Capillary flow in V-grooves
7.1 Introduction
* C h a p t e r 7 *
INFLUENCE OF BACKBONE CURVATURE ON CAPILLARY FLOW AND STABILITY IN OPEN V-GROOVE CHANNELS
analytical series solution for the streamwise velocity profile in a 2D cross-section of a fluid in a V-groove with a circular free surface. While not a full equation of motion, this result demonstrated that the friction factor coefficient 2D2h/w, where Dh is a hydraulic diameter proportional to the groove width and w is the mean streamwise velocity, increases with fluid contact angleθbut has a non-monotonic relationship with interior groove angleα. A dynamical model of viscous V-groove flow was later developed independently by Dong and Chatzis (1995), Romero and Yost (1996), and Weislogel and Lichter (1996; 1998), which described viscous liquids satisfying the Concus-Finn contact angle condition and which included the streamwise friction factor of Ayyaswamy et al. in determining a capillary-driven equation of motion. (see Chapter 4 for a review of this model).
Significant research has been devoted to the study of incompressible laminar flows in closed, curved pipes. Dean (1927) computed the equations of motion in a pipe of circular cross-section which is not straight but instead has a small constant curvature (i.e., the radius of curvature is much larger than the pipe radius) and a constant streamwise pressure gradient. Dean found that such curvature induces a secondary cross flow of magnitudeO(Redk), whereRe =ρud/µ is the Reynolds number, d is pipe diameter, and k is pipe curvature. The flow is then not unidirectional but instead forms twin helical structures within the pipe. The correction to the streamwise velocity has magnitudeO(Re2dk)and results in a given pressure gradient producing lower streamwise flux in a curved pipe than a straight pipe. This effect arises due to interaction between inertial and viscous forces. In inviscid fluids, Dean’s secondary flow does not arise (Berger et al., 1983), and in viscous fluids at zero Reynolds number (Re = 0), no cross flow whatsoever occurs and the solution is unidirectional (Chadwick, 1985; Dean, 1927). Similar results have been found in pipes with non-circular cross-sections. Rectangular pipes at nonzero Reynolds number have been shown to generate secondary flow similar to that of Dean (Norouzi and Biglari, 2013). Torsion (twisting) out of plane of the curvature in circular cross-section pipes has been considered in addition to curvature alone, and found to have a positive or negative effect on flux atO(ε2), withεbeing the ratio of pipe curvature to pipe diameter, depending on the Reynolds number (Wang, 1981b). Although no cross flow occurs in a gently curved pipe at Re = 0, the unidirectional flow solution does differ from that of a straight pipe, with the result that, at O(ε) (first order in the ratio of radius of curvature to pipe radius), the center of streamwise flow (the point of maximum flow speed) is shifted towards the interior of curves.
Furthermore, the cross-sectional flux is slightly enhanced at O(ε2) as compared to a straight pipe with identical pressure gradient (Chadwick, 1985). Analyses ofRe = 0flow in rectangular and elliptical curved pipes have shown that the center of streamwise flow shifts towards the interior of curves, as with circular pipes (Wang, 2012). However, whether flux is enhanced or suppressed as compared to straight pipes depends on the aspect ratio of the pipe cross-section.
In the analysis of the present work, inertial terms are negligible, and hence the Dean effect is also negligible. However, as in the results of Chadwick (1985) and Wang (2012), we will find that curvature of the V-groove induces a change in the flux via a modification of streamwise
flow.
Flows in closed pipes lack a free surface, limiting how directly they can be compared to V-groove flow, but free-surface flows in curved geometries have also been investigated. For example, Stokes et al. (2013) analyzed gravity-driven shallow inertial flow in helical open channels, ap- proximately0.5m in diameter. At this scale surface tension was ignored, and the change in shape of the interface was driven by inertial forces, with the result that the fluid layer was thicker on the outside of curves than on the inside. In the present work, the shape of the fluid interface will be affected by the groove curvature, but this effect will be driven by surface tension and not by inertia.
The effect of substrate curvature on capillary-driven free-surface flows has been investigated in the context of thin films coating curved substrates. A model for such curved thin film flows was first constructed by Schwartz and Weidner (1995), and a more accurate result was developed with rigorous perturbation theory by Roy et al. (2002). Rumpf and Vantzos (2013) rederived the result of Roy et al. from a different starting point by identifying the gradient flow form of the thin film equation. The direct influence of substrate curvature on fluid interface shape, and hence capillary pressure, and the existence of changes in local volume element and flux in a curvilinear coordinate system are all factors which also arise in the curved V-groove. The qualitative rule that fluid thins where substrate curvature is negative and thickens where it is positive (Roy et al., 2002) will be shown to apply to the V-groove system as well. And the general approach to developing the equations of motion by perturbative expansion in the low curvature limit is analogous to the derivation of Roy et al. (2002). However, the V-groove has a number of unique characteristics that make the thin film approach inapplicable. First, the fluid is confined between two walls and has only one long dimension, rather than being unconfined in two dimensions. Furthermore, the existence of these groove walls creates a unique fluid interface shape, which we will see has a small correction at the first perturbative order due to curvature.
This differs from the thin film on a curved substrate, in which the film is conformal to the substrate and experiences capillary pressure due to substrate curvature at zeroth order. A more detailed comparison of the two systems will be presented in Section 7.2.13.
Berthier et al. (2016) performed a preliminary experiment examining flow in open rectangular channels following a curved path, and examined the relative advance of liquid filaments in the channel corners ahead of the bulk flow. The experiment was qualitative in nature, and the curved V-grooves forming the channel corners had different lengths and complex boundary conditions, and were thus not directly comparable. That said, filaments in V-grooves with positive curvature appeared to be thicker than those in grooves with negative curvature. Based on static images in that paper, filaments on the negatively-curved groove appeared longer than those in the positively-curved groove. While strong conclusions cannot be drawn from these results, they indicate that curvature does change the behavior of groove flow, and suggest that an analysis of the problem is worth pursuing.
Wu et al. (2018) performed drop-tower microgravity experiments measuring flow in V-grooves with a curved backbone. The experiments were performed with a relatively large-scale system, having fluid film thickness of approximately 3cm and length of approximately 3-10cm, with groove radii of curvature of approximately 7-27cm. These scales placed the system in a regime in which inertial effects would be relevant and in which the aspect ratio is not slender, unlike this thesis, which considers viscous flow in slender grooves. Wu et al. found that increasing positive curvature enhanced the advance of flow in the groove (no experiments were performed with negative curvature). They further developed a semi-analytic model for flow in V-grooves with a backbone of constant curvature by adding an additional pressure contribution from the backbone, and modifying the friction factor in Weislogel’s (1996; 1998) model of viscous, straight, and slender V-grooves to include a correction polynomial in the Dean number, De = Re√
Dk, whereRe is the Reynolds number,D the “hydraulic diameter,” andk the backbone curvature.
This correction was approximated as a polynomial with coefficients derived empirically from computational fluid dynamics (CFD) simulations. Wu et al.’s semi-analytic model qualitatively captured the effect of positive curvature enhancing flow; however, the method of combining an inertia-free (Re = 0) base model with an inertial curvature correction likely limited the accuracy of their model.
In this work, a new physical model for viscous microgravity flow in V-grooves with curved backbones is developed. It is carried out to first perturbative order, at which vertical curvature is relevant but lateral curvature and torsion are not (these would enter as second order effects).
The model differs from that of Wu et al. (2018) in three key respects. First, although it is also based on Weislogel’s (1996; 1998) straight, viscous V-groove model, all curvature corrections are developed in terms of backbone curvature, rather than inertial factors such as Dean number.
Thus, for example, while the model of Wu et al. (2018) would include no flux corrections in a purely viscous (Re = 0) regime, we find through careful derivation that corrections to the flux term are in fact necessary. Second, our model does not require constant backbone curvature, and includes those additional terms which arise in grooves with varying curvature. Third, our model is derived analytically and rigorously from first principles (in a perturbative expansion, following Weislogel [1996; 1998]) and does not rely on ad-hoc addition of terms, or on CFD or empirical parameter fitting. The result is a partial differential equation which is structurally distinct from that of Wu et al. (2018).
The derivation of the model is based on the slender perturbation theory derivation of the straight V-groove equation by Weislogel (1996; 1998), as shown in Chapter 4. However, the backbone curvature is captured by using a generalized nonorthogonal curvilinear coordinate system, re- sulting in a significantly longer and more challenging derivation. We report this derivation in great detail, with the goal that a student could reproduce or build upon it.
In Section 7.2, we will derive the equations of motion of the new model. Section 7.3 will demonstrate the steady state and self-similar solutions of the model. In Section 7.4, we perform
a nonlinear stability analysis of steady state solutions and a non-normal linear stability analysis of self-similar solutions, as in Chapters 5 and 6. Section 7.5 will discuss numerical methods and caveats of the model.
7.2 Derivation of equations of motion