Bernoulli equation and negative pressure

[sinasahand]

Homework Statement:

bernoulli equation

Relevant Equations:

bernoulli equation and negetive pressure

In one pipeline with pressure P1 area A1 decrease to A2 we want to find P2 in area A2
we have bernoulli equation
p1+1/2 ρv^2=p2+1/2ρv^2
with low of conservation of mass A1V1=A2V2 that we can write V2=A1/A2 V1
if we keep in bernouli equation we have
P2=P1+1/2V2(1-(A1/A2)^2)
my quation is this if A2 goes to zero what will happen P2 will be negetive and -∞? How this will be correct

 

[TSny]

It's not clear if you are asking about a homework question. But, Bernoulli's equation can break down in certain situations. For example, the phenomenon of "cavitation" can occur if the pressure of the moving fluid is reduced too much. See for example

 

[kuruman]

The form of the continuity equation that I prefer is $Av=\beta$ where $\beta$ is the constant flow rate $\frac{dV}{dt}$. Then Bernoulli's equation can be written as$$p_2-p_1=\Delta P=\frac{1}{2}\rho \beta^2\left(\frac{1}{A_1^2}-\frac{1}{A_2^2}\right).$$ Note that everything on the right hand side is constant and independent of the speed. When $A_2<A_1$, $\Delta P$ is negative which means that the pressure in pipe 1 is greater than in pipe 2. The opposite is true when $A_2>A_1.$ That's all.

 

[TSny]

$$p_2-p_1=\Delta P=\frac{1}{2}\rho \beta^2\left(\frac{1}{A_1^2}-\frac{1}{A_2^2}\right).$$ Note that everything on the right hand side is constant and independent of the speed. When $A_2<A_1$, $\Delta P$ is negative which means that the pressure in pipe 1 is greater than in pipe 2. The opposite is true when $A_2>A_1.$ That's all.

Yes. But if $A_2$ is small enough compared to $A_1$, then the equation predicts that $p_2$ will be negative. I think this is what the OP was concerned about.