Manifolds with edges (contact lines)

S XLT(2)XL

Restriction 3: Fluids, kinematic condition, and massless interface

2.1.6 Manifolds with edges (contact lines)

We have now seen how to deal with edgeless manifolds, like the surfaces of raindrops and bubbles. Such surfaces were edgeless because they formed the interface between two distinct

V

_c

e L

_c

M

₍₁₎

M

₍₂₎

M

₍₃₎

ν ˆ

_j⁽²⁾

ν ˆ

_j⁽¹⁾

ν ˆ

_j⁽³⁾

θ

Figure 2.4: A tubular control volume V_c of radius ε enclosing a section L_c of a contact line, where three manifolds (M₍₁₎,M₍₂₎,M₍₃₎) meet. νˆ_j^(N)denotes the normal vector onLctangent to manifold M_(N). The contact angle between M₍₁₎ and M₍₂₎ is θ. The contact line Lc is depicted as straight but that need not be the case.

volumetric domains; when there are three volumetric domains, an edge will appear. For example, a water droplet sitting on a table has an edge where the water, air, and solid meet, often referred to as the contact line.

Consider such a contact line formed by intersection of three 2D manifolds, M₍₁₎, M₍₂₎, and M₍₃₎. Let the stress-mass tensor be restricted to the manifolds be given by S₍₁₎^ΞΛ, S₍₂₎^ΞΛ, and S₍₃₎^ΞΛ, and let the stress-mass tensor restricted to the line be given by B^ΞΛ (in particular, we ignore the line-bending moment a priori, as we will be assuming a massless line shortly). We can now proceed with an argument similar to that of the previous section, but with the roles of T and S replaced byS andB, respectively.

Consider a comoving tube of radiusεenclosing a sectionL_cof the contact line, as in Figure 2.4.

In the limit asε→0, Equation (2.5) reduces to d

dt Z

Lc

B^Ξtds= ^X

N=1,2,3

Z

Lc

S_(N)^Ξj −S^Ξt_(N)u^j_cνˆ_j^(N)ds+^h−B^Ξj+B^Ξtu^j_cˆt_jⁱ

∂Lc

+ lim

ε→0

X

N=1,2,3

Z

∂Vc

−T_(N)^Ξj +T_(N)^Ξtu^j_cn¯_jdA, (2.40)

whereνˆ_j^(N)is the outward-pointing unit vector normal to the contact line and tangent toM_(N), ds is the measure of an arc-length parametrization of the contact line Lc, and ˆtj is the unit tangent along L_c. Although the surface area of the tube goes to 0 as ε → 0, we left in the bulk stress tensorsT_(N)^Ξj because they can form a singularity at a moving contact line if a no-slip boundary condition is imposed. A number of resolutions are possible, including a slip boundary condition, elastic movement of the solid wall, and non-Newtonian stress tensors; a slip condition is perhaps the most plausible and leads to a complicated interface shape near the contact line (Cox, 1986). Let us simply assume that the contact line is motionless and continue from there.

In that case,

Z

Lc

∂_tB^Ξtds= Z

Lc



∂_s−B^Ξjˆt_j+ ^X

N=1,2,3

S_(N)^Ξjνˆ_j^(N)



ds. (2.41)

The most general form of B is B^ΞΛ=

"

−β −βu^µ

−βu^ν Betˆ^µˆt^ν−βu^µu^ν

#

, (2.42)

where β is the mass per unit length of the contact line. But let us restrict ourselves from the start to the case where the contact line is massless, so that

B^ΞΛ =

"

0 0

0 Beˆt^µtˆ^ν

#

. (2.43)

Note that B, called the line tension, is now a scalar since there is only one spatial degree of^e freedom on the line. This line tension is analogous to the surface tension, but on a 1-dimensional manifold instead of a 2-dimensional manifold.

Letting α be an arc-length coordinate tangent to the contact line (corresponding to ˆt^j), and lettingµrefer to coordinates normal to it, the conservation of tangent momentum and normal momentum are then

0 =−∇_s(B^αα) + ^X

N=1,2,3

S_(N)^αjνˆ_j^(N) (2.44a) 0 =−∇_s(B^µα) + ^X

N=1,2,3

S_(N)^µj νˆ_j^(N), (2.44b) or

0 =−∇_sBe+ ^X

N=1,2,3

S_(N)^αj νˆ_j^(N) (2.45a)

0 =−κnˆ^µB^e+ ^X

N=1,2,3

S_(N)^µj νˆ_j^(N), (2.45b) whereκ is the curvature of Lc andnˆ is its normal (parallel to ∂st).ˆ

In the simplest case of isotropic manifolds with S_(N)^ij = γ_(N)g^ij, the momentum conservation reduces further to

∇_sBe = 0 (2.46a)

κnˆ^µB^e = ^X

N=1,2,3

γ_(N)νˆ_(N)^µ . (2.46b)

Equation (2.46a) implies that in this case the line tension B^e must be constant. Furthermore, the balance of surface tensions determines the valid νˆ_(N), and in turn the contact angle. For the standard setup of a liquid drop on a flat solid surface, Equation (2.46b) reduces to

κB^e =γ_SG−γ_SL−γ_LGcosθ, (2.47)

whereθis the contact angle, and it is in this form that line tension is typically reported (Amirfazli and Neumann, 2004). Setting B^e = 0, i.e., assuming there is no line tension, recovers Young’s Law for the contact angle, Equation (2.2).

Debate remains over both the sign and magnitude of line tension in real systems, and it will not be considered for the work in this thesis. The interested reader is referred to the review article by Amirfazli and Neumann (2004).

References

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STABILITY ANALYSIS