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On the Capillary Problem for Compressible Fluids


Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are uniquely determined for any contact angle γ.

Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2. Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the present case under the presence of a mass constraint.

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Correspondence to Robert Finn.

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Finn, R., Luli, G.K. On the Capillary Problem for Compressible Fluids. J. math. fluid mech. 9, 87–103 (2007).

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Mathematics Subject Classification (2000).

  • 76B45
  • 53A10
  • 76M30
  • 76N99
  • 35J60


  • Capillarity
  • surface tension
  • mean curvature
  • compressible fluids
  • energy minimizer
  • nonlinear elliptic pde’s