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A mapping realizing a transformation of certain differential equations of mathematical physics to their linear form.
 
A mapping realizing a transformation of certain differential equations of mathematical physics to their linear form.
  
The [[Bernoulli integral|Bernoulli integral]] and the [[Continuity equation|continuity equation]] of a plane-parallel potential stationary motion of a barotropic gas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475101.png" />,
+
The [[Bernoulli integral|Bernoulli integral]] and the [[Continuity equation|continuity equation]] of a plane-parallel potential stationary motion of a barotropic gas $  ( \rho = F( p)) $,
 +
 
 +
$$
 +
\rho  = \rho _ {0} \left (
 +
1 -
 +
\frac{u  ^ {2} + v  ^ {2} }{2 \alpha }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475102.png" /></td> </tr></table>
+
\right )  ^  \beta  ,\ \
 +
 
 +
\frac{\partial  \rho u }{\partial  x }
 +
+
 +
 
 +
\frac{\partial  \rho v }{\partial  y }
 +
  = 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475103.png" /></td> </tr></table>
+
$$
 +
\alpha  = \
 +
 
 +
\frac{c  ^ {2} }{\gamma - 1 }
 +
,\ \
 +
\beta  = \
 +
 
 +
\frac{1}{\gamma - 1 }
 +
\ \
 +
( c  \textrm{ is }  \textrm{ the }  \textrm{ velocity }  \textrm{ of } \
 +
\textrm{ sound }  \textrm{ for }  \rho = \rho _ {0} ),
 +
$$
  
 
lead to the equation
 
lead to the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475104.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac \partial {\partial  x }
 +
\left [ \left (
 +
1 -
 +
\frac{v  ^ {2} }{2 \alpha }
 +
 
 +
\right )  ^  \beta  u \right ] +
 +
 
 +
\frac \partial {\partial  y }
 +
\left [ \left (
 +
1 -  
 +
\frac{v  ^ {2} }{2 \alpha }
 +
 
 +
\right )  ^  \beta  v \right ]  = 0,
 +
$$
  
 
which is used for determining the velocity potential
 
which is used for determining the velocity potential
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475105.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{\partial  \phi }{\partial  x }
 +
,\ \
 +
=
 +
\frac{\partial  \phi }{\partial  y }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475107.png" /> are the velocity components. By introducing new independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h0475109.png" /> equal to the slope of the angle made by the velocity vector with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751010.png" />-axis, equation
+
where $  u $
 +
and $  v $
 +
are the velocity components. By introducing new independent variables $  \tau = v  ^ {2} / 2 \alpha $
 +
and $  \theta $
 +
equal to the slope of the angle made by the velocity vector with the $  x $-
 +
axis, equation
  
 
is reduced to linear form:
 
is reduced to linear form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751011.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac \partial {\partial  \tau }
 +
\left [
 +
 
 +
\frac{2 \tau ( 1 - \tau ) ^ {\beta + 1 } }{1 - ( 2 \beta + 1) \tau }
 +
 
 +
\frac{\partial  \phi }{\partial  \tau }
 +
 
 +
\right ] +
 +
 
 +
\frac{( 1 - \tau )  ^  \beta  }{2 \tau }
 +
 
 +
\frac{\partial  ^ {2} \phi }{\partial  \theta  ^ {2} }
 +
  = 0.
 +
$$
  
 
This is the first hodograph transformation, or the Chaplygin transformation. The second Chaplygin transformation is obtained by applying the tangential [[Legendre transform|Legendre transform]]. The function
 
This is the first hodograph transformation, or the Chaplygin transformation. The second Chaplygin transformation is obtained by applying the tangential [[Legendre transform|Legendre transform]]. The function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751012.png" /></td> </tr></table>
+
$$
 +
\Phi  = x
  
is selected as the new unknown; it is expressed in terms of new independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751014.png" />, which replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751016.png" /> by the formulas
+
\frac{\partial  \phi }{\partial  x }
 +
+
 +
y
 +
\frac{\partial  \phi }{\partial  y }
 +
- \phi
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751017.png" /></td> </tr></table>
+
is selected as the new unknown; it is expressed in terms of new independent variables  $  u $
 +
and  $  v $,
 +
which replace  $  x $
 +
and  $  y $
 +
by the formulas
 +
 
 +
$$
 +
=
 +
\frac{\partial  \phi }{\partial  x }
 +
,\ \
 +
=
 +
\frac{\partial  \phi }{\partial  y }
 +
.
 +
$$
  
 
The equation
 
The equation
Line 35: Line 131:
 
assumes a linear form:
 
assumes a linear form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751018.png" /></td> </tr></table>
+
$$
 +
\left [ 1 -
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047510/h04751019.png" /></td> </tr></table>
+
\frac{v  ^ {2} }{2 \alpha }
 +
-
 +
 
 +
\frac \beta  \alpha
 +
v  ^ {2}
 +
\right ]
 +
 
 +
\frac{\partial  ^ {2} \Phi }{\partial  u  ^ {2} }
 +
+
 +
 
 +
\frac{2 \beta } \alpha
 +
u v
 +
 
 +
\frac{\partial  ^ {2} \Phi }{\partial  u \partial  v }
 +
+
 +
$$
 +
 
 +
$$
 +
+
 +
\left [ 1 -
 +
\frac{v  ^ {2} }{2 \alpha }
 +
- {
 +
\frac \beta  \alpha
 +
}
 +
u  ^ {2} \right ]
 +
\frac{\partial  ^ {2} \Phi }{\partial  v  ^ {2} }
 +
  = 0.
 +
$$
  
 
Hodograph transforms are employed in solving problems in the theory of flow and of streams of gases flowing around curvilinear contours.
 
Hodograph transforms are employed in solving problems in the theory of flow and of streams of gases flowing around curvilinear contours.
Line 43: Line 167:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chaplygin,  "On gas-like structures" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Kochin,  I.A. Kibel',  N.V. Roze,  "Theoretical hydrodynamics" , Interscience  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chaplygin,  "On gas-like structures" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Kochin,  I.A. Kibel',  N.V. Roze,  "Theoretical hydrodynamics" , Interscience  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Curle,  H.J. Davies,  "Modern fluid dynamics" , '''1–2''' , v. Nostrand-Reinhold  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Curle,  H.J. Davies,  "Modern fluid dynamics" , '''1–2''' , v. Nostrand-Reinhold  (1971)</TD></TR></table>

Latest revision as of 22:10, 5 June 2020


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 $ ( \rho = F( p)) $,

$$ \rho = \rho _ {0} \left ( 1 - \frac{u ^ {2} + v ^ {2} }{2 \alpha } \right ) ^ \beta ,\ \ \frac{\partial \rho u }{\partial x } + \frac{\partial \rho v }{\partial y } = 0, $$

where

$$ \alpha = \ \frac{c ^ {2} }{\gamma - 1 } ,\ \ \beta = \ \frac{1}{\gamma - 1 } \ \ ( c \textrm{ is } \textrm{ the } \textrm{ velocity } \textrm{ of } \ \textrm{ sound } \textrm{ for } \rho = \rho _ {0} ), $$

lead to the equation

$$ \frac \partial {\partial x } \left [ \left ( 1 - \frac{v ^ {2} }{2 \alpha } \right ) ^ \beta u \right ] + \frac \partial {\partial y } \left [ \left ( 1 - \frac{v ^ {2} }{2 \alpha } \right ) ^ \beta v \right ] = 0, $$

which is used for determining the velocity potential

$$ u = \frac{\partial \phi }{\partial x } ,\ \ v = \frac{\partial \phi }{\partial y } , $$

where $ u $ and $ v $ are the velocity components. By introducing new independent variables $ \tau = v ^ {2} / 2 \alpha $ and $ \theta $ equal to the slope of the angle made by the velocity vector with the $ x $- axis, equation

is reduced to linear form:

$$ \frac \partial {\partial \tau } \left [ \frac{2 \tau ( 1 - \tau ) ^ {\beta + 1 } }{1 - ( 2 \beta + 1) \tau } \frac{\partial \phi }{\partial \tau } \right ] + \frac{( 1 - \tau ) ^ \beta }{2 \tau } \frac{\partial ^ {2} \phi }{\partial \theta ^ {2} } = 0. $$

This is the first hodograph transformation, or the Chaplygin transformation. The second Chaplygin transformation is obtained by applying the tangential Legendre transform. The function

$$ \Phi = x \frac{\partial \phi }{\partial x } + y \frac{\partial \phi }{\partial y } - \phi $$

is selected as the new unknown; it is expressed in terms of new independent variables $ u $ and $ v $, which replace $ x $ and $ y $ by the formulas

$$ u = \frac{\partial \phi }{\partial x } ,\ \ v = \frac{\partial \phi }{\partial y } . $$

The equation

assumes a linear form:

$$ \left [ 1 - \frac{v ^ {2} }{2 \alpha } - \frac \beta \alpha v ^ {2} \right ] \frac{\partial ^ {2} \Phi }{\partial u ^ {2} } + \frac{2 \beta } \alpha u v \frac{\partial ^ {2} \Phi }{\partial u \partial v } + $$

$$ + \left [ 1 - \frac{v ^ {2} }{2 \alpha } - { \frac \beta \alpha } u ^ {2} \right ] \frac{\partial ^ {2} \Phi }{\partial v ^ {2} } = 0. $$

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)
How to Cite This Entry:
Hodograph transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodograph_transform&oldid=47241
This article was adapted from an original article by L.N. Sretenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article