontal (top panel) and vertical (bottom

wall) velocity components at different times. The rest of the parameter values are

ρr = 10−3, µr = 10−2, Re = 70, and We = 1. The grayscale bars for horizontal and verti-

cal velocity components are given for t = 100 only, which are the same for the other t values.

Fig. 6: The spatio-temporal evolution of horizontal (top panel) and vertical (bottom

wall) velocity components at different times. The rest of the parameter values are

ρr = 10−3, µr = 10−2, Re = 120, and We = 5. The grayscale bars for horizontal and verti-

cal velocity components are given for t = 100 only, which are the same for the other t values.

Fig. 7: The variation of yCG and Ar in the axial direction for different values of h:

(a,b) Re = 80 and We = 1, (c,d) Re = 140 and We = 5. The rest of the parameter values

are ρr = 10−3, µr = 10−2.

11

Fig. 8: The variation of yCG and Ar in the axial direction for channels having dif-

ferent values of Φ. The parameter values are Re = 60, We = 1, ρr = 10−3 and

µr = 10−2.

12

FIG. 1: Schematic diagram (not to scale) of the bubble motion inside a two-dimensional converging-

diverging symmetric channel. Fluid ‘B’ constitutes the continuous phase and fluid ‘A’ is initially

confined to a circular region (bubble) of radius, r.

13

(a) (b)

(c) (d)

FIG. 2: Comparison of shapes of the bubble (a,c) at x = 9.537 and (b,d) at x = 14.37 for two

different gird refinements. The panels (a,b) and (c,d) correspond to We = 10 and We = 25,

respectively. Grid− 1 (shown by solid line): minimum grid sizes near the boundary and the fluid

regions are 0.031 and 0.015, respectively. Grid−2 (shown by dashed line): minimum grid sizes near

the boundary and the fluid regions are 0.015 and 0.008, respectively. The rest of the parameter

values are ρr = 10−3, µr = 10−2 and Re = 100.

14

(a) (b)

10 20 30 40 50 60x

-1.2

-0.8

-0.4

0

yCG

607080

Re

10 20 30 40 50 60x

-0.1

-0.05

0

0.05

0.1

vCG

60

70

80

Re

(c)

10 20 30 40 50 60x

0.9

0.95

1

1.05

1.1

Ar

60

70

80

Re

FIG. 3: The variation of (a) yCG, (b) vCG, and (c) Ar in the axial direction for different values of

Re. The rest of the parameter values are ρr = 10−3, µr = 10−2 and We = 1.

15

(a) (b)

10 20 30 40 50 60x

-0.4

-0.2

0

0.2

0.4

0.6

yCG

100120140

Re

10 20 30 40 50 60x

-0.1

-0.05

0

0.05

0.1

vCG

100

120

140

Re

(c)

10 20 30 40 50 60x

0.8

1

1.2

Ar

100

120

140

Re

FIG. 4: The variation of (a) yCG, (b) vCG, and (c) Ar in the axial direction for different values of

Re. The rest of the parameter values are ρr = 10−3, µr = 10−2 and We = 5.

16

t = 20

t = 60

t = 80

t = 100

FIG. 5: The spatio-temporal evolution of horizontal (top panel) and vertical (bottom wall) velocity

components at different times. The rest of the parameter values are ρr = 10−3, µr = 10−2, Re = 70,

and We = 1. The grayscale bars for horizontal and vertical velocity components are given for

t = 100 only, which are the same for the other t values.

17

t = 20

t = 60

t = 80

t = 100

FIG. 6: The spatio-temporal evolution of horizontal (top panel) and vertical (bottom wall) velocity

components at different times. The rest of the parameter values are ρr = 10−3, µr = 10−2,

Re = 120, and We = 5. The grayscale bars for horizontal and vertical velocity components are

given for t = 100 only, which are the same for the other t values.

18

(a) (b)

10 20 30 40 50 60x

-1.5

-1

-0.5

0

0.5

1

1.5

yCG

00.250.5

h

10 20 30 40 50 60x

0.9

0.95

1

1.05

1.1

Ar

0

0.25

0.5

hh

(c) (d)

10 20 30 40 50 60x

-1

-0.5

0

0.5

1

yCG

00.250.5

h

10 20 30 40 50 60x

0.8

1

1.2

Ar

0

0.25

0.5

h

FIG. 7: The variation of yCG and Ar in the axial direction for different values of h: (a,b) Re = 80

and We = 1, (c,d) Re = 140 and We = 5. The rest of the parameter values are ρr = 10−3,

µr = 10−2.

19

(a) (b)

20 40 60x

-2

-1.5

-1

-0.5

0

yCG

Asymmetric (Φ = 0)

Asymmetric((Φ = π/2)

Symmetric(Φ = π)

20 40 60x

0.95

1

1.05

1.1

Ar

Asymmetric (Φ = 0)

Asymmetric((Φ = π/2)

Symmetric(Φ = π)

FIG. 8: The variation of yCG and Ar in the axial direction for channels having different values of

Φ. The parameter values are Re = 60, We = 1, ρr = 10−3 and µr = 10−2.

20