We were discussing the basics of drag
force &lift force and drag and lift coefficient in the subject of fluid
mechanics, in our recent posts.

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###

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We will discuss now a new topic i.e. compressible fluid
flow, in the subject of fluid mechanics, with the help of this post. Before
going in detail discussion about compressible flow, we must have basic
knowledge about various equations associated with the compressible flow.

Till now we were discussing the various concepts and
equations such as continuity equation Euler equation, Bernoulli’s equation and
momentum equation for in-compressible fluid flow. In same way we will have to
discuss above equations for compressible fluid flow too.

We have already seen the derivation of

**for compressible fluid flow in our previous post. We will start here our discussion about the compressible fluid flow with the derivation of Bernoulli’s equation for compressible fluid flow.***continuity equation*
Compressible flow is basically defined as the flow
where fluid density could be changed during flow.

###
**Bernoulli’s
equation for compressible fluid flow **

We will derive the Bernoulli’s equation for
compressible fluid flow with the help of Euler’s equation.

So, let us recall
the Euler’s equation as mentioned here.

In case of in-compressible fluid flow, the density of
fluid will be constant and therefore the integral of dp/ρ will be equivalent to
the P/ρ.

We are interested here for compressible fluid flow
and therefore the density of fluid will not be constant and therefore the
integral of dp/ρ will not be equivalent to the P/ρ.

In case of compressible fluid flow, the value of ρ
will be changing and hence value of p will also be changing. Change in ρ and p
will be dependent over the types of process during compressible fluid flow.

We will now consider the various types of processes
where pressure and temperature will be related with each other. We will secure
the value of ρ in terms of p with the help of equations of these processes and
we will use the value of ρ in above equation to secure the result of integral
of dp/ρ.

Bernoulli’s equation for isothermal process and for adiabatic
process will be different. Let us first consider a basic process i.e.
isothermal process.

###
**Bernoulli’s
equation for compressible fluid for an isothermal process**

We will secure here the value of ρ in terms of p with
the help of following equation of isothermal process.

PV = mRT, where temperature T will be constant

PV/m = RT = Constant

P/ ρ = Constant = C

_{1}
P/ ρ = C

_{1}
P / C

_{1}= ρ
Above equation will be the Bernoulli’s equation for
compressible fluid for an isothermal process. We can also write the Bernoulli’s
equation for compressible fluid for an isothermal process for two points 1 and
2 as mentioned here.

###
**Bernoulli’s
equation for compressible fluid for an adiabatic process **

We will secure here the value of ρ in terms of p with
the help of following equation of adiabatic process.

Above equation will be the Bernoulli’s equation for
compressible fluid for an adiabatic process. We can also write the Bernoulli’s
equation for compressible fluid for an adiabatic process for two points 1 and 2
as mentioned here.

Further we will go ahead to find out the momentum equation for compressible fluid flow, 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

Thanks for that, which is the clearest derivation on the Internet that I have found.

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