Capillary Number (Ca): Viscous Drag Versus Surface Tension in Multiphase Flow

The capillary number (Ca) expresses the ratio of viscous forces to surface-tension forces in a two-phase flow. Defined by Ca = μU / σ, where μ is dynamic viscosity, U is a characteristic velocity, and σ is interfacial tension, the quantity determines whether a flowing fluid drags an interface along or leaves it largely undisturbed. Understanding Ca equips engineers to design pore-scale displacement processes, microfluidic devices, and coating operations that require precise control of wetting patterns.

This guide traces how Ca emerged from early petroleum engineering experiments, elaborates the theoretical framework relating viscous shear to capillary pressure, surveys measurement approaches, and profiles applications across porous media, biomedical systems, additive manufacturing, and energy transition technologies. Cross-reference the Weber number explainer to delineate regimes where inertia supplants viscosity, and compare with the Bond number when gravity becomes relevant.

Definition, Dimensional Consistency, and Typical Values

Because Ca is dimensionless, it facilitates similarity analysis across experimental platforms, numerical simulations, and field operations. The numerator μU carries dimensions of N·m⁻¹ (or kg·s⁻²), matching the denominator σ. Typical Ca values span orders of magnitude: Ca ≪ 1 signifies surface-tension-dominated motion with nearly spherical droplets or bubbles; Ca ≈ 1 indicates comparable viscous and capillary stresses; and Ca ≫ 1 denotes viscous domination, producing elongated films and continuous phase displacement. Researchers often specify a microscopic capillary number based on pore-scale velocities, as well as a macroscopic variant using Darcy fluxes in porous media.

In stratified reservoirs, brine displacing oil at centimeters per second with μ ≈ 5 mPa·s and σ ≈ 30 mN·m⁻¹ yields Ca ≈ 1.7 × 10⁻³, insufficient for mobilizing residual oil. Polymer flooding increases effective viscosity to tens of mPa·s, raising Ca by an order of magnitude. Microfluidic emulsification often operates at Ca ≈ 0.1–10 to shear droplets into monodisperse streams, while high-speed coating flows can reach Ca above 100, generating uniform films as surface tension becomes negligible relative to viscous drag.

Historical Development and Standardisation

The capillary number emerged in the mid-20th century as petroleum engineers attempted to quantify why secondary oil recovery left significant hydrocarbons trapped in pores. Classic experiments by Hugh M. Muskat and colleagues in the 1930s documented the interplay of viscous forces and interfacial tension in core-flood tests, but a widely adopted dimensionless group crystallised later. In 1958, Donald Leverett introduced the Leverett J-function, linking capillary pressure to dimensionless parameters that implicitly contained the Ca balance. By the 1960s, Lake and others formalised Ca = μU/σ to collapse data for miscible and immiscible displacement.

Standardisation gained momentum as enhanced oil recovery (EOR) techniques matured. The Society of Petroleum Engineers embedded Ca reporting into recommended practices for surfactant and polymer floods, ensuring that laboratory chemical screening would translate to reservoir-scale performance. ISO 80000-11 now catalogues the capillary number among characteristic numbers for transport phenomena, specifying notation (Ca), definitions, and preferred symbol usage to maintain clarity across industries. Contemporary research extends the historical framework to carbon capture and storage (CCS), hydrogen storage, and groundwater remediation where viscous-capillary competition remains decisive.

Conceptual Foundations and Scaling Relationships

Linking viscous stress and capillary pressure

Surface tension generates a pressure jump Δp = σ κ across curved interfaces, where κ is mean curvature. In a pore throat of radius r, Δp ≈ 2σ/r resists displacement. Viscous stress scales with μU/L, where L approximates pore size or film thickness. Equating these magnitudes reveals Ca ≈ (μU/σ)(L/r). When L ≈ r, Ca quantifies the ability of shear to overcome capillary entry pressure. Deviations from unity capture geometry-induced effects, motivating empirical correlations such as the capillary desaturation curve that maps residual saturation versus log(Ca).

Interplay with other similarity parameters

The capillary number rarely acts in isolation. In porous media, the Reynolds number typically remains low, validating Darcy’s law and focusing analysis on Ca versus Bond number. In microchannels, Ohnesorge and Weber numbers demarcate whether droplet breakup is controlled by viscosity or inertia. Coating-flow theory introduces the Landau–Levich–Derjaguin scaling, where film thickness scales as L ∝ Ca^2/3, highlighting the predictive power of Ca for industrial processes.

Wettability, hysteresis, and contact-line physics

Real interfaces exhibit dynamic contact angles that differ from static values, giving rise to contact-line hysteresis. Empirical relations such as cos θ_d − cos θ_s ∝ Caⁿ (with n ≈ 1/3 to 1) connect Ca to dynamic wetting. Molecular-kinetic theory and hydrodynamic models both use Ca as a small parameter to expand contact-line velocity. For superhydrophobic surfaces, extremely low σ and pinned contact lines mean Ca must be carefully interpreted, often using apparent contact angles extracted from high-speed imaging.

Measurement Techniques and Data Reduction

Determining Ca requires accurate measurements of viscosity, velocity, and interfacial tension. Laboratories employ rotational rheometers or microrheology to characterise μ across relevant shear rates, especially for polymer or surfactant solutions whose viscosity may be shear-thinning. High-speed cameras or particle image velocimetry (PIV) quantify U at the interface; in porous media, Darcy velocity q = Q/A is substituted, recognising that local velocities exceed q by a factor linked to porosity. Interfacial tension σ is measured via pendant-drop, spinning-drop, or Wilhelmy plate methods, with corrections for temperature, salinity, and surfactant adsorption kinetics.

Data reduction emphasises uncertainty propagation. Because Ca scales linearly with μ, U, and σ⁻¹, relative uncertainties add in quadrature. Reporting should include measurement temperature, fluid compositions, and any time-dependent interfacial effects. For surfactant floods, σ may evolve as adsorption equilibrates; engineers therefore report an effective capillary number averaged over the displacement. Pairing Ca with residual oil saturation data or contact-angle hysteresis curves enables reproducible interpretation. The Reynolds number calculator helps document flow regimes alongside Ca in test reports.

Applications Across Industries

Enhanced oil recovery and subsurface storage

Surfactant and polymer floods aim to raise Ca to mobilise trapped hydrocarbons while maintaining sweep efficiency. Laboratory core floods map residual oil saturation versus Ca, informing chemical selection. In carbon dioxide sequestration, Ca controls residual trapping of CO₂ bubbles in brine-filled pores, influencing storage security. Hydrogen storage in depleted reservoirs similarly depends on Ca-mediated wettability alteration to prevent gas leakage.

Microfluidics, lab-on-chip, and biomedical devices

Droplet microfluidics relies on tuning Ca to toggle between dripping and jetting regimes, enabling generation of uniform droplets for diagnostics and drug delivery. Digital microfluidic platforms use Ca-based scaling to predict actuation voltages and electrode geometries. Capillary numbers also govern blood flow in microcirculation, where plasma–cell interfaces respond to shear and interfacial tension modified by surfactants or pharmaceuticals.

Coating, printing, and additive manufacturing

Slot-die coating, dip coating, and gravure printing all operate within defined Ca windows to ensure uniform film thickness and avoid defects such as ribbing or air entrainment. Additive manufacturing processes, including inkjet and aerosol jet printing, use Ca alongside Ohnesorge numbers to select inks and nozzle geometries that prevent satellite droplets. Battery electrode fabrication and perovskite solar cell coating similarly rely on Ca to balance solvent viscosity and surface tension.

Environmental and consumer-product engineering

Soil remediation technologies evaluate Ca when injecting surfactant solutions to mobilise non-aqueous-phase liquids (NAPLs). Household and industrial cleaners use surfactant packages to lower σ, boosting Ca at wiping velocities and improving contaminant removal. Agricultural spray design accounts for Ca to promote leaf coverage without excessive runoff, coupling with Weber-number criteria for droplet breakup.

Strategic Importance and Future Directions

The capillary number remains central to decarbonisation strategies that depend on subsurface fluid control, from CO₂ sequestration to geothermal energy production. Emerging research couples Ca with pore-scale imaging (micro-CT, confocal microscopy) to derive predictive models rooted in actual rock geometries. Machine-learning-assisted simulators incorporate Ca-dependent relative permeability curves to accelerate reservoir forecasts. In health care, Ca-guided design of lab-on-chip diagnostics and drug delivery carriers supports personalised medicine.

Looking ahead, integrating Ca with data-driven optimisation and advanced materials will expand its relevance. Adaptive surfactants that respond to shear or pH can dynamically tune σ, effectively modulating Ca in real time. Micro-robotics leverage Ca-aware control laws to navigate fluid interfaces for targeted therapies. Across these innovations, rigorous calculation, documentation, and communication of the capillary number ensure reproducible science and safe deployment. Pair this article with the ISO 80000-11 overview to keep symbol usage, units, and reporting conventions aligned.