The Flow of Liquids in Surface Grooves

Abstract

We have obtained detailed capillary kinetic data for flow of a series of alcohols with various surface tension to viscosity ratios, γ/µ, spreading in open V-shaped grooves cut in Cu with three different groove angles. The location of the three-phase contact line, z, with time always follows the formula z 2) K(R,θ)-[γh0/µ]t where R is related to the included groove angle β (R ) 90 -β/2), θ is the contact angle, and h0 is the groove depth. Two theoretical models which assume Poiseuille flow and static advancing contact angles were tested against the experimental data. One is a detailed hydrodynamic model with the basic driving force resulting from the pressure drop across a curved interface. The second depends on the total interfacial energy change, independent of the shape of the liquid interface. Both agree with the experimental data. In agreement with experiment, both models predict that the rate approaches zero as R f θ, and both require R -θ > 0. Both, including a physically unrealistic approximation by a cylindrical capillary, correctly scale the experimental data. Both predict numerical values in general agreement with experiment and with each other. Differentiation between the models is possible only in the K(R,θ) term which is shown to be only weakly dependent on the range of R,θ values studied. In the threshold region where the transition occurs between filled and empty regions of the groove, the liquid height decreases linearly with distance, within experimental limitations, and forms an angle which roughly scales as the contact angle for a significant fraction of the threshold region. On the basis of the present detailed experimental data for both kinetics and threshold profile, the differences between experiment and theory and between the theoretical models are insufficient to allow a clear choice between the models.

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“…According to the results of Seemann et al [32], under these conditions, an increase in the amount of water inside the grooves by coalescence will lead to a moving water front. Principally, a moving water front inside a groove is described by the kinetics of the Washburn law [33,34]. The detailed kinetics is dependent on the orientation of the groove and on the direction of the water flow with respect to gravity [35].…”

A similarity parameter for capillary flows

“…If we assume that the flow is fully developed and quasi-steady, the velocity profile will have a form similar to that of the Hagen-Poiseuille flow of equation (2) and the flow should follow the Washburn equation (3). Setting a = h 0 and comparing equations (17) and (3), it is evident that they are identical if the pressure drop is given by the expression Several analytical solutions for K have been found [18,19], each involving a different assumption regarding the radius of curvature of the top surface of the fluid, R top (see figure 1(c)). The solution for K that is generally closest to the experimentally determined value is based on a flat topped fluid (R top = ∞) and is given by…”

“…For example, liquids have been studied flowing in thin tubes [16], in surface grooves [17][18][19][20][21] and on microstrips [22]. All of these systems have been observed to follow Washburn-type dynamics, meaning the flow dynamics are functionally similar to equation (3) (i.e.…”

“…For fluids flowing in V-shaped grooves, we use data found in [19]. The geometry that was employed in these experiments (see figure 1(c)) consisted of a groove of angle β and height h 0 being fed from a liquid reservoir.…”

“…Multiple groove angles, groove heights, and liquids were tested, with each test's properties listed in table 3. A solution for the pressure drop over the length of the groove can be determined using the results of [19]. In that work, the authors found that the governing equation for the spreading could be written as…”