In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow.
is the speed of sound, determined by the equation of state of the fluid and conservation of energy.
For polytropic gases, we have
( γ − 1 ) =
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.
[2][3] 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
are with the equation of state
acting as third equation.
is the stagnation enthalpy,
is the magnitude of the velocity vector and
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
as dependent variables, we use a coordinate transform such that
become new dependent variables.
Similarly the velocity potential is replaced by a new function (Legendre transformation)[4] such then its differential is
, therefore Introducing another coordinate transformation for the independent variables from
= v cos θ
= v sin θ
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,
is the speed of sound.
Using the Bernoulli's equation we find where