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Note that subcritical and supercritical describe the stability of the outer lines of the pitchfork (dashed or solid, respectively) and are not dependent on which direction the pitchfork faces. For example, the negative of the first ODE above, x ˙ = x 3 − r x {\displaystyle {\dot {x}}=x^{3}-rx} , faces the same direction as the first picture ...
Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis. In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. [1]
Note the location of critical flow, subcritical flow, and supercritical flow. The energy equation used for open channel flow computations is a simplification of the Bernoulli Equation (See Bernoulli Principle ), which takes into account pressure head, elevation head, and velocity head.
One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas comes into a supercritical phase, and so cannot be liquefied by pressure alone.
Hydraulic jump in a rectangular channel, also known as classical jump, is a natural phenomenon that occurs whenever flow changes from supercritical to subcritical flow. In this transition, the water surface rises abruptly, surface rollers are formed, intense mixing occurs, air is entrained, and often a large amount of energy is dissipated.
Phase diagram of a generic fluid TQ diagram of heat introduction to a subcritical and transcritical cold source. In transcritical cycles, the pressure of the working fluid at the outlet of the pump is higher than the critical pressure, while the inlet conditions are close to the saturated liquid pressure at the given minimum temperature.
For example, “a fluid is considered to be ‘supercritical’ when its temperature and pressure exceed the temperature and pressure at the critical point”. In the studies of supercritical extraction, however, “supercritical fluid” is applied for a narrow temperature region of 1-1.2 T c {\displaystyle T_{c}} or T c {\displaystyle T_{c ...
Many pressurized gases are actually supercritical fluids. For example, nitrogen has a critical point of 126.2 K (−147 °C) and 3.4 MPa (34 bar). Therefore, nitrogen (or compressed air) in a gas cylinder above this pressure is actually a supercritical fluid. These are more often known as permanent gases.