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

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 $$d\Phi = xdv_x + y dv_y$$, 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{align} 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{align} $$

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


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

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.$$