Discrete Analog of the Burgers Equation

We propose the set of coupled ordinary differential equations dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.Comment: 6 pages, 5 figure

Evaporation and fluid dynamics of a sessile drop of capillary size

Theoretical description and numerical simulation of an evaporating sessile drop are developed. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. A shape of the rotationally symmetric drop is determined within the quasistationary approximation. Nonstationary effects in the diffusion of the vapor are also taken into account. Simulation results agree well with the data of evaporation rate measurements for the toluene drop. Marangoni forces associated with the temperature dependence of the surface tension, generate fluid convection in the sessile drop. Our results demonstrate several dynamical stages of the convection characterized by different number of vortices in the drop. During the early stage the street of vortices arises near a surface of the drop and induces a non-monotonic spatial distribution of the temperature over the drop surface. The initial number of near-surface vortices in the drop is controlled by the Marangoni cell size which is similar to that given by Pearson for flat fluid layers. This number quickly decreases with time, resulting in three bulk vortices in the intermediate stage. The vortices finally transform into the single convection vortex in the drop, existing during about 1/2 of the evaporation time.Comment: 23 pages, 12 figure

Evaporation and explosion of liquid drops on a heated surface

The literature pertinent to various aspects of drop evaporation on a heated surface is reviewed. Both the laser shadowgraphic and direct photographic methods are employed to study thermal stability and flow structures in evaporating drops in all heating regimes. It is revealed that four flow regions exist in stable and unstable type drops at low liquid-film type vaporization regime. As the surface temperature is raised, the flow regions reduce to two. In the nucleate-boiling type vaporization regime, the interfacial flow structure changes due to a reduction in the Marangoni number as well as the dielectric constant of the liquid. An evidence of bubble growth in the drops is disclosed. The micro explosion of drops is found to occur in the transition-boiling type heating range. No drop explosion takes place in the spheriodal vaporization regime except when the drop rolls on to a microscratch on the heating surface. It is concluded that the mechanisms for triggering drop explosion include the spontaneous nucleation and growth phenomena and the destabilization of film boiling.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47061/1/348_2004_Article_BF00266263.pd