Capillary flow in V-grooves
4.1 Background and motivation
With modern day advances in microfabrication techniques, capillary action is being used as a reliable form of flow control in numerous microfluidic devices as well, some involving capillary action through porous substrates like paper or polymer films, and others relying on a combination of capillary action, positive displacement pumping and electrophoresis. The number of such applications is multiplying rapidly with emphasis on disposable inexpensive platforms beneficial to global public health (Yager et al., 2006), drug discovery screening (Dittrich and Manz, 2006) and specialized fluid based logic circuitry (Prakash and Gershenfeld, 2007; Thorsen et al., 2002).
While flow in enclosed capillaries is the most well-known capillary action, wetting liquids will in fact rapidly and spontaneously creep along open surfaces containing grooves, interior corners, crevices, or roughened areas, a process known as wicking. Since the late 1960s, researchers have been incorporating this passive and reliable method of flow control in the design of novel propellant management devices able to store, channel, and meter fuel in microgravity envi- ronments (Hartwig, 2016, 2017; Jaekle, 1991; Levine et al., 2015; Rollins et al., 1985). Such systems have significantly extended mission lifetimes of spacecraft and satellites, enabling future interplanetary explorations as well. Modern propellant management systems consist of combi- nations of sponges, traps, troughs, vanes, and wicks to channel propellant flow by capillary action, systems which have been investigated extensively (Collicott and Chen, 2010; Darr et al., 2017; Jaekle, 1991, 1997; Weislogel, 2001; Weislogel et al., 2002). Shown in Figure 4.1 are two common structures designed in such a way that the liquid film thickness is much smaller than the streamwise flow distance, a limiting ratio which leads to considerable simplification of the governing equations of motion. However, even though the mechanism describing internal capillary flow, whereby a liquid column spontaneously fills the interior of a slender capillary tube, was well-understood by the 1920s, the mechanism driving spontaneous capillary flow along an open grooved channel required a half century more to be deduced.
Ongoing efforts to miniaturize fluid management systems for many different applications con- tinue to drive interest in the fundamentals of free-surface capillary flow along structured sub- strates. Besides the aforementioned propellant management applications, open capillary grooves are now being used for heat pipes to cool microelectronics, as open flows in each corner of a triangular cross-section heat pipe provide a large surface area from which to evaporate (Mallik et al., 1992). Open channels are also being explored for biomedical lab-on-a-chip devices. Ex- periments by Berthier et al. (2015) showed that even viscous fluids such as blood and alginate
(a)
(c) (d)
(b)
Figure 4.1: Two types of propellant management devices (PMDs) often found in satellite fuel tanks and used for passively routing propellant within open grooved channels. (a) Sketch and (b) fabricated structure showing vane-type PMDs. (c) Sketch and (d) fabricated structure showing sponge type PMDs (see Jaekle, 2011).
solutions are quickly wicked into V-shaped grooves. An overview of open-channel microfluidic biomedical devices may be found in the review article by Oliveira et al. (2019).
This chapter will first briefly review flow in closed capillaries, and then cover the derivation of the low-order V-groove flow equation developed by Weislogel (1996) and Romero and Yost (1996).
Subsequent chapters will introduce new work: the first comprehensive stability analysis of flow in straight V-grooves (Chapter 5), the behavior of conducting liquids in V-grooves under the influence of an external electric field (Chapter 6), and the behavior of liquids in V-grooves with curved backbones (Chapter 7).
4.1.1 Closed capillaries: Static equilibrium
The principle of capillary action in enclosed tubes has been well-known for centuries (Maxwell and Strutt, 1911), and is familiar to many of us from the classic drinking straw experiment. In that experiment, a straw of radius r is placed vertically in a large (radius ≫ r) container of water (see Figure 4.2).
In static equilibrium, the Navier-Stokes equations reduce to∇p=ρ⃗g,⃗g being the gravitational acceleration vector. Taking gravity to operate along the z-axis,p=pext−ρgz, wherez= 0 at the fluid interface of the container,gis the magnitude of gravitational acceleration andpextis the pressure of the surrounding vacuum or air; this is the well-known hydrostatic pressure (Landau and Lifshitz, 1987). (Since ρair ≪ ρwater, hydrostatic variations in pressure in the air can be neglected). Taking the fluid interface of the container (away from the straw) to have negligible mean curvature, the normal stress balance Equation (2.39) is simplypext−(pext−ρgz)|z=0 = 0.
z s
r
y z x
Figure 4.2: A closed capillary tube (or drinking straw) of radius r in a reservoir. zs measures the height of the fluid meniscus in the tube from the reservoir surface.
If the straw is wetting, i.e., is of a material that forms a contact angle of θ < 90◦ between water and air, then a concave interface will be formed. In static equilibrium, this interface conforms approximately to the surface of a sphere of radiusrsecθ, and hence has mean curvature κm ≈cosθ/r. The interface balance Equation (2.39) then implies pext−(pext−ρgz)|z=zg = 2κmγ ≈2γcosθ/r, wherezg is the height of the fluid in the straw. Hence,
zg= 2γcosθ
rρg . (4.1)
Note that details of how to measure the height of the fluid and the non-spherical surface curvature due to gravity were ignored; as we implicitly assumed r ≪ zg, or r ≪ pγ/(ρg), called the capillary length (Probstein, 1994). For water, which has ρwater = 103 kg/m3 and surface tensionγ ≈0.072N/m (Dean, 1999), the capillary length comes out to approximately 0.27 cm. For straws with larger radii, a more careful analysis would be necessary to compute an accurate height.
4.1.2 Closed capillaries: dynamics
In order to achieve the static equilibrium just described, fluid motion must occur. And, in the limit of zero gravity (more specifically, in the limit of zero Bond number, Bo = ρgr2/γ → 0), there is no steady state; the fluid will rise indefinitely.
Washburn (1921) appears to have been the first to derive the dynamics of capillary-driven flow of a Newtonian liquid in an enclosed tube. Neglecting inertia and gravity and assuming
unidirectional flow, the Navier-Stokes equations reduce to
0 =∂zw, (4.2a)
0 =∂xp=∂yp, (4.2b)
0 =−∂zp+µ(∂xx+∂yy)w, (4.2c)
wherew is velocity in thez direction, with boundary conditions
w|x2+y2=r2 = 0, (4.3a)
⟨w⟩z=zs =∂tzs, (4.3b)
where zs is the position of the fluid meniscus in z (the height of the fluid in the straw; see Figure 4.2) and ⟨w⟩ is the average velocity across the tube.1 The solution in the bulk is given by
w= (2/r2)(r2−x2−y2)∂tzs, (4.4a)
∂zp= −2γcosθ/r
zs ; (4.4b)
the velocitywhas the parabolic cross-section typical of Poiseuille flow and the constant pressure gradient is set by the capillary pressure divided by the fluid height. Balancing the momentum equation, Equation (4.2c), yields 2γcosθ/(rzs) =µ(8/r2)∂tzs, and hence
zs=
srcosθ 2
γ µ
t, (4.5)
a result now known as the Washburn relation. Washburn broadened the application of his relation by modeling porous media as collections of intertwined tubes and predicting that fluid fronts in such media would also be proportional toptγ/µ. Indeed, the Washburn scaling has since been successful in describing a variety of capillary transport phenomena, ranging from blood flow in microvascular hemodynamics (Jones, 1969; Skalak et al., 1989) to oil extraction from porous rocks (Wooding and Morel-Seytoux, 1976) to capillary water uptake in wood (Johansson and Kifetew, 2010), among others.
When gravity is included, the moving front no longer advances with the Washburn relation but instead as
zs=zg
h1 +W−e−1−(gr2ρt)/(8zgµ)i, (4.6) where zg is the asymptotic height from Equation (4.1) and W is the Lambert W function, or product log, defined byx= exp[W(x)]W(x)(Probstein, 1994). The time scale is thus revealed
1Note that imposing the no-slip boundary condition,w|x2+y2=r2 = 0, like this implies that the fluid cannot actually wet the walls of the tube! In reality, both the unidirectional flow and no-slip wall assumptions are simplifications, but as long as the length of the fluid in the tube is much greater than the radius of the tube (i.e., zs≫r), these issues can be ignored. For more information on the problem of the no-slip boundary condition and the advance of a fluid’s leading edge on a solid, see Cox (1986).
to betg = 8zgµ/(ρgr2) = 8γcosθµ/(ρg2r3). At early timest≪tg, the result can be expanded as
zs=
srcosθ 2
γ µ
t− ρgr2
12µt+O(t3/2), (4.7)
recovering the gravity-free solution at lowest order, while at late timest≫tg it asymptotically approaches zg as
zs=zgh1−e−1−(gr2ρt)/(8zgµ)i+Oe−2(gr2ρt)/(8zgµ). (4.8) 4.1.3 Open capillary flow
Two features of closed capillary flow stand out in the prior derivations. First, a threshold pressure pthreshold =−2γcosθ/r is required to push fluid into the capillary. Such a threshold pressure was achievable with a large fluid reservoir, but would make transfer of fluid from small capillaries to larger ones impractical. Second, when gravity acts parallel to the capillary, the height to which fluid can be transported is limited. This second issue could be avoided by means of tapering capillaries, but the threshold pressure requirement at the base would remain.
A simpler solution, both conceptually and from a manufacturing standpoint, is the use of open V-grooves (see Figure 4.3a). As shall be seen shortly, open V-grooves allow a fluid to reach effectively arbitrarily small radii of curvature, and hence arbitrarily low pressure, eliminating both the threshold pressure and the limit on flow height against gravity. Indeed, such grooves were proposed in by Concus and Finn (1969) to explain how trees can carry water to their topmost leaves despite the fact that their xylem conduits are too wide to allow closed capillary flow to great heights.
The theoretical study of flow in V-grooves appears to begin with Concus and Finn (1969), who studied the equilibrium shape of free fluid surfaces in sharp wedges. They found the contact angle conditions necessary for a fluid to fill a V-groove, and also discovered that the equilibrium surfaces are unbounded even in the presence of gravity, implying that fluid with with a contact angle below the requisite value can flow arbitrarily far into a V-groove, even against gravity. Ayyaswamy et al. (1974) then determined a semi-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 2Dh2/w, where Dh is a hydraulic diameter proportional to the groove width and w is the mean streamwise velocity, increases with fluid contact angle θ but has a non- monotonic relationship with interior groove half angle α. In the mid-1990s, Weislogel (1996), Romero and Yost (1996), and Dong and Chatzis (1995) independently developed a low-order, one-dimensional partial differential equation to describe the flow and found self-similar solutions in which the fluid advances as t1/2, incidentally the same power as the Washburn relation for closed capillaries. This differential equation was first order in time and second order in z, the axial variable, describing the height of the fluid in the groove as a nonlinear diffusion process.
α y
x h(z,t)
q
(b)
d
L
y x z
(a)
Figure 4.3: Diagrams of V-groove flow system. (a) Schematic of a wetting liquid film (0 ≤ θ < π/2) flowing within a slender open triangular groove with constant cross-section. The inlet midline film thickness at the origin is denoted by d= h(x = 0, z = 0, t) (note that we follow Weislogel’s convention of measuring the midline film thickness, rather than Romero and Yost’s convention of the film thickness at the wall) and the channel length byL whered/L≪1. (b) Cross-sectional view of the flow geometry depicted in (a) whereh(z, t)denotes the local midline film thickness,αis the groove internal half angle (note that we follow Weislogel’s convention of αbeing the internal groove half angle, rather than Romero and Yost’s convention ofαbeing the exterior groove angle), rc is the radius of curvature of the liquid interface andθ is the contact angle of the liquid wetting the channel sidewalls, which is assumed constant.
All three arrived at the same resulting equation by throwing out convective terms (i.e., assuming purely viscous flow), and making the key assumption that the groove is much longer than it is wide or deep, thus ignoring axial surface curvature. Romero and Yost (1996) explicitly assumed unidirectional flow; although this was an oversimplification, it provided a shortcut to the end result. Dong and Chatzis (1995) implicitly made a similar shortcut. Weislogel (1996) carried out a formal perturbation analysis, setting the model on firm theoretical footing.
Experimental studies of V-grooves have been performed with various fluids, including solder (Rye et al., 1996, 1998), silicone oil (Weislogel, 1996), methanol (Chen et al., 2009), acetone (Deng et al., 2014), and blood (Berthier et al., 2015), confirming thet1/2 spreading rate predicted by earlier theoretical studies.