Capillary flow in V-grooves
7.2 Derivation of equations of motion .1 Assumptions and method of derivation.1Assumptions and method of derivation
7.2.2 Coordinates and curvature Backbone coordinates
The line following the interior corner, or backbone, of an arbitrarily curved V-groove can be described as a space curve inR3 (Figures 7.2 and 7.3). A local coordinate system will first be set up on this curve. This coordinate system will be described in two stages: first, a coordinate triad will be constructed on the backbone; next, these coordinates will be extended to a region around the backbone. This is a common approach for constructing curvilinear coordinate systems around long narrow objects such as rods (Rubin, 2000), and has been used in fluid mechanics to describe flow in curving and twisting pipes (Tuttle, 1990; Wang, 1981b).
Let the backbone be described by a space curveβ(s), where⃗ sis an arc-length parametrization.
The backbone has a local tangent which makes an obvious choice for one leg of a local coordinate system,
sˆ= ∂ ⃗β
∂s. (7.1)
Note that because sis an arc-length parametrization ofβ,⃗ sˆis a unit vector.
The classical choice of normal and binormal vectors is the Frenet-Serret frame, which is based on the local curvature. This approach was employed, for example, by Wang (1981b) in an analysis of flow in helical pipes with circular cross-sections. The classical theory of space curves defines
x z
y
υ s ξ
Figure 7.2: The corner, or backbone of the V-groove is described by a space curve, with local coordinates {ξ,ˆ υ,ˆ ˆs}. Unlike global Cartesian coordinates ({x, y, z} in the diagram), local coordinates follow the space curve; ˆsis tangent to the curve, whileξˆandυˆare orthogonal toˆs.
total curvature and torsion, which are invariant quantities of the space curve, i.e., quantities which do not depend on coordinate systems. The curvature describes the bending of the curve, or the change in local tangent vectors. Torsion describes rotation or twisting of the normal andˆ binormal abouts, describing how curves twist out of their local plane. These quantities may beˆ constant or vary in s.
But the V-groove lacks the rotational symmetry of a circular pipe, and thus a more natural choice of coordinates exists, namely, settingυˆto be the normal vector which points “up” in the V, and then defining ξˆto be ˆs×υ. This way, the V-groove cross section will always look theˆ same in theξ-ˆυˆplane at constant s. Note that this set of coordinate vectors is an orthonormal triad on the backbone curve (Figure 7.2). Furthermore, rather than use invariant curvature quantities defined in terms of an unoriented space curve, it becomes natural to use kυ, the curvature projected onto the υ-ˆˆ s plane,kξ, the curvature projected onto the ξ-ˆˆs plane, and ȷ, the torsion of theυ vector. Mathematically,
kξ=−ˆs·∂sξ,ˆ (7.2a)
kυ =−ˆs·∂sυ,ˆ (7.2b)
ȷ= ˆυ·∂sξˆ=−ξˆ·∂sυ.ˆ (7.2c) kξ represents lateral curvature, i.e., curvature of the backbone in the ξ-ˆˆsplane. kυ represents vertical curvature, i.e., curvature of the backbone in the υ-ˆˆ s plane. ȷ describes the torsion or twisting of the V-groove orientation on the backbone curve. Figure 7.3 contains visualizations of kξ,kυ, andȷ.
u x s ux
s u x
s u x
s
k >0
xk <0
x(a) (b)
u x
s u x
s u x
s u xs
k >0
uk <0
u(c) (d)
ux s u
x s
u x s
ux s
ȷ <0 ȷ >0
(e) (f)
Figure 7.3: Diagrams distinguishing V-groove backbone curvature. (a)-(b): V-grooves with laterally curved backbones. (a) Lateral curvature kξ >0; (b) kξ <0.
(c)-(d): V-grooves with vertically curved backbones. (c) Vertical curvaturekυ >0; (d)kυ <0.
(e)-(f): V-grooves with torsion. (e) Torsionȷ >0; (f)ȷ <0.
Note that thescoordinate follows the backbone (the corner) of the groove, and theυcoordinate (Υ when nondimensionalized) always points “up” from the corner of the V-groove towards the opening of the groove, and the ξ coordinate is orthogonal to the υ coordinate in the cross- sectional plane.
Extended coordinates
Extending the triad off of the backbone in the most natural way leads to the coordinate system definition
⃗
x(ξ, υ, s) =β⃗(s) +ξξ(s) +ˆ υυ(s).ˆ (7.3) Whileξ,ˆ υ, andˆ sˆwere previously defined only on the backbone, a new coordinate triad can now be defined that exists in a region near the backbone. Without normalizing, these vectors are
∂ξ⃗x= ˆξ, (7.4a)
∂υ⃗x= ˆυ, (7.4b)
∂s⃗x= ˆs+ξ∂sξˆ+υ∂sυ.ˆ (7.4c) Note thatξˆandυˆstill form two legs of the triad, but∂s⃗ξis equal toˆsonly whenξ=υ= 0, i.e., on the backbone. Therefore, what was an orthonormal coordinate system on the backbone is a nonorthogonal coordinate system elsewhere. However, it will be seen that, to the perturbation order considered in this paper, the nonorthogonality can be ignored.
With these coordinates, a diagram of the curved V-groove can be seen in Figure 7.4. Just as in the straight groove, dis the fluid interface center height (fluid thickness) at the inlet of the groove, L is the characteristic length scale of the groove, α is the half angle of the groove corner, andθ is the contact angle of the groove with the wall. Now, however, υ is the vertical coordinate,ξ is the transverse coordinate, and sis the axial coordinate of the groove.
These coordinates have the convenient property that the scale factor is 1 in ξ and υ. That is, 1 unit of measurement inξ or υ corresponds to 1 unit in R3:
∂ξ⃗x·∂ξ⃗x= ˆξ·ξˆ= 1, (7.5a)
∂υ⃗x·∂υ⃗x= ˆυ·υˆ= 1. (7.5b) The scale factor tells how coordinate length relates to physical length. By analogy, in (r, θ) polar coordinates, measuring angular distance in radians is not equivalent to measuring radial distance in, say, meters. In that case,θhas a scale factor ofr, since multiplying the local angular distance by the radius gives physical distance. In the case of the curved-backbone V-groove, only thescoordinate has a non-unity scale factor, which will be denoted byf. Computing and simplifying this scale factor requires a few insights. First, since ξˆandυˆ are unit vectors, then
∂sξˆ·ξˆ=∂sυˆ·υˆ = 0. This means that ∂sξˆand ∂sυˆ have only the ˆsdirection in common, so that∂sξˆ·∂sυˆ= (ˆs·∂sξ)(ˆˆ s·∂sυ). Furthermore,ˆ ∂s( ˆξ·υ) =ˆ ∂s(0) = 0, so thatξˆ·∂sυˆ=−ˆυ·∂sξ.ˆ Using these facts,
f2 =∂s⃗x·∂s⃗x=sˆ+ξ∂sξˆ+υ∂sυˆ2
= (1−ξkξ−υkυ)2+ξ2+υ2ȷ2
=⇒ f = q
(1−ξkξ−υkυ)2+ (ξ2+υ2)ȷ2. (7.6)
d
L
(a)
α u
x h ( s,t )
q
(b)
R h ( s,t ) ˆ
u x s
Figure 7.4: Diagrams of curved-backbone 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 and curved backbone. The inlet midline film thickness at the origin is denoted by d=h(ξ = 0, s= 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 byLwhered/L≪1. sis an arc length coordinate following the backbone (the corner) of the groove; theυcoordinate points “up” from the corner of the groove to the groove opening; the ξ coordinate is orthogonal to the υ coordinate in the cross-sectional plane. (b) Cross-sectional view of the flow geometry depicted in (a) whereh(s, t)denotes the local midline film thickness, α is the groove internal half angle (note that we follow Weislogel’s convention ofα being the internal groove half angle, rather than Romero and Yost’s convention ofαbeing the exterior groove angle), Rhb is the radius of curvature of the liquid interface and θ is the contact angle of the liquid wetting the channel sidewalls, which is assumed constant.
The nonorthogonality of the system can be shown explicitly:
∂ξ⃗x·∂υ⃗x=⃗ξ·⃗υ= 0, (7.7a)
∂ξ⃗x·∂s⃗x= ˆξ·sˆ+ξ∂sξˆ+υ∂sυˆ=−υȷ, (7.7b)
∂υ⃗x·∂s⃗x= ˆυ·ˆs+ξ∂sξˆ+υ∂sυˆ=ξȷ. (7.7c) The fact that these dot products vanish at the backbone (where ξ = υ = 0) means that the coordinates are orthogonal there. It can be seen that the coordinate nonorthogonality arises not from curvature but from torsion, ȷ, just as in Wang’s (1981b) curved pipe. Noting this fact, Germano (1982) solved the same curved-pipe problem as Wang (1981b) in a new coordinate system with a frame of reference designed to rotate in such a way that the nonorthogonality was eliminated. Such an approach was effective due to the circular cross-section of the pipe, but would introduce significant challenges in describing a V-groove with a constantly changing orientation. In short, because the inconvenience of a rotating frame of reference in the V-groove would be greater than the inconvenience of a nonorthogonal coordinate system, we accept the latter.
The central methodology of this work is to treat backbone curvature as a first order perturbative correction to the straight V-groove equations of motion. Thus, in order for perturbation theory to be valid, some constraints on the magnitude of the backbone curvature must be enforced. In particular, it will be assumed that∂sξˆand∂sυˆare at mostO(ε/d), wheredis the characteristic thickness of fluid in the groove and
ε= d
L (7.8)
is the perturbation parameter, “characteristic thickness of fluid”/“characteristic length scale of the groove.” This implies that kξ, kυ, and ȷ are also O(ε/d). In other words, the radius of curvature must be much larger than the thickness of the fluid. Due to the symmetry of the problem under reversal of ξ, any corrections to the equations of motion must be at least quadratic inkξ, which will lead tokξ corrections being O(ε2) and thus irrelevant at first order (although kξ will be carried through much of the following calculations until it becomes clear exactly where it drops out). Thus, ultimately it will be found that onlykυ curvature contributes at first order. Furthermore, it will be seen later that the nonorthogonal terms (i.e., terms containing ȷ) contribute only O(ε2) corrections to the Navier-Stokes and boundary condition equations, and so these will also be irrelevant.