Hodograph transform
A mapping realizing a transformation of certain differential equations of mathematical physics to their linear form.
The Bernoulli integral and the continuity equation of a plane-parallel potential stationary motion of a barotropic gas ,
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where
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lead to the equation
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which is used for determining the velocity potential
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where and
are the velocity components. By introducing new independent variables
and
equal to the slope of the angle made by the velocity vector with the
-axis, equation
is reduced to linear form:
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This is the first hodograph transformation, or the Chaplygin transformation. The second Chaplygin transformation is obtained by applying the tangential Legendre transform. The function
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is selected as the new unknown; it is expressed in terms of new independent variables and
, which replace
and
by the formulas
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The equation
assumes a linear form:
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Hodograph transforms are employed in solving problems in the theory of flow and of streams of gases flowing around curvilinear contours.
References
[1] | S.A. Chaplygin, "On gas-like structures" , Moscow-Leningrad (1949) (In Russian) |
[2] | N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , Interscience (1964) (Translated from Russian) |
Comments
References
[a1] | N. Curle, H.J. Davies, "Modern fluid dynamics" , 1–2 , v. Nostrand-Reinhold (1971) |
Hodograph transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodograph_transform&oldid=17672