Capillary spreading

because we have assumed that a = 0, the governing equation, (6-31), for the shape function h becomes 

This equation must be solved subject to the constraint, (6-64), of constant volume. We should also expect the solution to satisfy the condition, (6-63), of compact support, but we shall need to return to discuss this point in more detail toward the end of this section. 

The first thing to notice about (6-77) is that it cannot be cast into the form of the nonlinear diffusion equation that was discussed in Section B. Hence, unlike the preceding case, here we cannotapply the solution ofthe diffusion equation from Subsection C.2. However, we can apply the same method of analysis in the hope of finding a self-similar form for the solution of the (/-dimensional symmetric version of (6-77). We begin by adopting the notation of (6-37) and/or (6 19) to express (6-77) in the form 

It was shown in the analysis of subsection B that the constraint of constant volume then requires that 

To this point, the analysis is identical to that of the preceding case. However, we now substitute the self-similar form for h(r,D into the governing equation, (6-78). The result is 

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Where the source is limited or ceases, capillary spreading eventually slows until further migration is limited and equilibrium is reached. This stable condition is attained when the leading edge of the laterally spreading light LNAPL fails to be... 

Another hypothesis was suggested by Kirdiyashkin et al. (1981) for the combustion synthesis systems characterized by melting of a reactant metal (e.g., Ti-C, H-B), where capillary spreading may control the combustion process (Shkiro and Borovinskaya, 1976). In these cases, it was suggested that an optimal density oc-ciffs where the volume of pores equals the volume of the molten metal. However, an analysis of the experimental data for the Ti-B system showed that this hypothesis may not be valid over the entire range of particle sizes investigated (Munir and Anselmi-Tamburini, 1989). 

Nekrasov, E. A., Maksimov, Y. M., Ziatriinov, M. K., and Shteinberg, A. S., Effect of capillary spreading on combustion-wave propagation in gas-free system. Combust. Explos. Shock Waves. 5,575 (1978). 

Darhuber A, Troian SM,ReisnerWW (2001) Dynamics of capillary spreading along hydrophilic microstripes. Phys Rev E 64(3) 031603(1-8). http // prola.aps.org/abstract/PRE/v64/i3/e031603... 

Another variant of the surface-tension-driven flow that is of potential interest to the microfluidics research community is the capillary-spreading phenomenon. Of particular interest can be the spreading of liquids on rough homogeneous substrates or on heterogeneous substrates with hydrophobic and hydrophilic stripes. For smooth homogeneous substrates, the radial advancement of a liquid of volume V has been observed by Tanner [4] as... 

By following an approach analogous to the above, Darhuber et al. [6] studied the dynamics of capillary spreading along hydrophobic microstripes. The smooth surface was processed chemically to create narrow hydrophilic stripes on a hydrophobic background. The equation governing the self-similar solutions, in that case, can be described as... 

Additionally, asymmetric fluid front may occur due to capillary spreading along the mold wall containing the inlet which is contacted first by the filling fluid. 

Similar to measuring the permeability and the anisotropy of liquid transport in the nonwoven fabric plane, the image analysis method was used to track the in-plane radial liquid advancing front to determine the rate of capillary spreading in a 2D fabric plane. Kawase et used a simple video camera to determine the capillary... 

T. Kawase, Y. Morimoto, T. Fujii, M. Minagawa, Spreading of liquids in textile assemblies, 1. Spreading of liquids in textile assemblies. I. Capillary spreading of liquids. Text Res J 56 (7) (1986) 409-414. 

Let be a new variable, which is a combination of r and f as in the previous case of capillary spreading h(t,r) should depend on this new single variable. Such solution is referred to as a similarity solution. 

FIGURE 3.4 Time evolution of the radius of spreading in log-log coordinate system (3). (1) capillary spreading (2) gravitational regime of spreading. 

In the earlier part of this section we showed that pure capillary spreading results in inconsistency of the mathematical treatment in the vicinity of the apparent moving contact line of the spreading droplet. This inconsistency is usually referred to as a singularity at the three-phase contact line. As we already mentioned in the Introduction to this chapter, this inconsistency is the result of... 

The time, values relevant for our experiments are calculated below. Let us estimate the Reynolds number during the capillary spreading stage. Equation 4.39 gives the velocity of spreading, which should be used for calculation of the Reynolds number. Simple rearrangement gives... 

The first stage is very short. Here, the capillary spreading prevails over the drop base shrinkage caused by the liquid imbibition into the porous substrate ... 

In the present section, we extend our study in Section 4.2 to the spreading of small silicone oil drops (capillary spreading regime) over different porous substrates, whose thickness is much bigger than the drop size. A number of similarities with the case of the spreading over thin porous substrates (Section 4.2) is found. 

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