Appearance in the Navier-Stokes equations
The Weber number appears in the incompressible Navier-Stokes equations through a free surface boundary condition.[3]
For a fluid of constant density and dynamic viscosity , at the free surface interface there is a balance between the normal stress and the curvature force associated with the surface tension:
Where is the unit normal vector to the surface, is the Cauchy stress tensor, and is the divergence operator. The Cauchy stress tensor for an incompressible fluid takes the form:
Introducing the dynamic pressure and, assuming high Reynolds number flow, it is possible to nondimensionalize the variables with the scalings:
The free surface boundary condition in nondimensionalized variables is then:
Where is the Froude number, is the Reynolds number, and is the Weber number. The influence of the Weber number can then be quantified relative to gravitational and viscous forces.