Washburn's equation

In physics, the Washburn's equation describes capillary flow in a bundle of parallel cylindrical tubes; it is extended with some issues also to imbibition into porous materials. The equation is named after Edward Wight Washburn; also known as Lucas–Washburn equation, considering that Richard Lucas wrote a similar paper three years earlier, or the Bell-Cameron-Lucas-Washburn equation, considering J.M. Bell and F.K. Cameron's discovery of the form of the equation fifteen years earlier.

In case of a fully wettable capillary, it is

\[ L^2=\frac{\gamma Dt}{4\eta}\]

where \(t\) is the time for a liquid of dynamic viscosity \(\eta\) and surface tension \(\gamma\) to penetrate a distance \(L\) into the capillary whose pore diameter is \(D\). In case of a porous materials many issues have been raised both about the physical meaning of the calculated pore diameter \(D\) and the real possibility to use this equation for the calculation of the contact angle of the solid.

The equation is derived for capillary flow in a cylindrical tube in the absence of a gravitational field, but according to physicist Len Fisher can be extremely accurate for more complex materials including biscuits (see dunk (biscuit)). Following National biscuit dunking day, some newspaper articles quoted the equation as Fisher's equation.^{[citation needed]}

In his paper from 1921 Washburn applies Poiseuille's law for fluid motion in a circular tube. Inserting the expression for the differential volume in terms of the length \(l\) of fluid in the tube \(dV=\pi r^2 dl\), one obtains

\[\frac{\delta l}{\delta t}=\frac{\sum P}{8 r^2 \eta l}(r^4 +4 \epsilon r^3)\]

where \(\sum P\) is the sum over the participating pressures, such as the atmospheric pressure \(P_A\), the hydrostatic pressure \(P_h\) and the equivalent pressure due to capillary forces \(P_c\). \(\eta\) is the viscosity of the liquid, and \(\epsilon\) is the coefficient of slip, which is assumed to be 0 for wetting materials. \(r\) is the radius of the capillary. The pressures in turn can be written as

\[P_h=h g \rho - l g \rho\sin\psi\] \[P_c=\frac{2\gamma}{r}\cos\phi\]

where \(\rho\) is the density of the liquid and \(\gamma\) its surface tension. \(\psi\) is the angle of the tube with respect to the horizontal axis. \(\phi\) is the contact angle of the liquid on the capillary material. Substituting these expressions leads to the first-order differential equation for the distance the fluid penetrates into the tube \(l\):

\[\frac{\delta l}{\delta t}=\frac{[P_A+g \rho (h-l\sin\psi)+\frac{2\gamma}{r}\cos\phi](r^4 +4 \epsilon r^3)}{8 r^2 \eta l}\]

References

de:Washburn-Gleichung