Weber number

The Weber number may be written as:

${\displaystyle \mathrm {We} ={\frac {\mbox{Drag Force}}{\mbox{Cohesion Force}}}=\left({\frac {8}{C_{\mathrm {D} }}}\right){\frac {\left({\frac {\rho \,v^{2}}{2}}\,C_{\mathrm {D} }\pi {\frac {l^{2}}{4}}\right)}{\left(\pi \,l\,\sigma \right)}}={\frac {\rho \,v^{2}\,l}{\sigma }}}$  

where

• ${\displaystyle C_{\mathrm {D} }}$ is the drag coefficient of the body cross-section.
• ${\displaystyle \rho }$ is the density of the fluid (kg/m³).
• ${\displaystyle v}$ is its velocity (m/s).
• ${\displaystyle l}$ is its characteristic length, typically the droplet diameter (m).
• ${\displaystyle \sigma }$ is the surface tension (N/m).

The modified Weber number,

${\displaystyle \mathrm {We} ^{*}={\frac {\mathrm {We} }{12}}}$  

equals the ratio of the kinetic energy on impact to the surface energy,

${\displaystyle \mathrm {We} ^{*}={\frac {E_{\mathrm {kin} }}{E_{\mathrm {surf} }}}}$,

where

${\displaystyle E_{\mathrm {kin} }={\frac {\pi \rho l^{3}v^{2}}{12}}}$  

and

${\displaystyle E_{\mathrm {surf} }=\pi l^{2}\sigma }$.

One application of the Weber number is the study of heat pipes. When the momentum flux in the vapor core of the heat pipe is high, there is a possibility that the shear stress exerted on the liquid in the wick can be large enough to entrain droplets into the vapor flow. The Weber number is the dimensionless parameter that determines the onset of this phenomenon called the entrainment limit (Weber number greater than or equal to 1). In this case the Weber number is defined as the ratio of the momentum in the vapor layer divided by the surface tension force restraining the liquid, where the characteristic length is the surface pore size.

• Weast, R. Lide, D. Astle, M. Beyer, W. (1989-1990). CRC Handbook of Chemistry and Physics. 70th ed. Boca Raton, Florida: CRC Press, Inc.. F-373,376.