Washburn's equation

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In physics, Washburn's equation describes capillary flow in porous materials.

It is

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

where t is the time for a liquid of viscosity η and surface tension γ to penetrate a distance L into a fully wettable, porous material whose average pore diameter is D.

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.

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 = πr2dl, 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 PA, the hydrostatic pressure Ph and the equivalent pressure due to capillary forces Pc. η is the viscosity of the liquid, and ε 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

Ph = hgρ − lgρsinψ
P_c=\frac{2\gamma}{r}\cos\phi

where ρ is the density of the liquid and γ its surface tension. ψ is the angle of the tube with respect to the horizontal axis. φ 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}

The solutions of this differential equation are also discussed in this paper.

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