Till now we were discussing the various concepts and
equations such as continuity
equation, Euler
equation, Bernoulli’s
equation and momentum
equation for incompressible fluid flow. In same way we have also
discussed above equations for compressible fluid flow.

We have already seen the derivation of

**,***continuity equation***,***Bernoulli’s equation***and velocity of sound in an isothermal process for compressible fluid flow in our previous posts.***momentum equation*
We will start here our discussion about the
compressible fluid flow with the derivation of expression for velocity of sound
in an adiabatic process.

### Expression for velocity of sound in adiabatic process

Before understanding the process to
derive the expression for velocity of sound in isothermal process, we must have
to study our previous post which shows the derivation of velocity of sound wave in a fluid
and velocity of sound in terms of bulk modulus.

For an adiabatic process, heat must be constant.

PV

*= Constant*^{γ}
P/ ρ

*= Constant*^{γ }
P/ ρ

*= Constant = C*^{γ }_{1 }
P ρ

^{-γ }= C_{1 }
Let us differentiate the above equation and we will
have following equation as mentioned here.

d (P ρ

^{-γ}) = 0
P (-γ) ρ

^{-γ-1}d ρ + ρ^{-γ }dP = 0
We will now divide the above equation with ρ

^{-γ}and we will have
- γ P ρ

^{-1}d ρ + dP = 0
γ P ρ

^{-1}d ρ = dP
γ P/ρ = dP /d ρ

dP /d ρ = γ RT

Let us recall the

*expression for the velocity of sound wave in a fluid*and we can write the above equation as mentioned here.
Where,
C is the velocity of sound

Therefore we will have following equation, as
mentioned here, which shows the expression for velocity of sound in adiabatic process.

Further we will go ahead to find out the basic concept of stagnation properties i.e. stagnation pressure, stagnation temperature and stagnation density in the subject of fluid mechanics, with the
help of our next post.

Do you have any suggestions? Please write in comment
box.

### Reference:

Fluid mechanics, By R. K. Bansal

Image courtesy: Google

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