Bernoulli Equation

Bernoulli Equation



The beginning of the principles of flight subject are some physical laws that a flight student will use all over the course. In this post we are talking about the basement of the aerodynamics.




To understand the bernoulli law, first we have to understand 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? 


continuity ecuation



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.

To conclude: If surface decreases the speed increases











Bernoulli realized that there are two kinds of pressures , static pressure  ( force/section) and dynamic pressure which depends on the speed ( 1/2*ρ*v^2).  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, while the static pressure or dynamic pressure changes. 

As you can see in the picture above if the surface is small, due to the continuity laws, the speed increases leading in a reduction static pressure, and increase of dynamic pressure.






 The bernoulli theorem for an uncompressible fluid (Mach number < 0.5 , so density is constant), but for compressible fluids , density is not constant, so 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 bernoulli ecuations depends on the Mach Number  (M).





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