The Weber number may be written as: :\mathrm{We} = \frac{\mbox{Inertial pressure}}{\mbox{Laplace pressure}} = \frac{\rho\,v^2}{ \left(\sigma/l\right)} = \frac{\rho\,v^2\,l}{\sigma} where • \rho is the
density of the fluid (
kg/
m3). • v is its
velocity (m/
s). • l is its characteristic
length, typically the droplet diameter (m). • \sigma is the
surface tension (
N/m). • \rho\,v^2 is the inertial or
dynamic pressure scale. • \sigma/l is the
Laplace pressure scale. The above is the force perspective to define the Weber number. We can also define it using energy perspective as the ratio of the kinetic energy on impact to the surface energy, :\mathrm{We}=\frac{E_\mathrm{kin}}{E_\mathrm{surf}}, where :E_\mathrm{kin} \propto \rho l^3 v^2 and :E_\mathrm{surf}\propto l^2 \sigma.
Appearance in the Navier-Stokes equations The Weber number appears in the incompressible
Navier-Stokes equations through a
free surface boundary condition. For a fluid of constant density \rho and
dynamic viscosity \mu, at the free surface interface there is a balance between the normal stress and the
curvature force associated with the surface tension: :\widehat{\bf n} \cdot \mathbb{T} \cdot \widehat{\bf n} = \sigma \left( \nabla \cdot \widehat{\bf n} \right) Where \widehat{\bf n} is the unit normal vector to the surface, \mathbb{T} is the
Cauchy stress tensor, and \nabla\cdot is the
divergence operator. The Cauchy stress tensor for an incompressible fluid takes the form: :\mathbb{T} = -pI + \mu\left[ \nabla {\bf v} + (\nabla {\bf v})^{T} \right] Introducing the dynamic pressure p_{d} = p - \rho {\bf g} \cdot {\bf x} and, assuming high
Reynolds number flow, it is possible to
nondimensionalize the variables with the scalings: :p_{d} = \rho V^{2}p_{d}', \quad \nabla = L^{-1}\nabla', \quad {\bf g} = g{\bf g}', \quad {\bf x} = L{\bf x}', \quad {\bf v} = V{\bf v}' The free surface boundary condition in nondimensionalized variables is then: :-p_{d}' + {1\over{\text{Fr}^{2}}}z' + {1 \over{\text{Re}}} \widehat{\bf n} \cdot \left[ \nabla'{\bf v}' + (\nabla'{\bf v}')^{T} \right] \cdot \widehat{\bf n} = {1\over{\text{We}}} \left( \nabla' \cdot \widehat{\bf n} \right) Where \text{Fr} is the
Froude number, \text{Re} is the Reynolds number, and \text{We} is the Weber number. The influence of the Weber number can then be quantified relative to gravitational and viscous forces. ==Applications==