Quantitatively-unexplained stationary waves or ridges often encircle icicles. Such waves form when roughly 0.1 mm-thick layers of water flow down the icicle. These waves typically have a wavelength of 1cm approximately independent of external temperature, icicle thickness, and the volumetric rate of water flow. In this paper we show that these waves can not be obtained by naive Mullins-Sekerka instability, but are caused by a quite new surface instability related to the thermal diffusion and hydrodynamic effect of thin water flow.

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The classical solution of bosonic d-brane in (d+1,1) space-time is studied. We work with light-cone gauge and reduce the problem into Chaplygin gas problem. The static equation is equivalent to vanishing of extrinsic mean curvature, which is similar to Einstein equation in vacuum. We show that the d-brane problem in this gauge is closely related to Plateau problem, and we give some non-trivial solutions from minimal surfaces. The solutions of d-1,d,d+1 spatial dimensions are obtained from d-dimensional minimal surfaces as solutions of Plateau problem. In addition we discuss on the relation to Hamiltonian-BRST formalism for d-branes.

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Particle diffusion in a two dimensional curved surface embedded in R3 is considered. In addition to the usual diffusion flow, we find a new flow with an explicit curvature dependence. New diffusion equation is obtained in ǫ (thickness of surface) expansion. As an example, the surface of elliptic cylinder is considered, and curvature dependent diffusion coefficient is calculated. 02.40.Hw, 02.40.Ma, 82.40.Ck I. MOTIVATIONThe particle motion on a given curved surface is old but interesting problem in wide range of physics. Especially the diffusion process of particles on such a manifold is still an open problem, and related to various kinds of phenomena.For example the motion of protein on cell membrane has great importance in biophysics. There are several research papers discussing on this problem. Some of them are treating this problem by using usual diffusion equation with curved coordinate, and discuss the curvature (Gauss curvature) dependence of its solution [1]. Other of them use the Langevin equation on curved surface and calculating the curvature dependence of diffusion coefficient [2]. The quantum mechanics of particle motion on such a curved manifold is also considered by many authors. This problem is usually explained by the Schroedinger equation with Laplace-Beltrami operator. However, when we treat the curved surface as embedded one in 3 dimensional Euclidean space, situation is changed and then we have a quantum potential term related to the curvature additional to the kinetic operator [3],[4], [5].Another example is in larger scale physics in which our consideration is devoted. Patterns of animal skins are well expressed by the reaction diffusion equation [6]. But the patterns are different for each parts even in one individual. For example, Char fish, the side part has white spot pattern, but the back part has labyrinth pattern. (For these two patterns, see for example [7].) One of the reasons might come from the curvature difference between side part and back part. If the diffusion is influenced by the curvature, this difference of patterns might be explained. Furthermore, the cross section of fish has form of ellipsoid and the surface can be approximated as the one of elliptic cylinder. In two dimensional space, we have only two kinds of curvature, one is Gauss curvature and other is mean curvature. Both are constructed from second fundamental tensor by taking determinant or trace. Gauss curvature can also be constructed only by * ogawanao@hit.ac.jp metric tensor and its derivatives, but this is not the case for the mean curvature. The elliptic cylinder, in which we have much interest, has zero Gauss curvature and nonzero mean curvature. Therefore to explain the pattern change of Char fish, solution of the diffusion equation should depend on mean curvature. This is impossible if we start from usual diffusion equation because it depends only on metric but not on second fundamental tensor. Therefore we need some new diffusion equation, which bring not only Gauss curvature but also mean curvat...