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A polytropic process is a thermodynamic process that obeys the relation: = where p is the pressure , V is volume , n is the polytropic index , and C is a constant. The polytropic process equation describes expansion and compression processes which include heat transfer.
The normalized density as a function of scale length for a wide range of polytropic indices. In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form = (+) / = + /, where P is pressure, ρ is density and K is a constant of proportionality. [1]
This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions. [1]
The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.
An example of a cycle of idealized thermodynamic processes which make up the Stirling cycle. A quasi-static thermodynamic process can be visualized by graphically plotting the path of idealized changes to the system's state variables. In the example, a cycle consisting of four quasi-static processes is shown.
One common class of barotropic model used in astrophysics is a polytropic fluid. Typically, the barotropic assumption is not very realistic. In meteorology, a barotropic atmosphere is one that for which the density of the air depends only on pressure, as a result isobaric surfaces (constant-pressure surfaces) are also constant-density surfaces.
An isentropic process is depicted as a vertical line on a T–s diagram, whereas an isothermal process is a horizontal line. [2] Example T–s diagram for a thermodynamic cycle taking place between a hot reservoir (T H) and a cold reservoir (T C). For reversible processes, such as those found in the Carnot cycle:
Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: + (+) = In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers' equation : ∂ u ∂ t + u ∂ u ∂ x = 0 ...