Bulk equations

Capillary flow in V-grooves

6.2 V-groove model with Maxwell stress

6.2.2 Bulk equations

The continuity and Navier-Stokes equations are given by

∇ ·⃗u= 0, (6.1)

ρ ∂⃗u

∂t + (⃗u· ∇)⃗u

=−∇p+µ∇²⃗u (6.2)

where the velocity field in Cartesian coordinates is represented by⃗u= (u, v, w), the fluid pressure by p(x, y, z), and the constant fluid density by ρ. Note that additional bulk forces due to the electric field would arise in Equation (6.2) if the permittivity within the fluid varied (e.g., due to density or temperature variations), or if there were free charges (such as in an ionic liquid) (Landau and Lifshitz, 1984); we do not consider these cases.

We nondimensionalize just as in Chapter 4, and relevant quantities are listed in Table 6.1. Note in particular that we set the characteristic velocity and time scales using the capillary pressure, rather than the Maxwell pressure, because of the requirement that capillary pressure thickness sensitivity dominate Maxwell pressure thickness sensitivity (a condition which will be satisfied if the capillary pressure is sufficiently large compared to the Maxwell pressure). The capillary pressure is thus the natural reference for nondimensionalizing velocity and other variables.

Quantity Scaling Rescaled variable

Slender parameter ε=d/L; ε²≪1

Thin film parameter

(film thickness/groove depth) δ=d/b; δ²≪1

Coordinates x_c=d X =x/x_c

y_c=d Y =y/y_c zc=L Z =z/zc

Polar coordinates r, β

Velocity uc=εwc U =u/uc

v_c=εw_c V =v/v_c w_c=εγCa/µ W=w/w_c

Streamwise flux q_c=d²w_c Q= [^R_Ωwdxdy]/q_c Stress tensor (in vacuum) τ_evac. (Eq. 6.6)

Stress tensor (in liquid) τ_eliq. (Eq. 6.7)

Electric potential (in vacuum) ψc Ψ =ψ/ψc

Electric potential (in liquid) Ψ^liq.=ψ^liq./ψc

Relative permittivity ϵ_r=ϵ_liq./ϵ₀

Interface electric field strength χ(Z) = [(WeR)/2](ψ^{b outer}₀ (z)/ψc)²Cb Electric field Ec= (ψc/b)(π/α)(d/b)^{[π/(2α)]−1}

Pressure pc=µwc/(εd) P =p/pc

Rescaled pressure P =PCaRb

Time tc=L/wc T =t/tc

Interface midline y_c=d H =h(z, t)/y_c

thickness

Interface shape y_c=d Σ(X, Z, T) =σ(x, z, t)/y_c

Stationary state HS(Z)

midline thickness

Self-similar variable η=Z/√

T

Self-similar state S(η)

midline thickness

Reynolds number Re =ρwcd/µ

Slender limit capillary number Ca =w_cµ/(εγ) = Φ(α, θ)

Electric Weber number We =ϵ₀E_c²d/γ= Caϵ₀E_c²(dε)/(w_cµ)

Maxwell pressure exponent m= (π/α)−2

Electric geometric factor (Eq. 6.36) C(α, θ,^b electrode geom.) =O(1) Table 6.1: Characteristic scalings and nondimensional vari-

ables used to describe dimensionless system shown in Figures 6.1 and 6.4. Note that ψ₀^outer(z) denotes the 0th order term in a cosine transform of the applied electric field at the top of the groove,r=b(see Section 6.3.2).

The dimensionless forms of Equations (6.1) and (6.2) are then given by

0 =∂_XU +∂_YV +∂_ZW, (6.3a)

ε³ReDU

DT =−∂_XP +ε^2h∂XX+∂Y Y +ε²∂ZZ

iU, (6.3b)

ε³ReDV

DT =−∂_YP+ε^2h∂_XX+∂_{Y Y} +ε²∂_ZZⁱV, (6.3c) εReDW

DT =−∂_ZP +^h∂XX+∂Y Y +ε²∂ZZ

i

W, (6.3d)

where D/DT is the dimensionless material derivative D

DT =∂_T +U ∂_X+V ∂_Y +W ∂_Z. (6.4)

In the limits whereε² ≪1,εRe≪1, the governing equations, Equation (6.3), then reduce to

∂XU +∂YV +∂ZW = 0, (6.5a)

∂_XP = 0 +O(ε²), (6.5b)

∂_YP = 0 +O(ε²), (6.5c)

∂_ZP =∂_XXW +∂_{Y Y}W +O(ε²). (6.5d) 6.2.3 Boundary conditions at the liquid interface

The two (dimensional) boundary conditions specifying the jump in normal and tangential stresses across the gas/liquid interface σ(x, z, t) can be written in terms of the stress tensor inside the liquid, τ_eliq., and the vacuum region stress tensor _eτvac.. (Note that τ_e here represents the full stress tensor including the pressure, not the deviatoric stress tensor.)

Letting the electric field in the vacuum region be represented byψ and the electric field within the liquid be represented byψ^liq., the stress tensors become

τevac.ij =ϵ₀

∇_iψ∇_jψ−1

2δij∇_kψ∇_kψ

(6.6) τeliq.ij =µ[∇_iuj+∇_jui]−pδij +ϵ_liq.

∇_iψ^liq.∇_jψ^liq.− 1

2δij∇_kψ^liq.∇_kψ^liq.

(6.7)

where δ_ij is the identity matrix (Saville, 1997). The external pressure has, without loss of generality, been set to a reference pressure of 0.

The interfacial stress balance conditions are then

[ˆn·(τ_eliq.−τ_evac.)·nˆ+γ(∇_s·n)]ˆ _y=σ(x,z,t)= 0, (6.8a) htˆi=1,2·(_eτ_liq.−τ_evac.)·nˆⁱ

y=σ(x,z,t)= 0, (6.8b)

where ∇_s = (∇ −n(ˆˆ n· ∇)) is the surface gradient operator, and the triad (ˆn,ˆtx,ˆtz) is the three unit vectors representing directions normal and tangent to the moving interface with the convention that nˆ points away from the fluid.

In rescaled units, the unit vectors are given by

Nˆ = 1

[1 + (∂XΣ)²+ε²(∂ZΣ)²]^1/2









−∂_XΣ 1

−ε∂_ZΣ









, (6.9a)

TˆX = 1 [1 + (∂XΣ)²]^1/2







 1

∂XΣ 0









, (6.9b)

TˆZ = 1

[1 +ε²(∂ZΣ)²]^1/2







 0 ε∂ZΣ

1









, (6.9c)

where Σ(X, Z, T) =σ(x, z, t)/d denotes the nondimensional interface function and subscripts denote differentiation with regard to the rescaled coordinates.

To orderO(ε²)then, the dimensionless normal boundary condition reduces to the form 0 =−P|_Y=Σ(X,Z,T)−Ca⁻¹ ∂_X²Σ

[1 + (∂_XΣ)²]^3/2

!

+We Ca

1

2|∇Ψ|²−Nˆ · ∇Ψ²

−ϵr

1 2

∇Ψ^liq.2−Nˆ · ∇Ψ^liq.2

Y=Σ(X,Z,T)

+O(ε²) (6.10a)

=^h−P−Ca⁻¹K(X, Z, T) +P_Maxwellⁱ

Y=Σ(X,Z,T)+O(ε²), (6.10b)

wherePMaxwell is the total Maxwell pressure (defined by the term with the coefficient We/Ca) (Landau and Lifshitz, 1984) andK(X, Z, T)is the dimensionless mean curvature of the interface functionΣ(X, Z, T) (defined to be positive for a wetting liquid). According to Equation (6.5b), the pressureP is independent of (X, Y), and thus the quantity(PMaxwell−Ca⁻¹K) must also be a function ofZ alone. In the absence of an electric field, this condition would imply thatK is a function of Z alone and thus that the interface is a circular section. However, that is no longer the case here, and a nonzero Maxwell pressure breaks the circularity ofΣ(X, Z, T).

The tangential boundary conditions meanwhile become 0 =

"

∂_YW −(∂_XΣ)∂_XW p1 + (∂_XΣ)²

#

Y=Σ(X,Z,T)

−We Ca

hε⁻¹Tˆ_Z· ∇Ψ Nˆ · ∇Ψ−ϵr

ε⁻¹Tˆ_Z· ∇Ψ^liq. Nˆ · ∇Ψ^liq.i

Y=Σ(X,Z,T)+O(ε²), (6.11a) 0 =^hTˆX· ∇Ψ Nˆ · ∇Ψ−ϵr

TˆX · ∇Ψ^liq. Nˆ · ∇Ψ^liq.i

Y=Σ(X,Z,T)+O(ε²). (6.11b) 6.2.4 Surface shape

In principle, we next must determine the shape of the fluid surface. In particular, the cross- sectional surface must satisfy the ordinary differential equation ∂_XP = 0, with the sides sat- isfying the contact angle condition ∂_XΣ(X, Z, T)|_wall = cot(α+θ)sign(X). If P_Maxwell were independent of Σ(X, Z, T) and X, then the result would be a segment of a circle. But for a conducting or dielectric film, that will not be the case.

However, it turns out to be sufficient for our case of interest (specifically, for a conducting fluid which is thin compared to the height of the groove walls) to approximate the surface as a circular segment. For a more accurate model, the surface shape would need to be computed numerically. Writing PMaxwell as a functional PMaxwell[Σ, X] and calling the midline interface thickness H, i.e., Σ(X = 0, Z, T) = H(Z, T), the surface would need be computed for each given H. Solving∂_X(P_Maxwell−Ca⁻¹K) = 0 with the aforementioned contact angle boundary conditions and the additional condition Σ(X= 0, Z, T) =H(Z, T) will yield the desired result, as long as P_Maxwell is not dependent on Z-derivatives of Σ(X, Z, T). While we will carry out such an analysis for the curved-backbone V-groove problem in Chapter 7, we do not do so here, as the resulting correction is small; see Section 6.7 for a numerical quantification of the errors induced by this approximation. From now on, we will assume that the surface is a circular section and the pressure is a function of H and Z alone, i.e., P =P[H, Z].