# Three-dimensional investigation of liquid slug Taylor flow inside a micro-capillary using holographic velocimetry

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## Abstract

Digital holography is an optical technique which is capable of providing instantaneous three-components of fluid flow velocity in three-dimensions (3D-3C) using a single camera. Digital holographic microscopy has been implemented in the present study to analyze liquid slug Taylor flow in a micro-channel of cross-sectional dimensions of 1,000 × 1,000 µm^{2}. The working fluids are water (liquid) and air (gas), with superficial velocities of liquid, *U* _{L} = 0.6 mm/s and gas, *U* _{G} = 1.2 mm/s, respectively. The corresponding Capillary number, *Ca* = 0.035 × 10^{−3} and Bond number, *Bo* = 0.144. The holographic velocimetry technique has been implemented and appropriately validated by comparing the velocity profile from present experiment with that from analytical velocity profile for single-phase flow. Complete flow field results, i.e., *u-*, *v-* and *w*-components of velocity inside the liquid slug volume, i.e., in both streamwise (*x*–*y*) and cross-stream (*y*–*z*) planes are presented. The present experiments on liquid slug Taylor flow show strong cross-stream velocity near the advancing and receding meniscus due to higher capillary pressure. The stream traces show converging and diverging radial flow in the cross-stream plane near the receding and advancing meniscus, respectively. Two three-dimensional recirculation bubbles are observed inside the liquid slug. Overall, this paper reports the complex three-dimensional flow field inside a liquid slug Taylor flow from the 3D-3C flow field measurements.

## Keywords

Capillary Number Particle Tracking Velocimetry Bubble Velocity Digital Holography Liquid Film Thickness## List of symbols

*I*_{H}Intensity distribution of hologram

*I*_{H,image}Intensity distribution of magnified hologram

*M*Magnification of microscope objective

- \( h_{{z_{\text{r}} }} \)
Free space impulse response function at distance

*z*_{r}from hologram plane*x*,*y**x*and*y*coordinates on hologram plane*x*_{r},*y*_{r}*x*and*y*coordinates on reconstruction plane*E*_{r}Complex amplitude distribution on reconstruction plane

*E*Complex amplitude distribution of plane collimated light

*λ*Wavelength of light

*z*_{o}Distance of object particle from hologram plane

*z*_{r}Distance of reconstructed particle from hologram plane

*I*_{r}Intensity distribution on reconstruction plane

- ∆
*t* Time between two successive holograms

*U*_{L}Liquid superficial velocity

*U*_{G}Gas superficial velocity

*U*_{TP}Two-phase velocity

*A*_{ch}Channel cross-sectional area

*Q*_{L}Liquid flow rate

*Q*_{G}Gas flow rate

*W*Width of channel along

*x*-direction*H*Height of channel along

*y*-direction*L*Depth of channel along

*z*-direction*L*_{s}Length of the liquid slug

*x**Non-dimensional coordinate along

*x*-direction (*x*/*W*)*y**Non-dimensional coordinate along

*y*-direction [*y*/(*H*/2)]*z**Non-dimensional coordinate along

*z*-direction [*z*/(*L*/2)]*U*_{B}Bubble velocity

*μ*_{L}Dynamic viscosity

*ρ*Density of fluid

- σ
Surface tension

*r*_{h}Hydraulic radius

*δ*Liquid film thickness between the channel wall and the gas bubble

*u*Streamwise velocity

*v*Transverse velocity

*w*Spanwise Velocity

*U*_{vw}Velocity on transverse plane

*U*_{abs}Absolute velocity

*U*_{rel}Relative velocity

*g*Gravitational acceleration

*d*_{p}Diameter of particle

*D*Distance between microscope objective and hologram plane

*Bo*Bond number, \( Bo = \frac{{(\rho_{\text{L}} - \rho_{\text{G}} )d_{\text{h}}^{2} g}}{\sigma } \)

*Ca*Capillary number, \( Ca = \frac{{\mu_{\text{L}} U}}{\sigma } \)

*Re*Reynolds number, \( \text{Re} = \frac{{\rho_{\text{L}} Ud_{\text{h}} }}{{\mu_{\text{L}} }} \)

*We*Weber number, \( We = \frac{{\rho_{\text{L}} U^{2} d_{\text{h}} }}{\sigma } \)

## Notes

### Acknowledgments

The authors thank Department of Science and Technology, Government of India for the financial support.

## Supplementary material

Supplementary material 1 (WMV 2169 kb)

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