Capillary flow in V-grooves
4.4 Discussion .1 Limitations.1Limitations
4.4.2 Rounded V-grooves (U-grooves)
Manufacturing V-grooves with perfectly sharp corners is challenging at the microscale, one of the reasons for which variations on the V-groove geometry have been investigated. Chen, Weis- logel, and Nardin (2006) generalized the model to V-shaped grooves with rounded bottoms (assumed to be circular sections) instead of sharp corners, which we will refer to as “U-grooves.”
Above the rounded bottom, the upper portion of each groove was assumed to have straight walls like the and V-groove. It was found that the governing equations remained similar to the V-groove equation, but included some nonlinear terms which could not be expressed in closed form (although the authors did construct an approximate closed form solution contain- ing only polynomial fractions). However, the authors identified several limiting cases, such as nearly-rectangular grooves and highly rounded “crescent-like” grooves, in which the model could be approximated with a simple polynomial expression. In these cases, the model remained a nonlinear diffusion equation similar to the V-groove equation, but with a different exponent.
Tang et al. (2015) considered asymmetrical V-grooves, with a sharp corner but with one wall slightly curved. This model also contained a nonlinear term with no closed form expression, and solutions were computed numerically.
These models are derived from the same ∂A/∂T = −∂Q/∂Z equation as the V-groove, but the dependence ofA andQonH are different. Most importantly, the cross-sectional shape of the fluid changes with H, unlike the V-groove, where the cross-sectional shape simply scales to larger or smaller sizes as H varies. This means that the cross-sectional area A is no longer quadratic in H, but instead follows some complicated (though still closed-form-expressible) nonlinear function ofH. The streamwise flux, however, must be computed numerically for each different cross-sectional shape, and hence leads to an inexpressible nonlinear function.
Despite the complexity of these models, their dynamics remain qualitatively similar to those of the V-groove. In particular, both exhibit self-similar solutions with the same t1/2 spreading behavior as the V-groove. The spreading profile in the U-grooves of Chen et al. (2006) appears steeper than that of the V-groove, resembling the terminating solutions 5 and 6 in Figure 4.7.
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* C h a p t e r 5 *
STABILITY OF V-GROOVE FLOW
Note: portions of this chapter are adapted from published work (White and Troian, 2019).