Universal self-scalings in a micro-co-flowing

Abstract

Experiments are carried out to explored the co-flows in a type of novel tapered rectanglar PMMA micro-channels. Variations of geometric parameter combinations bring great complexity to systematic descriptions of the co-flow behaviors. We find that the tapered co-flows are self-scaling processes. Based on theory of similitude, a self-similarity frame is buildup to reduce the complexity and dimension of the problem. Under such frame, universal scaling laws covering all the drop-detaching regimes (i.e., slug, dripping, thin jet) are derived, concisely and persuasively. The droplet length depends on the flow-rate-ratio and gives an exponential of nearly 1/2, which is featured as a typical ′Flow-Rate-Controlled-Flow′. The drop-detaching frequency depends on the two-phase capillary number and gives an exponential of nearly 3. Note that, these two formulas are valid in all the drop-detaching regimes over differed physical mechanisms. Moreover, the most astonishing thing is that, the former formula reproduce the scaling law of thin jet regime in straight circular co-flows, while the later one reproduce the scaling law of dripping regime in straight circular co-flows at the same time. Such highly consistency of scalings were rarely reported before. Furthermore, the clusters of the flow maps can be degenerated into one single flow diagram under our self-similarity frame. Such diagram can be divided naturally by machining learning into two parts, the Pan-dripping regime and the Pan-jetting regime, which are clear bordered at Wed≈1 and Cac≈0.28. These critical values are also consistent with the results in straight circular co-flows. This work can greatly facilitate the design and analysis method of devices for flow manipulation via simple formulas that catch the parameter range of liquid cells.