Capillary flow in V-grooves
7.2 Derivation of equations of motion .1 Assumptions and method of derivation.1Assumptions and method of derivation
7.2.13 Final reduced equation Putting all the pieces together,
7.2.13 Final reduced equation
groove with positive curvature (KΥ >0) will have lower pressure than one with negative cur- vature (KΥ <0). Therefore, a positive gradient in curvature (∂SKΥ >0) induces a negative gradient in pressure (∂SP < 0), and thus a positive contribution to the flux, Q. Comparing PbΥ/Pb0to the other geometric quantitiesAbΥ/Ab0andΓΥ/Γ0in Figure 7.8, it is immediately clear that this correction has the greatest magnitude for wide grooves; in particular, for grooves with α≳27◦. Thus, this term is often the most important correction for grooves with non-constant curvature.
The last correction is the ΓΥ/Γ0 term in the flux on the right-hand side of Equation (7.101).
This term combines the effects of the modified interface shape and the curvilinear coordinates on the flux Poisson equation. Note from Figure 7.8 (c) thatΓΥ/Γ0 is typicallyO(0.1)to O(1), and is positive for most grooves but changes sign and becomes negative for very wide grooves (α ≳ 72◦). Thus, for narrow grooves, positive backbone curvature enhances flux for a given H and ∂SH, while negative curvature suppresses it. For wide grooves, this is reversed (keep in mind that this result is predicated on fixed H and ∂SH, and does not describe nonlocal comparisons or pressure boundary conditions).
In the case of constant curvatureKΥ, the system follows the rule that pressure is high whenH is high and pressure is low whenH is low, so that fluid will flow from highH to low H. When KΥ varies inS, however, pressure is not necessarily monotonic in fluid thickness.
UnlikeRe = 0flow in circular, rectangular, or elliptical pipes with constant curvature (Chadwick, 1985; Wang, 2012), the flux correction (ΓΥ/Γ0 term) arises in the V-groove atO(ε) and thus scales linearly with the curvature. This occurs because of the asymmetry of the V-groove cross section (as noted earlier, corrections due to torsion or lateral curvature, KΞ, arise at second order due to the lateral symmetry of the groove). Furthermore, the pressure correction arising in the V-groove, PbΥ/Pb0, is entirely absent from closed-pipe analyses because there is no free surface to affect capillary pressure.
Comparison to thin film equation
The curvature corrections arising in Equation (7.101) are qualitatively similar to those in the equation for a thin film coating a curved substrate, first developed by Roy et al. (2002). Because the capillary pressure arises rather differently in the two cases (as the thin film equation has no bounding V-groove walls restricting its interface shape), it is easiest to compare the equations of motion without substituting in the capillary pressure. For the curved V-groove, Equation (7.101) in terms of a general pressure is
∂
∂T H2−εKΥH3AbΥ
Ab0
!
= (const.)×∂S
H4
1 +εH
ΓΥ Γ0KΥ
∂SP
+O(ε2), (7.102) with capillary pressure being
P =− 1 Ca
Pb0
H +εKfΥPbΥ
!
+O(ε2). (7.103)
Roy et al.’s (2002) equation of motion for a thin film on a curved substrate (in the form developed in Chapter 8) can be expressed as
∂
∂T
H−εKmH2=∇α (H3
3
egαβ +εH 1
2IIαβ−2Kmegαβ
∇βP )
+O(ε2), (7.104) with capillary pressure
P =− 1 Ca
n2Km+εh∇α∇αH+H4Km2−2KGio+O(ε2), (7.105) whereH is the fluid thickness normal to the substrate, egαβ is the substrate metric tensor, Km is the substrate mean curvature, KG is the substrate Gaussian curvature, IIαβ is the substrate shape tensor (second fundamental form), and ∇represents the covariant derivative.
In both equations, the time derivative of a volume element is balanced by the divergence of a flux driven by a pressure gradient. The curvature induces a negative correction to the volume element, although the thin film equation lacks the additional contribution due to change in interface shape that arises in the curved V-groove.
Looking at the right-hand side of each equation, in each case the curvature induces a correction to the flux in the form of a multiplier on the pressure gradient, arising from the modified flux integral. In the case of the V-groove, this integral is computed in a confined 2-D cross-section and has contributions both from the modified Poisson equation and from the modified interface shape. In the case of the thin film on the curved substrate, the integral is computed in 1-D but includes cross flow in two-dimensions, making the correction a two dimensional tensor instead of a scalar. In both the curved V-groove and the thin film on a curved substrate, this flux correction can be positive or negative depending on various parameters.
Both equations have a complicated capillary pressure correction which includes a term pro- portional to the negative mean curvature of the groove or substrate [the −εKΥPbΥ term in Equation (7.103) and the −2Km term in Equation (7.105)]. This results in the general rule of thumb that, for a given pressure, fluid in a positively curved V-groove or on a substrate with positive mean curvature will be thicker, and fluid in a negatively curved V-groove or on a substrate with negative mean curvature will be thinner.
Despite this rule of thumb, the differences between the pressure expressions in the two systems lead to substantial differences in behavior. The V-groove pressure is dominated by the effect of the groove walls, inducing the H−1 term, while the groove curvature is an O(ε) correction.
Thus, the thinning or thickening effect is tempered by the O(1) pressure term. In particular, groove curvature cannot cause the fluid to thin to the point of breaking, becauseH−1 diverges as H →0. But in the thin film, substrate curvature is theO(1) effect and the fluid interface thickness variation is the O(ε)correction. The substrate curvature is thus the dominant effect, and the fluid can become arbitrarily thin in regions of negative mean curvature.