Washburn's equation
Encyclopedia
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 a fully wettable capillary, it is
where
is the time for a liquid
of dynamic viscosity
and surface tension
to penetrate a distance 
into the capillary whose pore diameter is 
.
In case of a porous materials many issues have been raised both about the physical meaning of the calculated pore diameter
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.
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
of fluid in the tube 
,
one obtains
where
is the sum over the participating pressures, such as the atmospheric pressure 
, the hydrostatic pressure 
and the equivalent pressure due to capillary forces 
. 
is the viscosity
of the liquid, and
is the coefficient of slip, which is assumed to be 0 for wetting
materials.
is the radius of the capillary. The pressures in turn
can be written as
where
is the density of the liquid and 
its surface tension
.
is the angle of the tube with respect to the horizontal axis. 
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
:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Washburn's equation describes capillary flow
Capillary action
Capillary action, or capilarity, is the ability of a liquid to flow against gravity where liquid spontanously rise in a narrow space such as between the hair of a paint-brush, in a thin tube, or in porous material such as paper or in some non-porous material such as liquified carbon fiber, or in a...
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 a fully wettable capillary, it is
where
is the time for a liquidLiquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
of dynamic viscosity
and surface tensionSurface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
to penetrate a distance into the capillary whose pore diameter is .In case of a porous materials many issues have been raised both about the physical meaning of the calculated pore diameter
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
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
, but according to physicist Len Fisher can be extremely accurate for more complex materials including biscuits (see dunk (biscuit)
Dunk (biscuit)
To dunk is to dip a biscuit , bread, buttered toast, cake, or doughnut into a beverage, especially tea, coffee, or cold milk. Dunking releases more flavour from confections by dissolving the sugars, while also softening their texture...
). 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
of fluid in the tube ,one obtains
where
is the sum over the participating pressures, such as the atmospheric pressure , the hydrostatic pressure and the equivalent pressure due to capillary forces . is the viscosityViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
of the liquid, and
is the coefficient of slip, which is assumed to be 0 for wettingWetting
Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...
materials.
is the radius of the capillary. The pressures in turncan be written as
where
is the density of the liquid and its surface tensionSurface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
.
is the angle of the tube with respect to the horizontal axis. is the contact angle of the liquid on the capillary material.Substituting these expressions leads to the first-order differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
for
the distance the fluid penetrates into the tube
: