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Friday, 7 December 2018

CONTINUITY EQUATION FOR COMPRESSIBLE FLUID FLOW

We were discussing the basics of Boundary layer theorylaminar boundary layerturbulent boundary layerboundary layer thickness, displacement thickness and momentum thicknessenergy thickness and drag force &lift force, in the subject of fluid mechanics, in our recent posts. 

After understanding the fundamentals of drag and lift coefficient, we will discuss now a new topic i.e. Compressible 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 incompressible flow. In same way, we will start here our discussion about the compressible fluid flow with continuity equation. 

Compressible flow is basically defined as the flow where fluid density could be changed during flow. 

Continuity equation for compressible fluid flow 

As we know that continuity equation is based on the law of conservation of mass. 

According to the law of conservation of mass, matter could not be created and nor destroyed. In simple words, matter or mass will be constant. 

Therefore change in mass will be zero. Here, we will use this concept to find out the equation of continuity for compressible fluid flow. 

Let us write the equation now for conservation of mass for compressible fluid flow. Let us assume that fluid flow is one dimensional steady flow. 

Mass per second = Constant 
ρ AV = Constant 

Where, 
ρ = Density of fluid flow 
A = Area of cross-section 
V = Velocity of fluid flow 
Change of mass per second = 0 
d (ρ AV) = 0 
ρ d (AV) + AV dρ = 0 
ρ A dV +  ρ V dA + AV dρ = 0 

Now we will divide the above equation by term ρ A V 
Above equation is known as the continuity equation of compressible fluid flow. 

Further we will go ahead to find out the Bernoulli’s 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 

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