Wetting Phenomena in Chemical Engineering as a Rich Example Application of the Variational Calculus

1
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus

• Frédéric P.-A. Cortat
• Stanley J. Miklavcic

Linköping University, ITN
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Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Motivation

• Limitation of literature to illustrate
  variational
• calculus only elementary examples taken up
• (hanging cable), or linear systems with simple
• boundary conditions
• Single solution
• Only trivial boundary conditions considered
• Non-trivial boundary conditions
• not addressed

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
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Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Philosophy of the variational method

• Physical system select energetically most
• favourable state minimum
• Minima determined by standard derivative or
• by variational minimization of functional
• Drawback resulting equations lead to extrema,
• not necessarily minimum of energy
• If only one solution, always implied that
• solution is minimum

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
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Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Example equilibrium of a wetting meniscus

• More involved physics
• problem equilibrium shape
• of wetting meniscus
• Simple, physically intuitive
• Non-trivial boundary
• conditions
• At least two plausible solutions
• Combination gives possibility for no solution

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
5
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Model for the wetting system

• Energy functional

Contact energy with the solid
Mass of liquid displaced
Extension of interfacial area

• For acute contact angles, profile can be
  non-injective

Need parametric representation
Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
6
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Machinery of variational method

• Natural constraint
• Interface must be connected to solid surface
• Constraint included by creating a modified
• functional

• Lagrangian multiplier associated with
  constraint
• Natural constraint is new aspect to variational
• method add extra component to usual school
• problems

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
7
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Euler-Lagrange equations

• Euler-Lagrange differential equations

• Depend only on physical parameters,
• boundary information do not appear

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
8
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Natural boundary condition

• Very general boundary condition
• contact line

• Mathematically correct via variational approach
• Young-Dupré boundary condition
• contact angle independent of solid
  topology

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
9
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Particularities of the system

• System implemented for simulations (Matlab)
• For given physical parameters, numerical
• scheme gives in general a pair of solutions
• minimum and maximum of energy functional
• system has inherent stable
• and unstable solutions
• If solid raised too high, no more solution
• existence limit of solutions

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
10
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Stable profiles

• Profiles corresponding to minimum of energy
• functional behave as expected

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
11
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Unstable profiles

• Profiles corresponding to maximum of energy
• functional behave counter-intuitively

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
12
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
13
Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Summary and conclusions

• Problem of determining wetting meniscus rich in
• features clarifying variational method
• Illustrates how abstract boundary conditions
• are implemented
• Exhibits both minima (stable) and maxima
• (unstable)
• Establish criteria that earmarks existence limit
• of solutions
• System used for own research on interfaces

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25
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Wetting Phenomena in Chemical Engineering as a
Rich Example Application of the Variational
Calculus
Sources for further information

• F.P-A. Cortat and S.J. Miklavcic, Phis. Rev. E
  68, 052601 (2003)
• F.P-A. Cortat and S.J. Miklavcic, Langmuir 20,
  3208 (2004)
• F.P-A. Cortat and S.J. Miklavcic, ITN internal
  reports series
• F.P-A. Cortat and S.J. Miklavcic, J. Col. Surf.
  Sci, in revision
• F.P-A. Cortat and S.J. Miklavcic, Phis. Rev. E,
  submitted

Frédéric Cortat Linköping Universitet, ITN
freco_at_itn.liu.se
EMAC 2005, Melbourne 2005-09-25