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]