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
Here, is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have , where is the specific heat ratio and is the stagnation enthalpy, in which case the Chaplygin's equation reduces to
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 is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.
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 involving the variables fluid velocity , specific enthalpy and density are
with the equation of state acting as third equation. Here is the stagnation enthalpy, is the magnitude of the velocity vector and is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy , which in turn using Bernoulli's equation can be written as .
Since the flow is irrotational, a velocity potential exists and its differential is simply . Instead of treating and as dependent variables, we use a coordinate transform such that and become new dependent variables. Similarly the velocity potential is replaced by a new function (Legendre transformation)
such then its differential is , therefore
Introducing another coordinate transformation for the independent variables from to according to the relation and , where is the magnitude of the velocity vector and is the angle that the velocity vector makes with the -axis, the dependent variables become
The continuity equation in the new coordinates become
For isentropic flow, , where is the speed of sound. Using the Bernoulli's equation we find
where . Hence, we have
- ^ Chaplygin, S. A. (1902). On gas streams. Complete collection of works.(Russian) Izd. Akad. Nauk SSSR, 2.
- ^ Sedov, L. I., (1965). Two-dimensional problems in hydrodynamics and aerodynamics. Chapter X
- ^ Von Mises, R., Geiringer, H., & Ludford, G. S. S. (2004). Mathematical theory of compressible fluid flow. Courier Corporation.
- ^ Landau, L. D.; Lifshitz, E. M. (1982). Fluid Mechanics (2 ed.). Pergamon Press. p. 432.