Capillary flow in V-grooves
6.6 Stability analysis
Comparison to thin film instability
While the previous section proved stability of quiescent states mathematically, we still may wish to understand intuitively why the V-groove is stable under an electric field while a thin film on a flat plate is unstable. Turning to linear stability analysis of the quiescent state will help to provide intuition. The key is the fact that the V-groove capillary pressure is ∝ h−1 (the exponent is not important; the lack of derivatives is important), while the thin film capillary pressure is ∝ ∇2h. In both cases, the Maxwell pressure is proportional to a function of halone with no derivatives. Thus, the wavelength of a perturbation does not induce any difference between Maxwell and capillary pressure for the V-groove, while long wavelengths on thin films induce a weakening of the capillary pressure relative to the Maxwell pressure.
More explicitly, let us linearize the V-groove equation with a constant electric field strength χ(Z) =χ0 by writing H=H0+δH, with H0 constant. Then,
2H0
∂δH
∂T =−H04
− 1
H02 +χ0mH0m−1
∂2δH
∂Z2
=⇒ ∂δH
∂T =−H0
2
1−χ0mH0m+1K2δH, (6.66)
where we have performed a Fourier transform ofZtoK. Clearly the wavenumberKis irrelevant to the question of whether the system is stable or not; all that matters is the sign of (1− χ0mH0m+1, i.e., whether H0 is greater than or less thanHthresh..
But the thin film equation on a flat plate under an electric field is given by (Kim et al., 1992)
∂H
∂T =−1 3∇ ·
H3∇
ε2∇2H+We
2 ∇Ψ· ∇Ψ
, (6.67)
where we have applied the V-groove nondimensionalization for consistency of comparison (thus yielding the extra factor of ε2 in front of the capillary term). Linearizing about a sinusoidal perturbationδH yields a perturbed electric field Ψ0+ Ψ1δH, and hence
∂δH
∂T =−1
3H03ε2∇ ·h∇∇2δH+ε−2We{∇Ψ0· ∇Ψ1}δHi
=⇒ ∂δH
∂T = 1
3H03ε2h−K4+ε−2We∇Ψ0· ∇Ψ1K2iδH. (6.68) Thus, long wavelength (small wavenumber K) perturbations are unstable, due to the higher power of K on the capillary pressure term. For a full derivation of the stability of a thin film under an electric field, see de Surgy et al. (1993) (who did not use the thin film equation explicitly but performed a linear stability analysis of the full Navier-Stokes equations and took both thick and thin limits, yielding the thin film equation result in the latter case) or Kim et al.
(1992), who included inertial effects and gravity for a thin film on an inclined plane under an electric field.
While this linear analysis provides intuition, the nonlinear stability analysis presented above was more general, as it allowed non-infinitesimal perturbations and arbitrary electric fields.
6.6.2 Nonlinear stability analysis: Stationary interface with non-zero fluid flux and constant electric field
For the case of a stationary interface with non-zero fluid flux (or a stationary interface shape with different boundary values of the pressure), finding a Lyapunov functional is more challenging, since the boundary terms are no longer trivial. In this case, we consider only a constant external electric field with χ(Z) =χ0, and follow an analysis similar to that in Section 5.2.
LettingG=H2, Equation (6.43) becomes
∂G
∂T =∂Z
h−G2∂Z
G−1/2+χ0Gm/2i
=∂Z 1
2G1/2−m
2χ0G(m+2)/2
∂ZG
= 1 3∂Z
∂Z
G3/2−G3/2S − 3m m+ 4χ0
G(m+4)/2−G(m+4)/2S
, (6.69)
where GS = HS2 is the stationary solution for the given pressure, midline thickness, or flux boundary conditions. We now define a Lyapunov functionalFby
F[G] = Z Z2
Z1
2
5G5/2−G3/2S G+ 3 5G5/2S
− 3mχ0 (m+ 4)(m+ 6)
h
2G(m+6)/2−(m+ 6)G(m+4)/2S G+ (m+ 4)G(m+6)/2S i
dZ.
(6.70) The functional derivatives are given by
δF
δG =G3/2−G3/2S − 3m
m+ 4χ0G(m+4)/2−G(m+4)/2S (6.71) δ2F
δG2 = 3
2G1/2− 3m
2 χ0G(m+2)/2. (6.72)
Note thatF[GS] = 0and(δF/δG)|G=GS = 0. F is convex when(δ2FδG2)>0:
δ2F
δG2 >0 ⇐⇒ G <(mχ0)−2/(m+1)
⇐⇒ H <(mχ0)−1/(m+1) =Hthresh.. (6.73) Finally, Fis always decreasing:
∂F
∂T = Z Z2
Z1
δF δG
∂G
∂TdZ = Z Z2
Z1
δF
δG∂ZZδF δGdZ
= δF
δG∂Z
δF δG
Z2
Z1
−1 3
Z Z2
Z1
∂Z
δF δG
2
=−1 3
Z Z2
Z1
∂ZδF δG
2
<0. (6.74)
Note that the boundary term vanishes in the case of a pressure or thickness boundary condition because(δF/δG)|G=GS = 0, and in the case of a flux boundary condition because∂Z(δF/δG) =
−3(Q−QS),QS being the stationary flux.
Therefore, for the case of a constant electric field, Hthresh. is not only the threshold thickness below which the equation of motion is well-posed; it is also the threshold thickness below which the system exhibits Lyapunov stability around a stationary solution with pressure or flux boundary conditions.
6.6.3 Generalized linear stability analysis: Self-similar states
Following the methodology of Chapter 5, we perform a generalized linear stability analysis of self-similar states with constant electric field χ(Z) =χ0. We linearize according to
H(η, τ) =S(η) +δH(η, τ), (6.75) where τ = ln(T), S(η) is the self-similar state obeying the given boundary conditions, and δH(η, τ) represents the infinitesimal non-modal disturbance function. Substituting this into Equation (6.43) yields the governing linear disturbance equation to order δH,
∂δH
∂τ =LδH, (6.76)
with
L= S 2
h
1−mχ0Sm+1i∂ηη+ η
2 + 2S′−m(3 +m)χ0Sm+1S′
∂η
+ S′′
2 −m
2χ0Smh(m+ 1)(m+ 3)(S′)2+ (m+ 2)SS′′i. (6.77) The transient operator is then given by
L+L† 2 = S
2
h1−mχ0Sm+1i∂ηη+S′ 2
h1−m(2 +m)χ0Sm+1i∂η
−1 4
n1 +S′′+mχ0Smh(m+ 1)(m+ 2)(S′)2+mSS′′io. (6.78) Parameter sweeps testing linear stability for a variety of half angles α and initial and final fluid thickness conditions Sa = S(η = 0) and Sb = S(η = ηb) were performed. Fluid thickness was set to Sa = 1 and various Sb ∈ {0.01,0.1,0.5,0.8,1} for self-similar advancing solutions, and vice versa for receding solutions . System parameters were set to α from 15◦ to 75◦, and χ0 from 0 to 0.9×χthresh., where χthresh. is the maximum electric field strength for which S(0) = 1 < Hthresh. (see Equation (6.44)). In all cases, the system was both transiently and asymptotically stable, with numerical abscissa less than−0.18and spectral abscissa below−0.81 (see Chapter 3 and Chapter 5 for reviews of generalized linear stability analysis and numerical and spectral abscissae). In the nondimensionalized system, the timescale of comparison is 1; hence the numerical abscissa of O(−0.1)may be considered slightly weak stability, as perturbations decay on a timescale of O(10).
A subset of these results is shown in Figure 6.10, which displays plots of the numerical abscissa for advancing, (a), and receding, (b), systems, with χ0 = 0.9×χthresh.. The advancing case
shows little variation between different interior groove angles; this result is likely due to the fact that the majority of the fluid is very thin in the groove and hence feels little effect from the electric field. The slightly greater stability of narrow (small α) grooves is likely attributable to the faster decay of the electric field within those grooves. The receding states, on the other hand, have most of their fluid at a large thickness, and so are less stable than the advancing solutions. Furthermore, as noted in Section 6.5.4, receding solutions at large χ0 develop a sharp “corner” effect, where S′′ ≪ 0. An inspection of the constant term in the transient operator, Equation (6.78), suggests that as S′′ becomes more negative, the upper bound on the eigenvalue of the operator (i.e., the upper bound on the numerical abscissa) increases. This effect thus likely explains why receding states are less stable than advancing states, and why narrow grooves with small α (which have a sharper “corner”) should be less stable than wide grooves, as reflected in Figure 6.10 (b).
To gain some intuition for why the numerical abscissa remains near−(1/4)for advancing states, even asχ→χthresh., consider the simplest case ofS =S0 =const.. In this case, the transient operator is given by
L+L† 2 = S0
2
h1−mχ0Sm+10 i∂ηη−1 4
= S0 2
1− χ0 χthresh.
∂ηη−1
4. (6.79)
Thus, for χ0 ≤ χthresh., the numerical abscissa is at most −1/4. As the wavelength of per- turbations decreases, the eigenvalue of (L+L†)/2 becomes more negative. But as soon as χ0 > χthresh., suddenly short wavelengths instead increase the eigenvalue. Indeed, for any χ0 > χthresh., the numerical abscissa is unbounded, indicating an ill-posed PDE. For more com- plicated base statesS(η), the bounds vary slightly, but the same qualitative behavior applies as long asS′′ is not highly negative (i.e., the qualitative behavior applies for advancing states, not receding states).
While a full stability analysis was carried out only for χ0 ≤ 0.9χthresh., exploratory results suggest that the numerical abscissae for advancing states indeed remain negative up to χ0 = 0.99χthresh., but the numerical abscissae of certain receding states may become positive around χ0 ≳0.96χthresh.. A more detailed study would be required to determine the exact cutoff, with special care taken to accurately capture the increasingly sharp “corner.” It is safe to say, however, that the self-similar states are transiently and asymptotically stable forχ0≤0.9χthresh.. Linear stability analysis was performed in matlab by second-order central finite difference on a domain η ∈ [0,40] with 2000 points, based on the length and mesh fineness found to be sufficient for straight V-grooves in Section 5.3.2. The transient operator was constructed in matrix form using second-order finite difference operators, and the maximum eigenvalue was found using matlab’s built-in eig function, which computes exact matrix eigenvalues (the matrices were sufficiently small that approximate eigensystem methods were not required).
(a) (b)
Figure 6.10: Representative numerical abscissae for self-similar solutions of flow in V-grooves with constant applied electric field, computed by finding the largest eigenvalue of the transient linear operator in self-similar coordinates, Equation (6.78). Results for advancing solutions (a) and receding solutions (b) are shown Numerical abscissae are computed for interior groove half angle α ∈ {5◦,15◦,30◦,45◦,60◦,75◦,85◦} and electric field strength χ0(α)/χthresh.(α) = 0.9, whereχthresh.(α)is the (α−dependent) maximum electric field strength for whichS(0) = 1is in the well-posed regime [see Equation (6.46)]. Results are shown in nondimensional variables, and each case (each α) is nondimensionalized independently. Results were computed with domain length 40 and the mesh size was taken to be 0.02, following Chapter 5. The horizontal axis describes the fluid thickness boundary condition, while the vertical axis is the numerical abscissa, ω (see Chapter 3).
(a) Self-similar advancing solutions with Dirichlet fluid thickness boundary conditions S(0) = Sa= 1 andS(40) =Sb ∈ {0.01,0.1,0.5,0.8,1}.
(b) Self-similar receding solutions with Dirichlet fluid thickness boundary conditions S(0) = Sa∈ {0.01,0.1,0.5,0.8,1}and S(40) =Sb= 1.