Concepts inModeling ice dynamics as a thin-film Stefan problem
Stefan problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem (also Stefan task) is a particular kind of boundary value problem for a partial differential equation (PDE), adapted to the case in which a phase boundary can move with time.
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Icicle
An icicle is a spike of ice formed when water dripping or falling from an object freezes. Typically, icicles will form when ice or snow is melted by either sunlight or some other heat source (such as heat leaking from the interior of a heated building), and the resulting melted water runs off into an area where the ambient temperature is below the freezing point of water (0 °C/32 °F), causing the water to refreeze. Over time continued water runoff will cause the icicle to grow.
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Ice
Ice is water frozen into the solid state. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions. The addition of other materials such as soil may further alter the appearance. Ice appears in nature in forms of snowflakes, hail, icicles, glaciers, pack ice, and entire polar ice caps. It is an important component of the global climate, and plays an important role in the water cycle.
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Freezing
Freezing or solidification is a phase change in which a liquid turns into a solid when its temperature is lowered below its freezing point. The reverse process is melting. All known liquids, except liquid helium, freeze when the temperature is lowered enough. Liquid helium remains liquid at atmospheric pressure even at absolute zero, and can be solidified only under pressure.
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Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R ¿ for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
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Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions¿constants, one variable x, elementary operations of arithmetic (+ ¿ × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
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