Tschaplygin equation

The Tschaplygin equation , named after the Russian-Soviet aerodynamicist Sergei Alexejewitsch Tschaplygin , is an exactly linearized potential equation of a steady flat gas flow. The equation is given in polar coordinates of the hodograph plane . The flow is isentropic , without shock waves .

With the help of the Legendre transformation , the Tschaplygin equation gives:

${\displaystyle {\delta ^{2}\phi \over \delta \theta ^{2}}+{v^{2} \over 1-v^{2}/c^{2}}\cdot {\delta ^{2}\phi \over \delta v^{2}}+v{\delta \phi \over \delta v}=0}$

where the conjugate potential, the angle between the direction of the speed and the abscissa , the amount of the speed and the local speed of sound means. The solution of the non-linear potential equation has been reduced to the solution of a linear equation for the function , in which, however, the boundary conditions are not linear. ${\displaystyle \phi }$ ${\displaystyle \delta }$ ${\displaystyle v}$ ${\displaystyle c}$ ${\displaystyle \phi (c,\delta )}$

literature

Commons : Category: Chaplygin equation  - album with pictures, videos and audio files

• LD Landau, EM Lifschitz: Textbook of Theoretical Physics. - Volume VI. Hydrodynamics. Scientific publishing house Harry Deutsch, Frankfurt am Main 2007, ISBN 978-3-8171-1331-6 , pp. 563-567 ( limited preview in Google book search).
• Richard Lenk, Walter Gellert (Ed.): Brockhaus abc - Physics. Volume 2, Brockhaus, Leipzig 1972, p. 1590.