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Experiments in Fluids

, 60:75 | Cite as

An analytical interface shape approximation of microscopic Taylor flows

  • Ulrich MießnerEmail author
  • Thorben Helmers
  • Ralph Lindken
  • Jerry Westerweel
Research Article

Abstract

This paper presents a geometric model to approximate the interface shape of Taylor bubbles and droplets moving in rectangular microchannels. To retrieve the entire 3D interface geometry, the following apriori knowledge is required: the cross-sectional channel geometry, the droplet/bubble length, and the flow-related front and back cap deformation ratios retrieved, e.g., from instantaneous images of translating interfaces. The accuracy of the interface shape model is benchmarked with experimentally generated average images of the statistical distribution of a particulate tracer. In the experiment, the fluidic material system, the aspect ratio of the channel cross section, and the droplet/bubble length are varied. The mean overall deviation between the model and the experimental data remains below \(2\%\). The interface shape model is applicable for horizontal pressure-driven liquid–liquid and gas–liquid fluid systems in a range of Ca \(< 0.01\), \(Re \lesssim 1\) and \(Bo \ll 1\) with the liquid continuous phase wetting the wall. The model can be applied, e.g., to guide the 3D flow field reconstruction of multi-plane \(\upmu\)-PIV measurements of Taylor flows.

Graphical abstract

List of symbols

Acronyms

1D

One dimensional

2C

Two velocity components

3D

Three dimensional

DMSO

Dimethylsulfoxide

FF

Flow-focussing

Gly

Glycerin

PDMS

Polydimethylsiloxane

PS

Polystyrol

RI

Refractive index

FFT

Fast-Fourier transform

µ-PIV

Microscopic particle image velocimetry

Oct

Octanol

Dimensionless quantities

Bo

Bond number

Ca

Capillary number

Re

Reynolds number

Greek symbols

\(\alpha\)

Volume flow ratio

\(\varDelta\)

Difference

\(\delta\)

Wall film thickness

\(\eta\)

Dynamic viscosity

\(\lambda\)

Light wavelength

\(\rho\)

Density

\(\sigma\)

Interfacial tension

\(\varepsilon\)

Disperse phase fraction

\(\gamma\)

Surface tension

Roman symbols

A

Area

a

Semi-major axis

\(A_{\text {r}}\)

Channel aspect ratio

b

Semi-minor axis

d

Diameter

f

Frequency

g

Gravitation acceleration

H

Channel height

I

Intensity

\(k_{\text {b}}\)

Back-deformation ratio

\(k_{\text {f}}\)

Front-deformation ratio

\(k_{\text {g}}\)

Dimensionless gutter radius

L

Length

p

Pressure

Q

Volume flow rate

R

Radius

s

Scaling factor of gutter radii (\(L_{\text {d}} < L_{\text {d,crit}}\))

u

Velocity

V

Volume

W

Channel width

x

x-coordinate

y

y-coordinate

z

z-coordinate

Superscripts

\(*\)

Normalized

0

Superficial

dyn

Dynamic

j

Generalization of stat and dyn

max

Maximum

min

Minimum

shift

Shifted

stat

Static

Subscripts

1

Line 1, Fig. 1

2

Line 2, Fig. 1

3

Line 3, Fig. 1

4

Line 4, Fig. 1

a

Line a, Fig. 1

b

Line b, Fig. 1/droplet back

c

Continuous phase

cam

Camera

ch

Channel

crit

Critical

d

Disperse phase

drop

Droplet

f

Droplet front

g

Line g, Fig. 1/gutter

i

Generalization of f and b

LP

Laplace

mean

Mean value

min

Minimum

n

Image control variable

p

Tracer particle

pin

Pinhole

theo

Theoretical value

tot

Total

xy

xy-plane at origin

zy

zy-plane at origin

Notes

References

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Ulrich Mießner
    • 1
    Email author
  • Thorben Helmers
    • 1
  • Ralph Lindken
    • 2
  • Jerry Westerweel
    • 3
  1. 1.Department of Environmental Process Engineering (UVT)University of BremenBremenGermany
  2. 2.Institute of Thermodynamics and Fluid DynamicsBochum University of Applied SciencesBochumGermany
  3. 3.Laboratory of Aero- and Hydrodynamics (AHD)Delft University of TechnologyDelftThe Netherlands

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