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

[math]\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. }[/math]

Here, [math]\displaystyle{ c=c(v) }[/math] is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have [math]\displaystyle{ c^2/(\gamma-1) = h_0- v^2/2 }[/math], where [math]\displaystyle{ \gamma }[/math] is the specific heat ratio and [math]\displaystyle{ h_0 }[/math] is the stagnation enthalpy, in which case the Chaplygin's equation reduces to

[math]\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. }[/math]

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 [math]\displaystyle{ 2h_0 }[/math] 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 [math]\displaystyle{ (x,y) }[/math] involving the variables fluid velocity [math]\displaystyle{ (v_x,v_y) }[/math], specific enthalpy [math]\displaystyle{ h }[/math] and density [math]\displaystyle{ \rho }[/math] are

[math]\displaystyle{ \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} }[/math]

with the equation of state [math]\displaystyle{ \rho=\rho(s,h) }[/math] acting as third equation. Here [math]\displaystyle{ h_o }[/math] is the stagnation enthalpy, [math]\displaystyle{ v^2 = v_x^2 + v_y^2 }[/math] is the magnitude of the velocity vector and [math]\displaystyle{ s }[/math] is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy [math]\displaystyle{ \rho=\rho(h) }[/math], which in turn using Bernoulli's equation can be written as [math]\displaystyle{ \rho=\rho(v) }[/math].

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

[math]\displaystyle{ \Phi = xv_x + yv_y - \phi }[/math]

such then its differential is [math]\displaystyle{ d\Phi = xdv_x + y dv_y }[/math], therefore

[math]\displaystyle{ x = \frac{\partial \Phi}{\partial v_x}, \quad y = \frac{\partial \Phi}{\partial v_y}. }[/math]

Introducing another coordinate transformation for the independent variables from [math]\displaystyle{ (v_x,v_y) }[/math] to [math]\displaystyle{ (v,\theta) }[/math] according to the relation [math]\displaystyle{ v_x = v\cos\theta }[/math] and [math]\displaystyle{ v_y = v\sin\theta }[/math], where [math]\displaystyle{ v }[/math] is the magnitude of the velocity vector and [math]\displaystyle{ \theta }[/math] is the angle that the velocity vector makes with the [math]\displaystyle{ v_x }[/math]-axis, the dependent variables become

[math]\displaystyle{ \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} }[/math]

The continuity equation in the new coordinates become

[math]\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. }[/math]

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

[math]\displaystyle{ \frac{d(\rho v)}{d v} = \rho \left(1-\frac{v^2}{c^2}\right) }[/math]

where [math]\displaystyle{ c=c(v) }[/math]. Hence, we have

[math]\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. }[/math]

See also

• Euler–Tricomi equation