Microfluidics Part 1 – Design & Fabrication

Accessible Equation Supplement

Prof. Steven S. Saliterman

1. Energy Balance in a Liquid Column

Equation 1.1 — Energy Balance

$$2 \pi r_0 h (\gamma_{SG} - \gamma_{SL}) = \Delta p \,\pi r_0^2 h$$

• \(\gamma_{SG}\): Solid–gas interfacial tension (N/m)
• \(\gamma_{SL}\): Solid–liquid interfacial tension (N/m)
• \(\Delta p\): Pressure difference (Pa)
• \(r_0\): Radius of the column (m)
• \(h\): Height of the column (m)

2. Surface Tension and Contact Angle

Equation 2.1 — Young’s Equation

$$\gamma_{LG}\cos\theta + \gamma_{SL} = \gamma_{SG}$$

• \(\gamma_{LG}\): Liquid–gas interfacial tension (N/m)
• \(\gamma_{SL}\): Solid–liquid interfacial tension (N/m)
• \(\gamma_{SG}\): Solid–gas interfacial tension (N/m)
• \(\theta\): Contact angle (degrees)

3. Capillary Rise

Equation 3.1 — Capillary Rise Height

$$h = \frac{2\gamma \cos\theta}{\rho g r}$$

• \(h\): Capillary rise height (m)
• \(\gamma\): Surface tension (N/m)
• \(\theta\): Contact angle (degrees)
• \(\rho\): Density (kg/m\(^3\))
• \(g\): Acceleration due to gravity (m/s\(^2\))
• \(r\): Tube radius (m)