bernoulli equation

bernoulli equation

BERNUILLI ECUATION

CONTINUITY ECUATION :

To understand the bernouilli law, first we have to underdand the continuity ecuation.

The continuity law just says that the mass per time unit is constant, which means that not depends on the tube section. mass (M)/time( t) = constant. 

What changes from M1 to M2  of the picture above? 

On one hand we know that density ( ρ)= mass (M)/volume (V) , or  M=ρ * V .  On The other hand we know that Volume( V) = Section ( S) * Lenght(l), so the density ecuation can be written M=ρ*S*l.

The continuity law ( M /t= constant) can be written  (ρ*S*l)/t = constant, and                  Speed ( v)=  lenght(l)/time (t) . The continuity ecuation finish up as ρ*S*v= constant.

I know this explanation is very boring , but is the only way to demostrate what changes from M1 to M2 is the speed of the fluid . If surface decreases the speed increases

continuity ecuation

On the following video you will find the explanation :

BERNUILLI ECUATION :

Bernouilli realizes that there are two kinds of pressures , static pressure  ( force/section) and dynamic pressure which depends on the speed ( 1/2*ρ*v).  So he stated that total pressure ( Pt) is equal to Static pressure + dynamic pressure. 

So his main discovery was that the total pressure keeps the same value what changes is the static pressure or dynamic pressure. 

On the following video you will have it bettter explained:

 

It is the basement to explain the lift force, anemometer …..

Until this point we have explained the bernouilli theorem for an uncompressible fluid ( Mach number < 0.5 , so density is constant ), but for compressible fluids , density is not constant, the ecuation changes. What we have studied it is just one case of what Bernoilli stated, as its original formula was ΔP+ρVδv=0 . 

The person who solved this ecuation por Compressible fluids called Saint Venant, who realizes that in this particular case the bernuilli ecuations depends on the Mach Number  (M).

 

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