# Bernoulli equation and negative pressure

• sinasahand
In summary, the conversation discusses the use of Bernoulli's equation to find the pressure (P2) in area A2 when the area decreases from A1 to A2. The equation can break down in certain situations, such as when A2 goes to zero, resulting in a negative pressure. This phenomenon is known as cavitation and can occur if the pressure of the moving fluid is reduced too much.

#### 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

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

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.

kuruman said:
$$p_2-p_1=\Delta P=\frac{1}{2}\rho \beta^2\left(\frac{1}{A_1^2}-\frac{1}{A_2^2}\right).$$ 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.

## What is the Bernoulli equation?

The Bernoulli equation is a fundamental principle in fluid mechanics that describes the relationship between pressure, velocity, and elevation in a fluid. It states that in an ideal, incompressible fluid, the total energy of the fluid remains constant along a streamline.

## How does the Bernoulli equation relate to negative pressure?

The Bernoulli equation includes a term for pressure, and this pressure can be positive or negative. Negative pressure is usually associated with a decrease in fluid velocity, such as when a fluid flows through a constriction. This decrease in velocity leads to an increase in pressure, which can appear as a negative value in the Bernoulli equation.

## Why is negative pressure important in fluid dynamics?

Negative pressure plays a crucial role in many fluid dynamics phenomena, such as lift in airfoils and the flow of gases through pipes. In these cases, negative pressure gradients can cause fluids to accelerate, leading to important engineering applications.

## Is negative pressure always a desirable outcome in fluid dynamics?

No, negative pressure can also have negative consequences in fluid dynamics. For example, negative pressure gradients can cause cavitation, which is the formation and collapse of bubbles in a liquid due to low pressure. This can damage equipment and decrease the efficiency of fluid systems.

## How is the Bernoulli equation used in real-world applications?

The Bernoulli equation is used in many areas of science and engineering, including aerodynamics, hydraulics, and meteorology. It is often used to calculate pressure differences and fluid velocities in pipes, pumps, and airfoils. It also helps scientists understand and predict the behavior of fluids in various situations.