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

###
**Expression
for velocity of sound in isothermal 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 isothermal process, temperature must be
constant.

As we know the following equation, as mentioned
here, we will use this equation to derive the velocity of sound in an
isothermal process.

PV = mRT

PV/m = RT

P/ρ = RT

As we are discussing here the case of an isothermal
process and therefore the term RT will be constant and hence we can write the
above equation as mentioned here.

P/ρ = Constant = C

_{1}
P ρ

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

d (P ρ

^{-1}) = 0
- P ρ

^{-2}d ρ + ρ^{-1}dP = 0
We will now
divide the above equation with ρ

^{-1}and we will have
- P ρ

^{-1}d ρ + dP = 0
P ρ

^{-1}d ρ = dP
P/ ρ = dP /d ρ

dP /d ρ = 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 isothermal process.

Further we will go ahead to find out the velocity of sound in an adiabatic process, 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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