The effect of the contact line on droplet spreading

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The effect of the contact line on droplet
spreading

• A paper by
• Patrick J. Haley and Michael J. Miksis
• presented by
• Scott Norris
• Jan. 23, 2003

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What is a contact line?
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The problem

• No-slip BC causes non-integrable force
  singularity at contact line
• Contact line velocity depends on angle in a
  complicated way
• Monotonic function of theta
• Some theta produce no motion
• Model should resolve and include these factors

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This paper...

• Introduces a lubrication model which satisfies
  these requirements.
• Samples different models
• 3 slip conditions (to resolve force singularity)
• 3 theta/velocity relations
• Finds short-time asymptotic solutions for several
  cases.
• Numerically determines model and parameter
  effects on spreading rate.

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The characters

• Cylindrical co-ordinates (axi-symmetric)
• variables
• h height of drop surface
• contact angle
• R radius of drop

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The characters

• Constants
• equilibrium contact angle
• viscosity
• density
• g acceleration due to gravity

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The Equations

• Navier-Stokes in the interior.
• Some dimensionless parameters
• Ca Capillary number
• B Bond number
• Re Reynolds number

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The EquationsBoundary Conditions

• Three different theta/U relations

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The EquationsBoundary Conditions

• No normal velocity!
• Navier Slip condition
• Three different kinds

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The Lubrication Approximation

• Assume height ltlt width
• Benefits
• Creates small parameter (height/width)
• Can use asymptotic analysis to simplify problem

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Some Magic(leading order asymptotics)

• Following Analysis of Greenspan (1978)
• Vertical components now "small"
• Reduces to a 1-D PDE for h(r,t)

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Some new B.C.'s

• Symmetry and Smoothness require
• At r 0
• At r R

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Free boundary B.C.

• Need an additional B.C. for free boundaries

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An asymptotic analysis

• Small-time deviation from I.C.'s
• Steady-state shape profile, but...
• Initial contact angle differs from static CA
• Profile change driven only by CA deviation
• Different Models
• Both constant and singular slip \lambda(h)
• Both constant and linear f(\theta)

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Asymptotic Results(singular slip)

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Asymptotic Results(constant slip)

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A numerical method

• Chebyshev numerical method
• The Good
• FFT's to calculate derivatives (n log n)
• Many collocation points near contact line
• High accuracy near contact line
• The Bad
• Chebyshev scales poorly with high-order
• This problem is high-order.

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Results

• Drop profile for reference case

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Results

• Dependence on Bond number

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Results

• Dependence on Capillary number

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Results

• Dependence on slip model

constant
1/h
1/h2
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Results

• Dependence on theta/U relation model

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Comparison with Experiment

• Chen (1988) measured the spreading of a very thin
  liquid drop and found a spreading rate of
  t(1/10).
• Haley/Miksis recovered this using case (3) of the
  relation (cubic).

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Summary of Results
Linear
Quadratic
Cubic
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References

• Chen J., Experiments on a spreading drop and its
  contac angle on a solid, J. Colloid Interface
  Sci. (1988), vol. 122, pp. 60-72.
• Greenspan, H.P., On the motion of a small viscous
  droplet that wets a surface, J. Fluid. Mech
  (1978), vol. 84, pp. 125-143.
• Haley, P.J. and Miksis, M.J., The effect of the
  contact line on droplet spreading, J. Fluid.
  Mech. (1991), vol. 223, pp. 57-81.

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