# Lucas-Washburn equation

Entry by Meredith Duffy, AP225, Fall 2011

The Lucas-Washburn equation, also known simply as Washburn’s equation, describes the rate dl/dt of fluid flow through a cylindrical capillary of radius r as a function of the driving pressure. Making the assumptions that flow is laminar viscous and incompressible and that the capillary is much longer than it is wide, Washburn applies Poiseuille’s Law for the pressure drop in a fluid flowing through a cylinder to derive the following equation

where η is viscosity and ΣP is the sum of atmospheric pressure (zero if the ends of the capillary are open), hydrostatic pressure, and capillary pressure. ϵ is the coefficient of slip, taken to be zero for a fully wettable surface. Capillary pressure is

where γ is surface tension and θ is the solid-liquid contact angle. In the cases of horizontal or vertical capillaries, Washburn writes, hydrostatic and atmospheric pressure are negligible.

Washburn goes on to extend the equation to fluid flow through a porous medium by comparing it to flow through n cylindrical pores of varying radii. Again neglecting external pressures, he finds that the total pore volume V penetrated in time t is proportional to √(γt/η). Despite the assumption of cylindrical pores, he finds high accuracy of this proportionality for several liquids and materials.

## References:

[1] Washburn E.W., The Dynamics of Capillary Flow, Physical Review 1921, 17, 273.

## Keyword in References:

Imbibition in Porous Membranes of Complex Shape: Quasi-stationary Flow in Thin Rectangular Segments