Nonisothermal spreading of liquid drops on horizontal plates

1
Non-isothermal spreading of liquid drops on
horizontal plates

• A paper by P. Ehrhard and S. Davis
• Presented by Lael Fisher
• Tuesday January 28, 2003

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Why is this important?

• coating technology
• mould filling
• performance of fuel tanks in space
• cited 66 times (Web of Science)
• innovative heat pipe systems (Zhang 2001)
• spin coating (Kitamura 2001)
• fluid dispensing process (Chen et. al. 2000)

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Whats new?

• angle v. speed generalized U ? (?-?A)m
• gravity
• plate uniformly heated or cooled

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Mobility Exponent

• found empirically
• m 1 (Greenspan)
• m 3 (Schwartz Tejeda)
• m infinity (constant contact angle)

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Problem Formulation B.C.s

• Navier - Stokes
• Continuity Equation
• Energy Equation
• no-slip at rigid boundary relaxed to avoid the
  appearance of shear-stress singularity at moving
  contact line

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Procedure

• I.C.s
• symmetry conditions
• smoothness conditions
• conservation of volume
• contact angle conditions
• non-dimensionalize variables
• O(1) problem gives lubrication approximation
• dimensionless parameters appear

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Evolution Equation

• lubrication approx. gives evolution eqn
• solve for velocity field
• experiment agreement
• Marangoni number
• 0
• gt 0
• lt 0

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Results Isothermal Spreading

• M 0
• modified Bessel functions (1st kind)
• gravity
• no fluid motion at steady state

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Results Isothermal Spreading
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Results Non-Isothermal Spreading (heating)

• fluid flow always present
• no gravity
• as M increases, heating slows drop spreading and
  limits drop size

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Results Non-Isothermal Spreading (Cooling)

• thermocapillary flow reversed
• flatter drops
• no gravity
• aids spreading

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ResultsCooling and Heating

• top ?A 0
• bottom ?A 0.5
• top line on each figure is an approximation
  (which breaks down)

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Molecular Connection (de Gennes)

• small-scale physics of contact lines
• model long-range van der Waals repulsion
• no contact line nearby drop
• precursor film from main drop
• do not consider film edge contact line

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Conclusions

• non-isothermal
• heating retards spreading
• cooling augments spreading
• ?A 0
• isothermal drop goes to infinity
• plate heated, drop will spread finitely far
• plate cooled, drop goes to infinity faster
• gravity
• important at large times
• determines power law for unlimited spreading

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Bibliography

• P. Ehrhard S.H. Davis 1991 Non-isothermal
  spreading of liquid drops on horizontal plates.
  Journal of Fluid Mechanics. 229 365-388
• Gennes, P.J. DE 1985 Wetting statics and
  dynamics. Rev. Mod. Phys. 57 827
• Zhang N.L. Innovative heat pipe systems using a
  new working fluid. Int Commun Heat Mass. 28(8)
  1025-1033 Nov 2001
• Kitamura A. Thermal effects on liquid film flow
  during spin coating. Phys. Fluids 13(10)
  2788-2794 Oct. 2001
• Chen X.B., Schoenau G., Zhan W.J. Modeling of
  time-pressure fluid dispensing processes. IEEE T
  Electron Pa M. 23 (4) 300-305 Oct. 2000