BERNOULLI’S EQUATION FOR COMPRESSIBLE FLUID FLOW

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.

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 continuity equation 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.

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 = C₁

P/ ρ = C₁

P / C₁ = 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.

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