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Taylor slug gas flow

The Taylor flow is a special case of slug flow where the bullet-shaped bubbles (Taylor bubbles) are separated by free-gas-entrained liquid slugs.

The key non-dimensional number is the capillary number $C a = μ U / γ$ (µ = water viscosity, U = average water velocity, γ = surface tension).

For well-defined Taylor slugs Ca should be < 0.01
Source : https://doi.org/10.3390/pr9050870

Name	Symbol	Definition	Description
Archimedes	Ar	$\frac{ρ_{l} (ρ_{l} - ρ_{g}) g d^{3}}{μ_{l}^{2}}$	The ratio of the gravitational to the viscous effects
Bond or Eötvös	Bo Eo	$\frac{{gd}^{2} (ρ_{l} - ρ_{g})}{4 σ}$	The ratio of the gravitational (buoyancy) and the capillary force scales
Cahn	Cn	$\frac{ξ}{d}$	The ratio of the interface width and the tube diameter or any other length scale
Capillary	Ca	$μ U / σ$	The ratio of the viscous forces and the capillary forces
Ca/Re	$Ca / Re$	$μ^{2} / (ρ d σ)$	(N/A)
Froude	Fr	$U / \sqrt{gd}$	The ratio between the flow inertia and the external field
Laplace	La	${[\frac{σ}{g d^{2} (ρ_{l} - ρ_{g})}]}^{0.5}$	The ratio of the capillary and the gravitational (buoyancy) effects
Ohnesorge	Oh	$\frac{\sqrt{W e}}{Re} = \frac{μ}{\sqrt{σ ρ d}}$	The ratio of the viscous force to the inertia and the surface tension forces
Reynolds	Re	$ρ Ud / μ$	The ratio between the inertia and the viscous forces
Suratman	Su	$\frac{{Re}^{2}}{{We}^{2}} = \frac{1}{{Oh}^{2}} = \frac{σ ρ d}{μ^{2}}$	The ratio of the surface tension to the viscous forces
Weber	We	$CaRe = \frac{ρ U^{2} d}{σ}$	The ratio of the inertial forces to the interfacial forces

TABLEAU : Nombres adimensionnels pour un flux multiphasé

Slug regime

In the slug regime, interfacial tension is greater than inertial forces, and the Weber numbers are $7.61 \times 10^{- 6} < W e_{w s} < 4.87 \times 10^{- 2}$ and $5.94 \times 10^{- 6} < W e_{k s} < 5.94 \times 10^{- 4}$

Map of regime as a function of the two flows :

_Source : Chung, P.M.-Y., Kawaji, M., 2004. The effect of channel diameter on adiabatic two-phases flow characteristics in microchannels. International Journal of Multiphase Flow 30, 735–761

Attention pour des pressions importantes il faut utiliser un Luer Lock :

At Ca ≲ 0.01 and Reynolds number Re < 100 the interface sees mostly surface-tension forces. The T-junction pinches off the incoming air whenever the growing plug spans the channel, giving highly monodisperse “unit cells” of gas and liquid—a regime first mapped by Thorsen et al. and subsequently parameterised in many studies of T-junction droplet generators.

Longeur des train d'onde de gouttes

The scaling law that predicts the final droplet length, $L_{d}$ , at the point of detachment is

L_{d} = w_{c} + w_{n} \frac{Q_{d}}{Q_{c}}

where $Q_{c}$ and $Q_{d}$ are the flow rates of the continuous and dispersed phases respectively and $w_{c}$ and $w_{n}$ are the widths of the neck and of the main channel, equal for two identical capillaries. In non-dimensional form, this can be expressed as

L_{d} / w_{c} = 1 + α Q_{d} / Q_{c}

where $α = w_{n} / w_{c}$ is a positive constant that depends on the geometry of the T-junction (Garstecki et al., 2006). According to Xu et al. (2008), the blocking length is not necessarily equal to $w_{c}$ but can be generalised as $ε w_{c}$ as it depends on the geometry of the channel. Therefore, the scaling relation (5) can be modified as

L_{d} / w_{c} = ε + α Q_{d} / Q_{c}

where $ε$ is a fitting parameter related to the geometry of the microchannel. This scaling law suggests that the droplet length depends only on the variation of the flow rate ratio of the two immiscible fluids. However, we also observed the variations in droplet length with capillary number in our simulations, analogous to those obtained by Christopher et al. (2008).

SOURCE : Investigation of pressure profile evolution during confined micro-droplet formation using a two-phase level set method

Further reading : Von Karman vortex streets

Variation-of-the-Von-Karman-vortex-streets-as-a-function-of-the-Reynolds.png