- Washburn's equation
In
physics , Washburn's equation describes capillary flow inporous materials.It is
:L^2=frac{gamma Dt}{4eta}
where t is the time for a
liquid ofviscosity eta andsurface tension gamma 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 physicistLen Fisher can be extremely accurate for more complex materials including biscuits (seedunk (biscuit) ). FollowingNational biscuit dunking day , some newspaper articles quoted the equation as "Fisher's equation".In his [http://prola.aps.org/abstract/PR/v17/i3/p273_1 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 forwetting materials. r is the radius of the capillary. The pressures in turncan be written as:P_h=h g ho - l g hosinpsi:P_c=frac{2gamma}{r}cosphi
where ho 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-orderdifferential equation forthe distance the fluid penetrates into the tube l::frac{delta l}{delta t}=frac{ [P_A+g ho (h-lsinpsi)+frac{2gamma}{r}cosphi] (r^4 +4 epsilon r^3)}{8 r^2 eta l}
[Edward W. Washburn. "The Dynamics of Capillary Flow" (1921). Physical Review, volume 17, issue 3, p. 273 - 283. ]
References
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