**2. Lucas-Washburn theory**

The so-called wicking (or absorbency) rate is of great importance for both scientific and practical uses. The Lucas-Washburn idea provides a more scientific definition of the wicking rate. This theory deals with the rate of a liquid drawn into a circular tube via capillary action. A capillary like this is a severely simplified representation of a pore in a true fibrous media with a complex structure. For laminar viscous flows, theory is a specific form of the Hagen-Poiseuille law state Landau and Lifshitz. According to this law, the volume dV of a Newtonian liquid with viscosity μ that wets through a tube of radius r and length h during dt is given by the relation:

$$\text{dV/dt} = \pi \text{r} \mathbf{4} \ (\mathbf{p1} \text{-p2}) / 8 \mathbf{h} \mu \text{ } \tag{1}$$

where p1 – p2 is the pressure difference between the tube ends. The capillary force and gravitation both contribute to the pressure difference here. The angle between the tube axis and the vertical direction is denoted by β, while the contact angle of the liquid against the tube wall is denoted by θ. The value of the capillary pressure p1 is:

$$\mathbf{p1} = 2\mathbf{\dot{\gamma}} \cos \theta / \mathbf{r} \tag{2}$$

While hydrostatic pressure p2 is:

$$\mathbf{p2} = \mathbf{h} \,\mathsf{J} \,\mathsf{g} \,\cos\,\mathsf{f},\tag{3}$$

#### *Absorbency and Wicking Behaviour of Natural Fibre-Based Yarn and Fabric DOI: http://dx.doi.org/10.5772/intechopen.102584*

where γ denotes the liquid surface tension, ζ is liquid density, g is the gravitational acceleration, and h, in this case, is the distance travelled by the liquid measured from the reservoir along the tube axis. This distance obviously is the function of time, h = h (t), for a given system. The Lucas-Washburn equation is obtained by substituting the values p1, p2 and h(t) into Eq. (1), expressing the liquid volume in the capillary V as πr 2 h:

$$\mathbf{dV}/\mathbf{dt} = \mathbf{r}\,\,\mathbf{\hat{s}}\,\,\cos\,\theta/4\mu\mathbf{h}\text{-}r2\zeta\,\,\mathbf{\hat{g}}\,\,\cos\,\beta/8\mu\tag{4}$$

For a given system, parameters such as r, γ, θ, ζ, g and β remain constant. We can then reduce the Lucas-Washburn equation (4) by introducing two constants,

$$\mathbf{K}' = \mathbf{r} \,\,\mathbf{y} \,\,\cos\,\theta/4\,\,\mu \,\text{and}\,\mathbf{L}' = \mathbf{r} \,\,\zeta \,\mathbf{g} \,\,\cos\,\theta/8\mu\tag{5}$$

into a simplified version,

$$\mathbf{ch}/\mathbf{dt} = \mathbf{K}'/\mathbf{h} \text{-L}'\tag{6}$$

The above relation is a non-linear ordinary differential equation that is solvable only after ignoring the parameter L<sup>0</sup> ; this has a physical interpretation when either the liquid penetration is horizontal (β = 900 C) or r is small or the rising liquid height h is low that K<sup>0</sup> /h > > L<sup>0</sup> or L<sup>0</sup> ! 0, and the effects of the gravitational field are negligible and the acceleration g vanishes. The Lucas-Washburn equation (6) could thus be solved with ease:

$$\mathbf{h} = \sqrt{2\mathbf{K}}\mathbf{'}\mathbf{t}.\tag{7}$$

The result satisfies the initial condition h = 0 for t = 0.

Despite the intricate, noncircular, non-uniform and nonparallel nature of the pore spaces, the Lucas-Washburn technique provides an approximation tool for investigating the wicking and wetting behaviour of textiles. In the field of liquid sorption in a porous area, Washburn (1921) has expressed the following equation:

$$\mathbf{H} = \mathbf{C}\mathbf{t}^{0.5} \tag{8}$$

where H is the wicking height (m); C, the capillary liquid transport constant. The capillary force causes liquid to flow through a capillary channel, as shown in the equation above. The radius of the capillary channel, the contact angle between the liquid and the capillary channel and the rheological qualities of the liquid all influence the capillary force [5].
