Chaplygin's equation

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

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

• Euler–Tricomi equation