Differential equations

Hicks equation

In fluid dynamics, Hicks equation or sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after William Mitchinson Hicks, who derived it first in 1898. The equation was also re-derived by Stephen Bragg and William Hawthorne in 1950 and by Robert R. Long in 1953 and by Herbert Squire in 1956. The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842. The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation. Representing as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by , the stream function that defines the meridional motion can be defined as that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by where where is the total head, c.f. Bernoulli's Principle. and is the circulation, both of them being conserved along streamlines. Here, is the pressure and is the fluid density. The functions and are known functions, usually prescribed at one of the boundary. (Wikipedia).

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