Other Dimensionless Numbers
A number of other dimensionless numbers are useful for describing fluid flows on the microscale. These numbers are summarized in Table 2-3.
Table 2-3: Dimensionless Numbers useful for characterizing microfluidic flows.
Dimension less Number
Symbol
Definition
Notes
Bond number/ Eötvös number
Bo
Eo
Ratio of surface tension forces to body forces (usually gravity) for two-phase flow. Density can refer to density difference when considering buoyancy. At high Bond numbers surface tension has a minimal effect on the flow, (), at low Bond numbers, surface tension dominates ().
Capillary number
Ca
Ratio of viscous forces to surface tension for two phase flow. At low capillary numbers () flow in porous media is dominated by surface tension.
Dukhin number
Du
Ratio of the surface conductivity contribution to the fluid bulk electrical conductivity contribution in electrokinetic phenomena. For low Dukhin numbers (), surface conductivities can be neglected when modeling electrokinetic flows.
Knudsen number
Kn
Ratio of the gas mean free path to the characteristic length scale. Continuum flow occurs when Kn<0.01, slip flow occurs in the range 0.01<Kn<0.1, transitional flow occurs in the range 0.1<Kn<10 and Free Molecular Flow occurs when Kn>10. For more details see Modeling Rarefied Gas Flows.
Mach number
Ma
Ratio of the convective speed to the speed of sound in the medium. The Laminar Flow interface in the Microfluidics Module can handle compressible flows with . For higher Mach number flows the CFD Module is required.
Marangoni number
Mg
Ratio of thermal surface tension forces to viscous forces. In the low Marangoni number range () viscous forces dominate over surface tension gradients.
Ohnesorge /Laplace /Surataman numbers
Oh
La
Su
 
Relates the inertial and surface tension forces to the viscous forces. Used to describe the breakup of liquid jets and sheets: at low Ohnesorge and Reynolds numbers the Rayleigh instability occurs, at high Ohnesorge and Reynolds numbers atomization occurs.
Weber number
We
Ratio of inertial forces to surface tension forces. For small Weber numbers () surface tension dominates the flow.
Symbol definitions: ρ is the fluid density, g is the body force acceleration (usually the acceleration due to gravity), L is the characteristic length scale for the problem, σ is the surface tension coefficient, μ is the fluid viscosity, v is the characteristic velocity, Kσ is the surface conductivity, KB is the bulk conductivity, λ is the mean free path, α is the thermal diffusivity (α=κ/(ρcp)), cp is the heat capacity at constant pressure, κ is the thermal conductivity, and T is the absolute temperature (ΔT is the characteristic temperature difference).