Weber number

 ( Fluid Dynamics ) 

The Weber number ( We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for with strongly curved surfaces.

It is named after Moritz Weber (1871–1951).

(2012). 9781461432654, Springer Science & Business Media. . ISBN 9781461432654

It can be thought of as a measure of the relative importance of the fluid's inertia compared to its surface tension. The quantity is useful in analyzing thin film flows and the formation of droplets and bubbles.

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Mathematical expression The Weber number may be written as:

$\backslash mathrm\{We\}\; =\; \backslash frac\{\backslash mbox\{Inertial\; pressure\}\}\{\backslash mbox\{Laplace\; pressure\}\}$

= \frac{\rho\,v^2}{ \left(\sigma/l\right)} = \frac{\rho\,v^2\,l}{\sigma}

where

• $\backslash rho$ is the density of the fluid (kilogram/meter³).
• $v$ is its velocity (m/second).
• $l$ is its characteristic length, typically the droplet diameter (m).
• $\backslash sigma$ is the surface tension (N/m).
• $\backslash rho\backslash ,v^2$ is the inertial or dynamic pressure scale.
• $\backslash sigma/l$ 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,

$\backslash mathrm\{We\}=\backslash frac\{E\_\backslash mathrm\{kin\}\}\{E\_\backslash mathrm\{surf\}\}$,

where

$E\_\backslash mathrm\{kin\}\; \backslash propto\; \backslash rho\; l^3\; v^2$  

and

$E\_\backslash mathrm\{surf\}\backslash propto\; l^2\; \backslash sigma$.

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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 $\backslash rho$ and dynamic viscosity $\backslash mu$, at the free surface interface there is a balance between the normal stress and the curvature force associated with the surface tension:

$\backslash widehat\{\backslash bf\; n\}\; \backslash cdot\; \backslash mathbb\{T\}\; \backslash cdot\; \backslash widehat\{\backslash bf\; n\}\; =\; \backslash sigma\; \backslash left(\; \backslash nabla\; \backslash cdot\; \backslash widehat\{\backslash bf\; n\}\; \backslash right)$

Where $\backslash widehat\{\backslash bf\; n\}$ is the unit normal vector to the surface, $\backslash mathbb\{T\}$ is the Cauchy stress tensor, and $\backslash nabla\backslash cdot$ is the divergence operator. The Cauchy stress tensor for an incompressible fluid takes the form:

$\backslash mathbb\{T\}\; =\; -pI\; +\; \backslash mu\backslash left$

Introducing the dynamic pressure $p\_\{d\}\; =\; p\; -\; \backslash rho\; \{\backslash bf\; g\}\; \backslash cdot\; \{\backslash bf\; x\}$ and, assuming high Reynolds number flow, it is possible to nondimensionalize the variables with the scalings:

$p\_\{d\}\; =\; \backslash rho\; V^\{2\}p\_\{d\}\text{'},\; \backslash quad\; \backslash nabla\; =\; L^\{-1\}\backslash nabla\text{'},\; \backslash quad\; \{\backslash bf\; g\}\; =\; g\{\backslash bf\; g\}\text{'},\; \backslash quad\; \{\backslash bf\; x\}\; =\; L\{\backslash bf\; x\}\text{'},\; \backslash quad\; \{\backslash bf\; v\}\; =\; V\{\backslash bf\; v\}\text{'}$

The free surface boundary condition in nondimensionalized variables is then:

$-p\_\{d\}\text{'}\; +\; \{1\backslash over\{\backslash text\{Fr\}^\{2\}\}\}z\text{'}\; +\; \{1\; \backslash over\{\backslash text\{Re\}\}\}\; \backslash widehat\{\backslash bf\; n\}\; \backslash cdot\; \backslash left\; \backslash cdot\; \backslash widehat\{\backslash bf\; n\}\; =\; \{1\backslash over\{\backslash text\{We\}\}\}\; \backslash left(\; \backslash nabla\text{'}\; \backslash cdot\; \backslash widehat\{\backslash bf\; n\}\; \backslash right)$

Where $\backslash text\{Fr\}$ is the Froude number, $\backslash text\{Re\}$ is the Reynolds number, and $\backslash text\{We\}$ is the Weber number. The influence of the Weber number can then be quantified relative to gravitational and viscous forces.

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Applications 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.

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Further reading

• 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.

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