Why the pressure is small at point 2? [closed]

Question

In the given venturimeter, area of cross-section is small at 2 so pressure should be large as it is equal to force/area, but it is given that the pressure is low at 2, how?

Answer 1

It is important to remember that this system is in steady state. So after the water started flowing the system had some time to settle and this is the end result after everything has settled down. This might be obvious, but it has helped me understand these kinds of problems in the past.

We can reason that this is the case using only three things: (1) conservation of mass (no fluid is lost in this system), (2) $F=ma$ and (3) forces in fluid are caused by pressure gradients $F\propto-\frac{dp}{dx}$

So start with (1). We first have to ask the question: how much fluid volume is passing every second through an imaginary surface located at 1? The answer is $Av$ where $A$ is the cross-sectional area at position 1 and $v$ is the (average) velocity across this area. How do we know this? Imagine you time a stopwatch at $t=0$, you wait for $\Delta t$ seconds and then you time the stopwatch again. Which part of the fluid has passed through the surface? The answer is all the fluid which is within $v\Delta t$ to the left of our surface (see picture). The fluid has moved $v\Delta t$ to the right, so everything which is less than that distance from the surface has had a chance to pass through.

In steady state, the amount of fluid that passes past a certain point in the pipe has to be equal everywhere. If at point 1 more fluid passes than at point 2, there must be fluid leaking or getting destroyed between those two points. This implies that $A_1v_1=A_2v_2$. Since at position 2 the area is smaller, the fluid must speed up.

So how does the fluid do this? Newtons' second law says that if something accelerates this must be done by a force. Which force has accelerated the fluid between point 1 and 2? In this case, the pressure gradient force. Because there is high pressure at point 1 and low pressure at point 2 (=pressure gradient), there is a force in the right direction in the region between them (see second picture). This causes the fluid to accelerate, resulting in a higher velocity at point 2.

We can also view this as energy conservation using Bernoulli's' principle. Pressure is a form of energy and so is kinetic energy. Pressure energy is converted to kinetic energy and vice versa.

The reasoning above might sound backwards to you. We first reasoned the water should speed up and then we reasoned that there should be a pressure difference between point 1 and 2. How did this pressure difference get there in the first place? The answer is that it is complicated. You might imagine the fluid being stationary first and then we start a pump so the water starts flowing. Generally speaking, the pressure and velocity field will be complicated during this stage so the pressure at point 2 could be higher at some times. After the system has reached steady state, the reasoning above holds and we know there must be a pressure difference.