Lucas-Washburn

Post-printing nip capillary sorption of ink and ink vehicles is discussed using Lucas-Washburn theory and the influence of the rate of capillary sorption on ink holdout, show through and set off are discussed. Finally, the long-term migration of oil vehicles over fibre surfaces by spreading with the attendant loss of paper opacity is described. 

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 this chapter, we first investigate the conditions for spontaneous capillary flow in open or confined microchannels, composite or not, and we show that a generalized Cassie angle governs the onset of SCF [6]. Then we present the dynamics of the capillary flow with a generalized Lucas-Washburn-Rideal expression for the flow velocity and travel distance [7-9]. Finally, we focus on the particular effect of precursor capillary filaments— sometimes called Concus-Finn filaments [10,11]—that sometimes exist in sharp corners, depending on the wettability of the walls. 

The Lucas-Washburn-Rideal (LWR) Law for Confined Cylindrical Channels... 

The study of the dynamics of capillary flows started as early as the 1910s with the developments made by Bell, Cameron, Lucas, Washburn and Rideal... 

In this section, the general expression for the determination of the velocities of spontaneous capillary flows in composite, confined microchannels of arbitrary shapes is presented. This expression generalizes the conventional Lucas-Washburn-Rideal model, which is valid for cylindrical channels. It will be shown that the use of an equivalent hydraulic diameter in the Lucas-Washburn-Rideal model introduces a bias when the shape of the channel cross section differs notably from a circle. 

The travel distance varies as the square root of the time, in agreement with the Lucas-Washburn-Rideal (LWR) model for capillary flows inside cylinders [7-9]. The liquid velocity can be readily derived from (1.39)... 

Consider first the case of confined cylindrical channels. This was the first configuration studied. Lucas, Washburn and Rideal gave a non-inertial solution as early as 1921 [7-9]. In (1.46), the value of A can be determined by considering the value of the hydraulic resistance per unit length =S/Ijthe wetted perimeter = 2 nR, and the cross-sectional surface area and using (1.45). One obtains A = Rf 4. After substitut-... 

The dynamics of capillary flow in confined channels, is now well known in the case of composite walls. In the case of a cylindrical channel, the Lucas-Washburn-Rideal (LWR) expression produces the relation between the travel distance or velocity with the time. But using an equivalent hydraulic diameter for describing the motion in an arbitrary cross section channel introduces a bias, which is important if the cross-section differs notably from the circle. An expression generalizing the LWR expression has been derived, which details the dynamics of the capillary flow under the condition that inertia is negligible, which is often the case at the microscale. In such a case, the advancing contact angle has been found to be constant, and close to the static value. 

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). 

Assiuning that the Lucas-Washburn law works for the nanowebs, the velocity of front propagation (dL/dt) is expressed as ... 

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