Systematic asymptotic expansions are used to find the leading-order equations for the pressure-driven flow of a thin sheet of viscous fluid. Assuming the fluid geometry to be slender with non-negligible curvatures, the Navier–Stokes equations with appropriate free-surface conditions are simplified to give a ‘shell-theory’ model. The fluid geometry is not known in advance and a time-dependent coordinate frame has to be employed. The effects of surface tension, gravity and inertia can also be incorporated in the model.
A model for species diffusion is presented, with evaporation at a moving free boundary. The resulting problem resembles a one-phase Stefan problem with superheating, but the usual Stefan condition at the moving boundary is replaced by a version which, in the classical setting, would violate conservation of energy. In the fast evaporation limit, however, the problem reduces to a classical nonlinear Stefan problem with negative latent heat.
A model is presented for the diffusion-driven drying
of a polymeric solution such as liquid
paint. Included is a stress build-up and relaxation in the polymer network
of the viscoelastic
material, which influences the diffusion process. The behaviour of the
(one-dimensional) model
is analysed by means of the maximum principle and illustrated with numerical
calculations.
A description of the adiabatic decay of the Lamb dipolar vortex is motivated by a variational characterization of the dipole. The parameters in the description are the values of the entrophy and linear momentum integrals, which change in time due to the dissipation. It is observed that the dipole dilates during the decay process [radius R∼(νt)1/2], while the amplitude of the vortex and its translation speed diminish in time proportional to (νt)−3/2 and (νt)−1.
Abstract. Different approaches are discussed of variational principles characterizing coherent vortex structures in two-dimensional flows. Turbulent flows seem to form ordered structures in the large scales of the motion and the self-organization principle predicts asymptotic states realizing an extremal value of the energy or a minimum of enstruphy. On the other hand the small scales take care of the increase of entropy, and asymptotic results can be obtained by applying the theory of equilibrium statistical mechanics.
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