Notable solutions

Capillary flow in V-grooves

4.3 Notable solutions

α (

degrees

)

0 5 10 15 20

0

˚

20 ₄₀

˚

₆₀

˚ ˚

0 20 40 60 80

A

θ =

00 0.5 1 1.5 2 2.5 3

20 40 60 80

Γ

0

˚

20 ₄₀

˚

₆₀

˚ ˚

θ =

0 20 40 60 80

0 0.01 0.02

0.03 20 0₄₀

˚ ˚

₆₀

˚ ˚

θ =

Φ R

0

˚

20 ₄₀

˚

₆₀

˚ ˚

θ=

0 5 10 15 20

0 20 40 60 80

α (

degrees

)

(b)

(a) (d)

(c)

Figure 4.4: H-independent functions pertinent to capillary flow of a Newtonian liquid film with constant contact angle θ in a slender open V-groove with half opening angle α satisfying the Concus-Finn condition α+θ < π/2. Plotted are the functions (a) A(θ, α), (b)^b R(θ, α), (c)^b Γ(θ, α), and (d) Φ(θ, α) described in the text.

the values of const and Q_o. For a V-groove with fixed corner angle and liquid contact angle (i.e., constant value of A(α, θ)) extending between endpoints^b Z1 and Z2, stationary solutions H_S correspond to

HS=

"

H₁³− 3QS

A(α, θ)b (Z−Z1)

#1/3

(4.33) for Dirichlet and Neumann conditions HS(Z1) =H1 andQ2 =QS, respectively. Alternatively, Dirichlet conditions imposed at both endpoints, such that H_S(Z₁) = H₁ and H_S(Z₂) = H₂, yield

H_S(Z) =

H₂³ Z−Z₁

Z₂−Z₁ +H₁³ Z₂−Z Z₂−Z₁

1/3

. (4.34)

Note from Equation (4.17) and the scaling for the streamwise flow speed w_cthat specification of the boundary film thickness is equivalent to specification of the boundary fluid pressure.

Solutions H_S to Equation (4.32) can also be generated subject to constant flux Q_S at one boundary (Q₁ = Q_S or Q₂ = Q_S) and conservation of volume V_S, which sets the constant valueCS in the implicit relation :

V_S=Ab Z Z2

Z1

H²(Z)dZ (4.35)

= 3^5/3A^b1/3 5QS

h

(CS−QSZ1)^5/3−(CS−QSZ2)^5/3i. (4.36) Likewise, constant volume and fixed liquid height at one endpoint yield similar forms.

Representative solutionsH_S(Z)are plotted in Fig. 4.5 for Dirichlet conditions which pin the inlet height to H1 = 1.0and pin the outlet heightH2 to the five values shown. From the expression for the flux given by Equation (4.30), it is evident that the solution with H₂ = 1.33, which exhibits interface slope values dH/dZ which are everywhere positive, describes a stationary solution with net streamwise fluxQS <0, i.e., net flow directed from right to left. The uniform solution H₂(Z) = 1.0 clearly then represents a case with no flux and no subsurface flow, i.e., a quiescent liquid filament. The remaining curves with negative interface slopes throughout the domain correspond to stationary solutions with positive flux, i.e., net flow directed from left to right. Other boundary conditions yield similar shapes with characteristic scaling Z^1/3.

Z

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

= 0.01

= 0.33

= 0.67

= 1.00 H_S (Z = 3) = 1.33

H

_S

Figure 4.5: Representative stationary solutions subject to Dirichlet conditionsHS(Z1) =H1= 1.0and H_S(Z₂) =H₂ = 0.01, 0.33, 0.67, 1.00, and 1.33 for the range Z ∈[Z₁, Z₂] = [0,3.0].

4.3.2 Self-similar spreading and draining solutions with fixed boundary pressure Previous studies have delineated the conditions leading to existence and uniqueness of self-similar solutions as well as the attraction of spatially confined initial distributions toward self-similar base states (Vázquez, 2007). Here we focus on volume non-conserving positive statesS(η)consistent with time-independent Dirichlet boundary conditions imposed at the domain endpoints, namely S(0) = 1 and S(η → η_B) = const where η_B denotes a location far downstream of the origin.

The Dirichlet condition at the origin can be set to unity without loss of generality since as evident from Equation (4.42), a rescaling involving a multiplicative factor of S(0) leaves the governing equation unchanged.

In general, for self-similarity to hold, there can be no intrinsic length or time scale imposed on the flow, in contrast to the steady state solutions examined in the previous section which depend on the groove lengthZ₂−Z₁ . A simple scaling analysis of Equation (4.29) reveals that self-similar solutions may be possible wheneverT << L²/H ∼O(L/ε). To find such solutions, it is convenient to expand and rewrite Equation (4.29) in the form

∂H

∂T −H 2

∂²H

∂Z² − ∂H

∂Z 2

= 0. (4.37)

The ansatzH_sim(η, T) defined by

H_sim(η) =T^2β−1S(η) where (4.38) η= Z

T^β forβ >0, (4.39)

allows for a large class of self-similar solutions (Vázquez, 2007; Weislogel and Lichter, 1998)

satisfying the general second order nonlinear differential equation S

2S_ηη+ (S_η)²+βηS_η + (1−2β)S = 0. (4.40) Inspection of the asymptotic behavior of S(η) as η → ∞ helps ascertain what range of ex- ponents β are required for bounded non-terminating (i.e., S > 0) states such that S_ηη and Sη asymptotically approach zero as η → ∞ . While the first two terms on the left side of Equation (4.40) then vanish identically, care must be taken with regard to the third term which couples an increasingly large value of η with an increasingly small term S_η. Balancing the third and fourth terms yields the proper asymptotic scaling, namelydS/S∼[(2β−1)/β]dη/η, and hence S(η → ∞) ∼ η^(2β−1)/β. Therefore, only the range 0 < β ≤ 1/2 yields bounded non-terminating self similar states.

Boundary conditions also impose constraints on the allowable values of the exponent β. For example, enforcement of constant liquid volume V ≃ ^R_Z^Z2

1 H²dZ = T^5β−2R_Z^Z₁²S²dη is only consistent withβ = 2/5. According to Equation (4.30), enforcement of a constant flux boundary condition Q =−A(α/θ)Hb ²(∂H/∂Z) = const =−A(α/θ)Tb ^3β−2S²Sη is only consistent with β = 3/5. Clearly then, a constant flux boundary condition (Neumann condition) is therefore inconsistent with bounded solutions.²

In what follows, we restrict attention to the value β = 1/2, which accords with the Washburn relation and allows enforcement of time-independent Dirichlet boundary conditions. For this category of solutions, the nondimensional flux defined in Equation (4.23) is represented by

Q(η, T) =−A(α, θ)^b T^1/2 S²∂S

∂η. (4.41)

The self-similar solutionS(η) then satisfies the equation:

SSηη+ηSη+ 2(Sη)² = 0. (4.42) To ascertain the interface shape of these solutions, we numerically solved Equation (4.42) by rewriting the second order equation as a system of first order equations and using the ODE45 solver in Matlab (Mat, 2015). Shown in Fig. 4.6 are representative solutions for receding, uniform, advancing, and terminating states S(η) satisfying the far field Dirichlet conditions shown. According to Equation (4.41), solutions with Sη > 0 correspond to states with net liquid flux to the left, designated receding states, while solutions with S_η <0 correspond to a net liquid flux to the right, designated advancing states. The solution for which Sη vanishes everywhere, which corresponds to a zero flux solution in self-similar coordinates, is designed a

2When gravity in theZ-direction is included [see Equation (4.28)], the only admissible self-similar constant is β = 1/3. This indeed leads to bounded self-similar states, but requires a boundary condition of the form P(0)∝T^1/3 orQ(0)∝T^−4/3. If gravity is perpendicular to the flow direction, the governing equation picks up corrections similar to those in Chapter 7 and admits onlyβ = 1/2(see Appendix C). Thus, the Washburn-like T^1/2 spreading would be maintained with Dirichlet pressure boundary conditions.

S (η)

η

1

2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.5 1.0

1.5 Receding

Advancing Uniform

Termina ting

Figure 4.6: Representative self-similar solutions S(η) with pressure boundary condition, for terminating (1-7), advancing (8-13), uniform (14) and receding (15-17) states. The compu- tational domain used in numerically solving for these solutions was [0 ≤ η ≤ 80]. (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 conditionS(η= 0) = 1at the origin. Solutions 1-7 exhibit interface slopes at the origin given by S_η(0) =-10, -2, -0.8, -0.5, -0.4, -0.36, and -0.3492. Solutions 8-13 satisfy far field Dirichlet conditions given, respectively, byS(80)= 0.01, 0.17, 0.33, 0.5, 0.67 and 0.83. (Resulting values for the interface slopes at the origin are S_η(0) = -0.3491, -0.3418, -0.3185, -0.2745, -0.2090, and -0.1185, respectively.) The line denoted by 14 represents a uniform solution whereS(η) = 1.0. Solutions 15-17 satisfy far field Dirichlet conditions given, respectively, by S(80) = 1.17, 1.33 and 1.5. (Resulting values for the interface slopes at the origin areS_η(0)= 0.1485, 0.3283 and 0.5451, respectively.)

uniform state. It represents an exception in that it is the only solution which satisfies volume conservation. Solutions whose advancing front are characterized by a vanishing value ofS(η)are likewise designated terminating states. The numerical solutions indicate that solutions undergo termination only when the interface slope at the origin S_η(0) ≲ −0.349. It should be noted that there is a unique terminating solution with finite slope (and hence finite flux) at S = 0;

all other terminating solutions are not physically accessible without some additional physics to describe behavior at the termination point.

4.3.3 Self-similar converging and receding solutions with fixed boundary flux

Satisfying a flux condition at η = 0 requires self-similar exponent β = 3/5. In this case, Equation (4.40) reduces to

SQS_Qηη+ 6

5ηS_Qη+ 2(S_Qη)²−2

5SQ= 0. (4.43)

Because SQ=T^−1/5H, there is no solution with limη→∞H =const.∈(0,∞). In fact, there is also no smooth, nontrivial solution withlim_η→∞S_Q=const.∈[0,∞); this can be seen from Equation (4.43) itself, as theS_QS_Qηη andS_Q²_η terms must vanish in such a limit, which in turn forces S_Qη∝η^1/3.

0.0 0.5 1.0 1.5

S

_Q

(η)

0.0 1.0 2.0 3.0 4.0 5.0 6.0

η

Term ina

ting 1

2 3 4 5 6

7 8 9 10 11 13 14

15 16 Receding 12

Converging

Figure 4.7: Representative self-similar solutions S_Q(η) with flux boundary condition for termi- nating (1-5), converging (6-13), and receding (14-16) states. The computational domain used in numerically solving for these solutions was [0 ≤ η ≤ 80]. (Only the range [0 ≤ η ≤ 6.0]

is shown in the figure). Solutions satisfy S_Q(η = 0) = 1 at the origin. Solutions 1-5 exhibit interface slopes at the origin given byS_Qη(0)=-10, -1, -0.59, -0.55, and -0.5398. Solutions 6-13 exhibit interface slopes at the origin given, respectively, byS_Qη(0)= -0.5396, -0.5358, -0.5215, -0.4931, -0.4467, -0.3778, -0.2817, and -0.1537. And solutions 14-16 have exhibit initial slopes S_Qη(0) = 0, 0.2187, and 0.4733.

Solutions therefore fall into three categories, shown in Figure 4.7. Terminating solutions, like those in the β = 1/2 system, reach S_Q = 0 at finite η, and only one such solution has finite flux at the termination point. Second, converging solutions have negative slope (and positive flux) at η = 0 but positive slope (negative flux) at largeη. Receding solutions have uniformly positive slope, and hence represent flux from large η back toη = 0.

4.3.4 Capillary rise against gravity

While flow in a closed capillary of constant radius reaches a finite heightz_g= (2γcosθ)/(rρg) described in Equation (4.1), the front of liquid flow in a V-groove rising against gravity has no finite limit. Indeed, it is this ability to transport liquid to arbitrary heights that motivated Concus and Finn (1969) to suggest it as a mechanism of transpiration in trees.

The static form of Equation (4.28) is given byH_G²∂ZHG+BH_G⁴ = 0, and hence has solution HG(Z) = H₀

1 +BZH0

, (4.44)

whereH0 is the fluid thickness atZ = 0 (Verbist et al., 1996). AlthoughlimZ→∞HG(Z) = 0, H_G is positive for all finiteZ. And the total volume remains finite:

v_G= (Ld²) Z ∞

Z=0

AHb _G²dZ = (Ld²)AHb ₀/B= (AΓ/b R)h^b ₀γ/(ρg), (4.45) which is comparable to the closed-capillary volume of(2πcosθ)rγ/(ρg). In other words, the V- groove and the closed capillary have similar limits on the amount of fluid they can raise against

gravity, but the V-groove distributes that constant volume in a long, narrow thread which can reach arbitrary heights.

4.4 Discussion