Chaplygin's equation

In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow.^[1] It is

${\displaystyle {\frac {\partial ^{2}\Phi }{\partial \theta ^{2}}}+{\frac {v^{2}}{1-v^{2}/c^{2}}}{\frac {\partial ^{2}\Phi }{\partial v^{2}}}+v{\frac {\partial \Phi }{\partial v}}=0.}$

Here, ${\displaystyle c=c(v)}$ is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have ${\displaystyle c^{2}/(\gamma -1)=h_{0}-v^{2}/2}$, where ${\displaystyle \gamma }$ is the specific heat ratio and ${\displaystyle h_{0}}$ is the stagnation enthalpy, in which case the Chaplygin's equation reduces to

${\displaystyle {\frac {\partial ^{2}\Phi }{\partial \theta ^{2}}}+v^{2}{\frac {2h_{0}-v^{2}}{2h_{0}-(\gamma +1)v^{2}/(\gamma -1)}}{\frac {\partial ^{2}\Phi }{\partial v^{2}}}+v{\frac {\partial \Phi }{\partial v}}=0.}$

The Bernoulli equation (see the derivation below) states that maximum velocity occurs when specific enthalpy is at the smallest value possible; one can take the specific enthalpy to be zero corresponding to absolute zero temperature as the reference value, in which case ${\displaystyle 2h_{0}}$ is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.^[2][3]

Derivation

For two-dimensional potential flow, the continuity equation and the Euler equations (in fact, the compressible Bernoulli's equation due to irrotationality) in Cartesian coordinates ${\displaystyle (x,y)}$ involving the variables fluid velocity ${\displaystyle (v_{x},v_{y})}$, specific enthalpy ${\displaystyle h}$ and density ${\displaystyle \rho }$ are

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}(\rho v_{x})+{\frac {\partial }{\partial y}}(\rho v_{y})&=0,\\h+{\frac {1}{2}}v^{2}&=h_{o}.\end{aligned}}}$

with the equation of state ${\displaystyle \rho =\rho (s,h)}$ acting as third equation. Here ${\displaystyle h_{o}}$ is the stagnation enthalpy, ${\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}}$ is the magnitude of the velocity vector and ${\displaystyle s}$ is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy ${\displaystyle \rho =\rho (h)}$, which in turn using Bernoulli's equation can be written as ${\displaystyle \rho =\rho (v)}$.

Since the flow is irrotational, a velocity potential ${\displaystyle \phi }$ exists and its differential is simply ${\displaystyle d\phi =v_{x}dx+v_{y}dy}$. Instead of treating ${\displaystyle v_{x}=v_{x}(x,y)}$ and ${\displaystyle v_{y}=v_{y}(x,y)}$ as dependent variables, we use a coordinate transform such that ${\displaystyle x=x(v_{x},v_{y})}$ and ${\displaystyle y=y(v_{x},v_{y})}$ become new dependent variables. Similarly the velocity potential is replaced by a new function (Legendre transformation)^[4]

${\displaystyle \Phi =xv_{x}+yv_{y}-\phi }$

such then its differential is ${\displaystyle d\Phi =xdv_{x}+ydv_{y}}$, therefore

${\displaystyle x={\frac {\partial \Phi }{\partial v_{x}}},\quad y={\frac {\partial \Phi }{\partial v_{y}}}.}$

Introducing another coordinate transformation for the independent variables from ${\displaystyle (v_{x},v_{y})}$ to ${\displaystyle (v,\theta )}$ according to the relation ${\displaystyle v_{x}=v\cos \theta }$ and ${\displaystyle v_{y}=v\sin \theta }$, where ${\displaystyle v}$ is the magnitude of the velocity vector and ${\displaystyle \theta }$ is the angle that the velocity vector makes with the ${\displaystyle v_{x}}$-axis, the dependent variables become

${\displaystyle {\begin{aligned}x&=\cos \theta {\frac {\partial \Phi }{\partial v}}-{\frac {\sin \theta }{v}}{\frac {\partial \Phi }{\partial \theta }},\\y&=\sin \theta {\frac {\partial \Phi }{\partial v}}+{\frac {\cos \theta }{v}}{\frac {\partial \Phi }{\partial \theta }},\\\phi &=-\Phi +v{\frac {\partial \Phi }{\partial v}}.\end{aligned}}}$

The continuity equation in the new coordinates become

${\displaystyle {\frac {d(\rho v)}{dv}}\left({\frac {\partial \Phi }{\partial v}}+{\frac {1}{v}}{\frac {\partial ^{2}\Phi }{\partial \theta ^{2}}}\right)+\rho v{\frac {\partial ^{2}\Phi }{\partial v^{2}}}=0.}$

For isentropic flow, ${\displaystyle dh=\rho ^{-1}c^{2}d\rho }$, where ${\displaystyle c}$ is the speed of sound. Using the Bernoulli's equation we find

${\displaystyle {\frac {d(\rho v)}{dv}}=\rho \left(1-{\frac {v^{2}}{c^{2}}}\right)}$

where ${\displaystyle c=c(v)}$. Hence, we have

${\displaystyle {\frac {\partial ^{2}\Phi }{\partial \theta ^{2}}}+{\frac {v^{2}}{1-{\frac {v^{2}}{c^{2}}}}}{\frac {\partial ^{2}\Phi }{\partial v^{2}}}+v{\frac {\partial \Phi }{\partial v}}=0.}$

See also

• Euler–Tricomi equation

References

1. Chaplygin, S. A. (1902). On gas streams. Complete collection of works.(Russian) Izd. Akad. Nauk SSSR, 2.
2. Sedov, L. I., (1965). Two-dimensional problems in hydrodynamics and aerodynamics. Chapter X
3. Von Mises, R., Geiringer, H., & Ludford, G. S. S. (2004). Mathematical theory of compressible fluid flow. Courier Corporation.
4. Landau, L. D.; Lifshitz, E. M. (1982). Fluid Mechanics (2 ed.). Pergamon Press. p. 432.