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

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

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

$\frac{{\partial }^2\mathrm{\Phi }}{\partial {\theta }^2}+v^2\frac{2h_0-v^2}{2h_0-(\gamma +1)v^2/(\gamma -1)}\frac{{\partial }^2\mathrm{\Phi }}{\partial v^2}+v\frac{\partial \mathrm{\Phi }}{\partial v}=0.$

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

$\begin{array}{rl}\frac{\partial }{\partial x}(\rho v_x)+\frac{\partial }{\partial y}(\rho v_y)& =0,\\ h+\frac{1}{2}{v}^2& =h_o.\end{array}$

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 $\varphi$ exists and its differential is simply $d\varphi =v_{x}dx+v_{y}dy$. 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)

$\mathrm{\Phi }=x{v}_x+y{v}_y-\varphi$

such then its differential is $d\mathrm{\Phi }=xd{v}_x+yd{v}_y$, therefore

$x=\frac{\partial \mathrm{\Phi }}{\partial v_x},y=\frac{\partial \mathrm{\Phi }}{\partial v_y}.$

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

$\begin{array}{rl}x& =\cos \theta \frac{\partial \mathrm{\Phi }}{\partial v}-\frac{\sin \theta }{v}\frac{\partial \mathrm{\Phi }}{\partial \theta },\\ y& =\sin \theta \frac{\partial \mathrm{\Phi }}{\partial v}+\frac{\cos \theta }{v}\frac{\partial \mathrm{\Phi }}{\partial \theta },\\ \varphi & =-\mathrm{\Phi }+v\frac{\partial \mathrm{\Phi }}{\partial v}.\end{array}$

The continuity equation in the new coordinates become

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

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

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

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

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

See also

• Euler–Tricomi equation