Drying a liquid paint layer

Subject of this study is the free boundary problem of a liquid layer that is dried by evaporation. Using a Stefan type problem, we model the diffusion driven drying of a layer of liquid paint consisting of resin and solvent. The effect of a small perturbation of the flat boundary is considered. We include the discussion of evaporation constant as a free parameter. For both small and big wavenumber, the high speed of evaporation can lead to instability. We first recognize this instability in the linearized equation. Using numerical calculations, we show that the instability also happens in the full equation

A free boundary problem for evaporating layers

The subject of this paper is a free boundary problem for a liquid layer that is dried by evaporation. Using a Stefan type problem, we model the diffusion driven drying a layer of a liquid paint consisting of resin and solvent. For the one-dimensional case, the movement of the free boundary is found in terms of a short time asymptotic analysis. When including a fluid flow and the levelling of the surface in a two-dimensional model, two small parameter cases are discussed. The first one concerns the levelling by surface tension under the assumption of a small aspect ratio, where the thin film equation appears in the free boundary condition. The second one concerns the effect of a small perturbation of the flat free boundary that shows different decay for long and short wavelength surface elevations

The viscous modulation of Lamb’s dipole vortex

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

Quasi-homogeneous critical swirling flows in expanding pipes, part I

Time-independent swirling flows in rotationally symmetric pipes of constant and varying diameter are constructed using variational techniques. Critical flows in pipes of uniform cross section are found by extremizing the cross-sectional energy at constrained value of the cross-sectional helicity and the axial how rate. Discrete classes of hows are found explicitly, parameterized by the radius and the value of the cross-sectional helicity. These flows are steady Beltrami-type hows and are independent of the axial co-ordinate. Solution branches can show the phenomenon of flow-reversal at the central axis of the pipe: the axial velocity becomes negative with increasing radius. An approximation for swirling flows in a pipe with varying circular cross section is constructed in a quasi-homogeneous way as a succession in the axial direction of critical flows in uniform pipes. With the changing value of the radius, the value of the cross-sectional helicity is determined by the requirement that the approximation satisfies the correct energy and flux conservation. It will be shown that these requirements are equivalent to satisfying the necessary solvability conditions that arise from a perturbation analysis around the quasi-homogeneous approximation. The approximate flows have the property that in lowest order approximation the local cross-sectional energy density increases linearly with R, while the local cross-sectional helicity density is constant