Washburn-Lucas equation

In 1923, Bosanquet pointed out that in Lucas—Washburn equation the velocity goes to infinity for t = 0. This physically impossible situation was a result of neglecting inertial effects. He described the motion of the meniscus with the following differential equation (obtained from the one-dimensional Navier-Stokes equations)... 

Fig. 10. Variation of the liquid height with time for pentane in a capillary tube (1) experimental points (2) Lucas—Washburn equation taking into account the hydrostatic pressure (3) Lucas-Washburn equation neglecting the hydrostatic pressure. From Ref. 91.

NOTE g/d=grams per denier L-W=Lucas-Washburn Equations PET=poly(ethyleneterphalate). 

Gillespie (8) on the other hand developed an equation of the Lucas-Washburn type without specific reference to an explicit pore model on the basis of D Arcy s law (6). Assuming that AP was constant Gillespie derived the following equation for two dimensional radial spreading of a liquid drop... 

In 1917, Lucas determined the position of the front of liquid being wicked by a porous material as a function of time. He found that it propagates as a behavior which was previously reported by Bell and Cameron. A few years later, in 1921, Washburn explained Lucas observations assuming (a) that the porous material can be seen as a system of parallel capillaries (b) that the flow in each capillary tube is stationary and axisymmetric and (c) that the flow is well described by a Poiseuille profile with the ptessure difference across the interface given by the Laplace equation. Under these conditions, the position of the meniscus inside a capillary tube varies with time as... 

Poiseulle s equation states that the flow rate in a tube is inversely related to the distance of the liquid movement. Based on Poiseulle s equation, Lucas [40] and Washburn [41] developed an equation for flow rates in capillaries ... 

The limitations of the Washbum-Lucas equation are frequently overlooked. The equation assumes incorrectly a constant advancing contact angle 6U for the moving meniscus [42,43]. The Washburn-Lucas equation (19) does not take in account the inertia of the flow [25], and implies that at time t = 0 and / = 0, the flow rate is infinite. In spite of these limitations, a variety of liquids have obeyed the Washbum-Lucas wicking kinetics [44]. Other forms of the Washbum-Lucas equation have been suggested [45-51]. 

---