The critical capillary number in a microchannel is a dimensionless number that characterizes the balance between viscous and capillary forces in determining the flow behavior of fluids in microscale channels.

Md. Al-Amin, why did you explain the capillary number? My question is about the critical value of this parameter.

The critical capillary number in a microchannel is a dimensionless number that characterizes the transition between different flow regimes dominated by the interplay between viscous and capillary forces. It doesn't have a single, fixed value and can vary depending on several factors, including:

• Channel geometry: The cross-sectional shape and dimensions of the microchannel can significantly influence the critical capillary number. For example, rectangular channels generally have a lower critical capillary number than circular channels.
• Fluid properties: The viscosity and surface tension of the fluids involved play a crucial role in determining the critical capillary number. Higher viscosity or lower surface tension will lead to a higher critical capillary number.
• Flow conditions: The flow rate and pressure can also affect the critical capillary number. Higher flow rates or pressures will generally lead to a higher critical capillary number.

In microfluidics, the critical capillary number (Ca​)cr for a common mini channel (larger than typical micro channels) can vary based on specific factors such as channel dimensions, fluid properties, and surface tension. However, for mini channels, the transition from viscous-dominated to capillary-dominated flow might occur in the range of 0.01 to 0.1.

For a common micro channels it might occur in the range of 0.001 to 0.1

• Droplet formation: For droplet formation in T-junction microchannels, the critical capillary number typically lies between 0.01 and 0.1. Below this range, continuous flow persists, while above it, discrete droplets start to form.
• Thread-to-drop transition: In some microfluidic devices, a transition from a continuous thread-like liquid flow to the formation of separate droplets can occur. The critical capillary number for this transition can vary depending on the specific geometry and flow conditions, but it is typically in the range of 0.1 to 1.
• Capillary fingering: When a less viscous fluid displaces a more viscous fluid in a porous medium, a phenomenon called capillary fingering can occur, where the less viscous fluid fingers its way through the medium. The critical capillary number for the onset of capillary fingering typically lies between 10^-4 and 10^-6.

In discussions regarding the appropriate Capillary Number (Ca) for ignoring capillary effects in microchannel studies, it's essential to consider the context of the interface interactions. Typically, for water flow inside silicone microchannels, many studies, including mine, use the water-air surface tension value of 0.0726 N/m. This is generally because the dominant interface under investigation is between the water and air, not between the water and the silicone walls, unless specific surface modifications of the silicone suggest otherwise.

Furthermore, concerning the critical value of Ca, while the standard threshold is often set at Ca = 1, I believe this may not be universally applicable. In my study, which investigates forced flow with Reynolds numbers between 125 and 1300, it appears that capillary effects could be non-negligible even above this threshold. This observation leads me to suggest that the traditional threshold of Ca = 1 might need adjustment based on specific experimental conditions and flow behaviors. Such an approach allows for a more nuanced understanding of when viscous forces indeed dominate over capillary forces in practical scenarios

Farshid Hesami