Chaplygin's equation explained

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

(\partial ^{2} \Phi)/(+ \partial \theta ^{2}) (v ^{2})/(1- v ^{2/c} ^{2}) (\partial ^{2} \Phi)/(\partial v ^{2}) + v (\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(\partial ^{2} \Phi)/(\partial \theta ^{2}) + v ^{2 (2 h ^{2} _{0- v})/(2 h ^{2 /(\gamma -1 )} _{0- (\gamma +1 )v})} (\partial ^{2} \Phi)/(\partial v ^{2}) + v (\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

2 h _{0} is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.^{[2] [3]}

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} (\partial)/(\partial x) (\rho v _{x)} + (\partial)/(\partial y) (\rho v _{y)} & =0 ,\\ h + (1)/(2) v ^{2} & = h _{o. \end{align}} \rho = \rho(s,h) acting as third equation. Here h _{o} is the stagnation enthalpy, v ^{2} = v ^{2} _{x} + v ^{2} _{y} 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)^[4] \Phi = xv _{x} + yv _{y} - \phi

such then its differential is

d\Phi = xdv _{x} + y dv _{y} , thereforex = (\partial \Phi)/(\partial v _{x}) , y = (\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 (\partial \Phi)/(- \partial v) (\sin\theta)/(v) (\partial \Phi)/(\partial \theta) ,\\ y & = \sin\theta (\partial \Phi)/(+ \partial v) (\cos\theta)/(v) (\partial \Phi)/(\partial \theta) ,\\ \phi & = - \Phi + v (\partial \Phi)/(\partial v) . \end{align}

The continuity equation in the new coordinates become

(d(\rho v))/(\left( dv) (\partial \Phi)/(\partial v) + (1)/(v) (\partial ^{2} \Phi)/(\partial \theta ^{2}) \right) + \rho v (\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(d(\rho v))/(d v) = \rho \left( 1- (v ^{2})/(c ^{2}) \right)

where

c = c(v) . Hence, we have(\partial ^{2} \Phi)/(+ \partial \theta ^{2}) (v ^{2})/(1- (v ^{2})/(c ^{2})) (\partial ^{2} \Phi)/(\partial v ^{2}) + v (\partial \Phi)/(\partial v) =0 .

See also

• Euler–Tricomi equation

Notes and References

1. Chaplygin, S. A. (1902). On gas streams. Complete collection of works.(Russian) Izd. Akad. Nauk SSSR, 2.
2. Sedov, L. I., (1965). Two-dimensional problems in hydrodynamics and aerodynamics. Chapter X
3. Von Mises, R., Geiringer, H., & Ludford, G. S. S. (2004). Mathematical theory of compressible fluid flow. Courier Corporation.
4. Book: Landau. L. D.. Lev Landau. Lifshitz. E. M.. Evgeny Lifshitz. Fluid Mechanics. 2. 1982. Pergamon Press. 432.