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

∂

2

Φ

∂

θ

2

+

v

2

1

−

v

2

/

c

2

∂

2

Φ

∂

v

2

+

v

∂

Φ

∂

v

=

0.

{\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.}

Here,

c

=

c

(

v

)

{\displaystyle 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

/

(

γ

−

1

)

=

h

0

−

v

2

/

2

{\displaystyle c^{2}/(\gamma -1)=h_{0}-v^{2}/2}

, where

γ

{\displaystyle \gamma }

is the specific heat ratio and

h

0

{\displaystyle h_{0}}

is the stagnation enthalpy, in which case the Chaplygin's equation reduces to

∂

2

Φ

∂

θ

2

+

v

2

2

h

0

−

v

2

2

h

0

−

(

γ

+

1

)

v

2

/

(

γ

−

1

)

∂

2

Φ

∂

v

2

+

v

∂

Φ

∂

v

=

0.

{\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.}

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

{\displaystyle 2h_{0}}

is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.

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