[Paper Review] Effects of capillary number and flow rates on the hydrodynamics of droplet generation in T-junction microfluidic systems

Akepogu Venkateshwarlu, Ram Prakash Bharti|arXiv (Cornell University)|May 21, 2021

Innovative Microfluidic and Catalytic Techniques Innovation69 references54 citations

TL;DR

This study numerically investigates droplet generation in T-junction microfluidic devices using finite element simulations to analyze the effects of capillary number (10⁻⁴ ≤ Cac ≤ 1) and flow rate ratio (0.1 ≤ Qr ≤ 10) on droplet hydrodynamics. It identifies six flow regimes, reveals that the squeezing regime persists for all Cac values when 2 ≤ Qr ≤ 10, and proposes a predictive power-law correlation (f_dd = 2.3 Q_r^0.417 Ca_c^0.685, R² = 0.9634) for droplet frequency in the dripping regime.

ABSTRACT

The hydrodynamics of droplets is significant in wide-ranging applications involving immiscible fluids and emulsions in food and pharmaceutical. The control and manipulation of droplets are primarily a function of flow governing and geometrical parameters. The finite element and level set approaches are used in this work to explore the influences of capillary number (Ca) and flow rate ratio (Qr) of dispersed and continuous phases on hydrodynamics of droplet generation in two-phase flow through T-junction cross-flow microfluidic device. A mathematical model based on a mass continuity, Navier-Stokes, and level set equations are solved computationally using the Eulerian framework for Ca = 1e-4 - 1 and Qr =0.1 - 10. Both immiscible phases, having equal density and unequal viscosity, flow (Re=0.1) through equal-sized channels. In particular, instantaneous phase flow field, droplet size, droplet detachment time and generation frequency are presented and discussed as a function of governing parameters (Ca and Qr). Considered parametric space is characterized as squeezing, first transition, dripping, second transition, parallel, and jet flow regimes. In contrast to threshold Ca ~ 0.01 in earlier studies, squeezing regime exists for all Ca and Qr = 2- 10. Flow regimes are also mapped into droplets and non-droplet zones. Threshold interfacial Ca, defining the boundary between droplet and non-droplet zones, scales quadratically with Qr. Droplet dynamics shows a complex dependence on Ca and Qr. Droplet length varies linearly with Qr in squeezing regime whereas power-law variation with Ca and Qr in dripping regime. Droplet frequency shows a power-law function of Ca and Qr in droplet zone. Present results compare excellently with earlier limited experimental and numerical studies. Finally, present results and predictive correlations can guide engineering and design of droplet microfluidics devices.

Motivation & Objective

• To understand the influence of capillary number (Cac) and flow rate ratio (Qr) on droplet hydrodynamics in T-junction microfluidic systems.
• To map flow regimes—squeezing, dripping, jetting, and others—based on Cac and Qr.
• To identify the transition between droplet and non-droplet zones using a threshold capillary number that scales quadratically with Qr.
• To develop predictive correlations for droplet size, detachment time, and generation frequency.
• To guide engineering design of droplet microfluidic devices through validated numerical simulations.

Proposed method

• Finite element method (FEM) is used to solve the Eulerian framework of a mathematical model based on mass conservation, Navier-Stokes, and conservative level set equations.
• The conservative level set method tracks the fluid-fluid interface with reinitialization and stabilization parameters to maintain sharp interfacial resolution.
• Simulations are conducted at a fixed Reynolds number (Rec = 0.1), focusing on low-inertia, viscous-flow conditions.
• The capillary number (Cac = uμ/σ) and flow rate ratio (Qr = Qd/Qc) are varied systematically across 10⁻⁴ ≤ Cac ≤ 1 and 0.1 ≤ Qr ≤ 10.
• The droplet detachment frequency (f_dd), droplet size (L/wc), and detachment time (τdd) are computed and analyzed as functions of Cac and Qr.
• A new predictive correlation f_dd = αQ_r^β Ca_c^γ is derived using regression analysis with R² = 0.9634 for the dripping regime.

Experimental results

Research questions

• RQ1 How do capillary number (Cac) and flow rate ratio (Qr) influence the formation and dynamics of droplets in T-junction microfluidic systems?
• RQ2 What are the distinct flow regimes (e.g., squeezing, dripping, jetting) and under what Cac and Qr conditions do they occur?
• RQ3 What is the threshold capillary number (Cac,trans) that separates droplet formation from non-droplet flow, and how does it scale with Qr?
• RQ4 How does droplet size vary with Qr in the squeezing regime, and with Cac and Qr in the dripping regime?
• RQ5 Can a predictive correlation be developed for droplet generation frequency (f_dd) in the dripping regime based on Cac and Qr?

Key findings

• The squeezing regime exists for all Cac values when 2 ≤ Qr ≤ 10, contradicting the commonly reported threshold Cac ≈ 10⁻².
• A transitional capillary number (Cac,trans) that separates droplet and non-droplet zones scales quadratically with Qr, expressed as Car,trans = βQ_r².
• In the squeezing regime, droplet length varies linearly with Qr, indicating direct control over droplet size via flow rate ratio.
• In the dripping regime, both droplet size and frequency follow a power-law dependence on Cac and Qr, with R² = 0.9874 for droplet size and R² = 0.9634 for frequency.
• A predictive correlation f_dd = 2.3 Q_r^0.417 Ca_c^0.685 is proposed for droplet frequency in the dripping regime (1/10 ≤ Qr ≤ 1/2, 10⁻² ≤ Cac ≤ 0.1), with excellent agreement between numerical and predicted values as shown in the parity plot.
• The study maps the entire parameter space into droplet and non-droplet zones, enabling reliable prediction of droplet formation for given Cac and Qr values.