Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’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 equationBernoulli’s equation and momentum 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.

### 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 = C1
P ρ-1= C

Let us differentiate the above equation and we will have following equation as mentioned here.

d (P ρ-1) = 0
- P ρ-2d ρ + ρ-1dP = 0

We will now divide the above equation with ρ-1 and we will have

- P ρ-1d ρ + dP = 0
P ρ-1d 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, 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