Capillary flow in V-grooves
6.5 Analysis of equation of motion .1 Effect of electric field on flux.1Effect of electric field on flux
6.5.4 Symmetry analysis and self-similar solution
initial condition H(Z, T0) = Hi(Z) is a part of a self-similar solution only if Tν/[ν+2(m+1)]
0 ×
Hi(T(m+1)/[ν+2(m+1)]
0 η) satisfies the self-similar equation of motion, Equation (6.58).
A boundary condition on J corresponds to a boundary condition on HTν/[ν+2(m+1)]. Hence it is only for particular values of ν that useful physical boundary conditions will fit the self- similarity. In particular, the action of v2 on P yields v2P = 1+mν P, while v2H = −1+mν H andv2Q=−3ν+(m+1)m+1 Q. A boundary condition is compatible with the self-similarity when the action of v2 on the relevant quantity vanishes, i.e., when the boundary condition is invariant under the symmetry (Hydon, 2000). Thus, Dirichlet pressure or interface thickness boundary conditions requireν = 0, and constant flux boundary conditions require ν =−(m+ 1)/3. The latter is somewhat “fine-tuned”; one would have to carefully implement an electric field which decays in Z at the correct rate in order for a constant-flux boundary condition to yield a self- similar solution. But the Dirichlet pressure or thickness boundary condition is compatible with ν = 0, or a constant electric field, which is likely to arise often in applications. Furthermore, with a constant external electric field, η = Z/T1/2, i.e., the Washburn-like T1/2 spreading is produced. In the case of a constant electric field strength χ(Z) =χ0, the self-similar equation Equation (6.58) simplifies to
S′′=− S′
S
η+ [2−m(3 +m)χ0Sm+1]S′ 1−mχ0Sm+1
=− S′
S η+ 2S′+ mχ0
1−mχ0Sm+1
η−(m+ 1)S′
, (6.59)
where we have denoted the self-similar solution byS(η) =H(Z/T1/2), and denotedη-derivatives byS′ =∂ηS.
Representative solutions of Equation (6.59) are shown in Figure 6.7. Advancing solutions (blue) represent flow from the origin Z = 0 towards Z =∞, and have a well defined finite limit at S(∞). Receding solutions (orange) represent flow from Z = ∞ back towards Z = 0. The uniform solution with constant S = 1 (gray) has no flux. Terminating solutions (red) begin with S >0 but reachS = 0 at a finite value ofη. It should be noted that for each unique χ0
andα there is a unique terminating solution with finite slope (and hence finite flux) atS = 0;
all other terminating solutions are not physically accessible without some additional physics to describe behavior at the termination point.
Each plot of Figure 6.7 uses a different set of initial slope conditions∂ηS in order to construct a representative set of self-similar solutions for the relevant electric field strength χ0 and internal groove half angle α. The top row represents solutions with internal groove half angle α= 30◦, corresponding to m= 4 andχthresh.= 0.25 (computed with respect to an assumed Smax = 1, for consistency with the rest of this work). The bottom row represents solutions with internal groove half angle α = 60◦, corresponding to m = 1 and χthresh. = 1 (also computed with respect to an assumedSmax= 1, for consistency). Each column represents self-similar solutions with a different electric field strength, χ0 ∈ {0,0.4,0.8} ×χthresh.. χthresh. is computed with
respect to Smax = 1, for consistency; it is for this reason that only one receding solution is shown for α = 30◦ with χ0 = 0.8χthresh. (higher receding solutions would exit the well-posed regime).
While the self-similar solutions are qualitatively similar with and without an electric field, some differences stand out. While the terminating solutions and lower advancing solutions have negative second derivative (∂ηηS < 0) for small η when no electric field is applied, stronger electric fields flatten out the slope and even push it to ∂ηηS > 0. This effect is larger for the narrowα= 30◦than for the α= 60◦ groove, due to the larger exponentm (4 vs. 1). Receding solutions also differ; as the electric field increases, the distance over which receding solutions transition from increasing slope to nearly flat becomes shorter, leading to an increasingly sharp
“corner.” This effect is again more pronounced in the narrow, α = 30◦ groove, and will be discussed more shortly.
Figure 6.8 shows plots of self-similar solutions under a constant external electric field [i.e., solutions of Equation (6.59)], with various Dirichlet thickness (S) boundary conditions and electric field strengths, for α= 30◦. Figure 6.9 shows similar plots for α= 60◦. In the case of advancing solutions [plots (a)-(d)], for whichS(0)> S(∞), stronger electric fields yield thinner films, just as with the stationary solutions. Furthermore, stronger electric fields yield a greater
∂ηηS near η= 0; in particular, it can be seen in plots (c)-(d) that the solution with no electric field has negative axial interface curvature (∂ηηS <0) while the solution with an electric field has positive axial interface curvature (∂ηηS >0) nearη= 0.
Receding solutions [plots (e)-(f)] show a different behavior, in which solutions with electric fields approach the thicknessS= 1faster than the solution without electric field. As the electric field strength,χ0, increases the transition from a positive slope to a nearly flat interface occurs over a shorter distance, yielding an increasingly sharp “corner.” Mathematically, this effect arises due to the denominator(1−mχ0Sm+1)in Equation (6.59) becoming smaller as S→Hthresh., leading to large-magnitude negative value of ∂ηηS. Put another way, the self-similar equation can be written as
0 = 1
2η∂η(S2) +∂η
"
S4∂P
∂S∂ηS
#
=ηSS′+ ∂
∂S
"
S4∂P
∂S
#
(S′)2+S4∂P
∂SS′′. (6.60) As(∂P /∂S)→0, the equation approaches a singular perturbation problem (Bender and Orszag, 1999), leading to a small region with a different scaling in which S′′ is very large.
The self-similar solutions in the V-groove with α= 60◦ (Figure 6.9) are qualitatively similar to those with α = 30◦ (Figure 6.8). But, as with the stationary solutions, the gap between the χ0 = 0interface and theχ0= 0.9χthresh.interface is larger for the wide60◦ groove than for the narrow30◦ groove. This is attributable to the electric field exponent difference;m(α= 30◦) = 4 andm(α = 60◦) = 1, so that the electric field decays faster in the narrow groove than the wide groove.
Note that the results in (c) and (d) for advancing states thinning to S = 0.1 and S = 0.01, respectively, decrease to those thicknesses and do not rupture. Depending on the system pa- rameters, S = 0.1 or S = 0.01 may be approaching the regime in which Van der Waals forces between the fluid and wall become relevant; such effects are omitted from the present model but would typically contribute to preventing rupture.
χ0 = 0 χ0 = 0.4χthresh. χ0 = 0.8χthresh.
α=30◦ α=60◦
Figure 6.7: Representative self-similar solutionsS(η)with Dirichlet fluid thickness (S) boundary condition, for terminating (red), advancing (blue), uniform (gray) and receding (orange) states.
The computational domain used in numerically solving for these solutions was[0≤η≤5]. (Only the range[0≤η≤3.0]is shown in the figure since the downstream behavior remains essentially unchanged beyond that value.) All solutions satisfy the Dirichlet condition S(η = 0) = 1 at the origin. Each plot uses a different set of initial slope conditions∂ηS in order to construct a representative set of self-similar solutions for the relevant electric field strength χ0 and internal groove half angle α.
The top row represents solutions with internal groove half angle α = 30◦, corresponding to m = 4 and χthresh. = 0.25 (computed with respect to an assumed Smax = 1, for consistency with the rest of this work). The bottom row represents solutions with internal groove half angle α= 60◦, corresponding to m= 1andχthresh. = 1(also computed with respect to an assumed Smax= 1, for consistency).
Each column represents self-similar solutions with a different electric field strength, χ0 ∈ {0,0.4,0.8} ×χthresh.. χthresh. is computed with respect to Smax = 1, for consistency; it is for this reason that only one receding solution is shown for α = 30◦ with χ0 = 0.8χthresh.
(higher receding solutions would exit the well-posed regime). Note also that the boundary conditions are not chosen consistently between the different cases; instead, a set of boundary conditions was chosen in each case in order to produce a clear set of differing solutions.
(a) (b)
(c) (d)
(e) (f)
Figure 6.8: Representative self-similar solutions of flow of conducting liquids in V-grooves with constant applied electric field, with internal groove half angleα= 30◦, and Dirichlet fluid thick- ness (S) boundary conditions, according to Equation (6.59). Results are plotted for grooves with 5 different values of nondimensional electric field, χ0 ∈ {0,0.05,0.1,0.15,0.2,0.225} cor- responding to (χ0/χthresh.) ∈ {0,0.2,0.4,0.6,0.8,0.9}, where χthresh. is the maximum electric field strength for whichS(0) = 1< Hthresh. [see Equation (6.44)]. The horizontal axis isZ, the axial coordinate along the groove, and the vertical axis is S, the nondimensional midline fluid thickness. Results were computed forZ ∈[0,40], but plotted only for Z ∈[0,3], because the remainder of the solution is nearly constant.
Plots (a)-(d) depict advancing solutions, withS(0) = 1, and varying boundary conditionS(40):
(a)S(40) = 0.8, (b)S(40) = 0.5, (c) S(40) = 0.1, (d)S(40) = 0.01.
Plots (e)-(f) depict receding solutions, with S(40) = 1, and varying boundary condition S(0):
(e): S(0) = 0.5, (f): S(0) = 0.01.
(a) (b)
(c) (d)
(e) (f)
Figure 6.9: Representative self-similar solutions of flow of conducting liquids in V-grooves with constant applied electric field, with internal groove half angle α = 60◦, and Dirichlet fluid thickness (S) boundary conditions, according to Equation (6.59). Results are plotted for grooves with 5 different values of nondimensional electric field, χ0 ∈ {0,0.2,0.4,0.6,0.8,0.9}
corresponding to(χ0/χthresh.)∈ {0,0.2,0.4,0.6,0.8,0.9}, whereχthresh.is the maximum electric field strength for whichS(0) = 1< Hthresh. [see Equation (6.44)]. The horizontal axis isZ, the axial coordinate along the groove, and the vertical axis is S, the nondimensional midline fluid thickness. Results were computed forZ ∈[0,40], but plotted only for Z ∈[0,3], because the remainder of the solution is nearly constant.
Plots (a)-(d) depict advancing solutions, withS(0) = 1, and varying boundary conditionS(40):
(a)S(40) = 0.8, (b)S(40) = 0.5, (c) S(40) = 0.1, (d)S(40) = 0.01.
Plots (e)-(f) depict receding solutions, with S(40) = 1, and varying boundary condition S(0):
(e): S(0) = 0.5, (f): S(0) = 0.01.