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. It is

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

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

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

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 $2h_{0}$ is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.

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 $(x,y)$ involving the variables fluid velocity $(v_{x},v_{y})$, specific enthalpy $h$ and density $\rho$ are

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

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

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

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

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

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

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

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

The continuity equation in the new coordinates become

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

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

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

where $c=c(v)$. Hence, we have

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

See also

• Euler–Tricomi equation