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.<ref>Chaplygin, S. A. (1902). On gas streams. Complete collection of works.(Russian) Izd. Akad. Nauk SSSR, 2.</ref> It is
- <math>
\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>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>c^2/(\gamma-1) = h_0- v^2/2</math>, where <math>\gamma</math> is the specific heat ratio and <math>h_0</math> is the stagnation enthalpy, in which case the Chaplygin's equation reduces to
- <math>
\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>2h_0</math> is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.<ref>Sedov, L. I., (1965). Two-dimensional problems in hydrodynamics and aerodynamics. Chapter X</ref><ref>Von Mises, R., Geiringer, H., & Ludford, G. S. S. (2004). Mathematical theory of compressible fluid flow. Courier Corporation.</ref>
DerivationEdit
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>(x,y)</math> involving the variables fluid velocity <math>(v_x,v_y)</math>, specific enthalpy <math>h</math> and density <math>\rho</math> are
- <math>
\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>\rho=\rho(s,h)</math> acting as third equation. Here <math>h_o</math> is the stagnation enthalpy, <math>v^2 = v_x^2 + v_y^2</math> is the magnitude of the velocity vector and <math>s</math> is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy <math>\rho=\rho(h)</math>, which in turn using Bernoulli's equation can be written as <math>\rho=\rho(v)</math>.
Since the flow is irrotational, a velocity potential <math>\phi</math> exists and its differential is simply <math>d\phi = v_x dx + v_y dy</math>. Instead of treating <math>v_x=v_x(x,y)</math> and <math>v_y=v_y(x,y)</math> as dependent variables, we use a coordinate transform such that <math>x=x(v_x,v_y)</math> and <math>y=y(v_x,v_y)</math> become new dependent variables. Similarly the velocity potential is replaced by a new function (Legendre transformation)<ref>Template:Cite book</ref>
- <math>\Phi = xv_x + yv_y - \phi</math>
such then its differential is <math>d\Phi = xdv_x + y dv_y</math>, therefore
- <math>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>(v_x,v_y)</math> to <math>(v,\theta)</math> according to the relation <math>v_x = v\cos\theta</math> and <math>v_y = v\sin\theta</math>, where <math>v</math> is the magnitude of the velocity vector and <math>\theta</math> is the angle that the velocity vector makes with the <math>v_x</math>-axis, the dependent variables become
- <math>
\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>\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>dh=\rho^{-1}c^2 d\rho</math>, where <math>c</math> is the speed of sound. Using the Bernoulli's equation we find
- <math>\frac{d(\rho v)}{d v} = \rho \left(1-\frac{v^2}{c^2}\right)</math>
where <math>c=c(v)</math>. Hence, we have
- <math>
\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>