Can you explain why Pressure is inversely proportional to velocity without using the bernoulli's equation?

Answer 1

In simpler terms, think of a fluid (like water or air) moving through a space, such as a pipe or open air. The key idea is that this fluid has a certain amount of energy that it can use in different ways. One way it uses this energy is by moving quickly, which is its velocity, and another is by exerting force on its surroundings, which is its pressure.

Now, imagine you have a fixed amount of energy to divide between running fast (high velocity) and pushing against something (high pressure). If you decide to run faster, you'll have less energy to push against things, which means your pressure decreases. Similarly, in fluid flow, when the fluid speeds up (increases its velocity), it has less energy to exert as pressure. So, as the speed of the fluid goes up, the pressure it can exert goes down, and vice versa. This happens because the fluid can only use its energy for one thing at a time: either moving quickly or pushing hard, but not both at maximum capacity.

Answer 2

Certainly! The relationship between pressure and velocity in fluid flow can be understood without directly invoking Bernoulli's equation by considering the conservation of mass and the principle of continuity.

1. Conservation of Mass:

When a fluid is in motion, the mass flow rate must remain constant along a streamline. This is based on the principle of conservation of mass. The mass flow rate m_dot is given by the product of the cross-sectional area A and the velocity v: m_dot = A × v.

If the fluid is incompressible (density remains constant), an increase in velocity must be accompanied by a decrease in cross-sectional area, and vice versa.

2. Principle of Continuity

The principle of continuity states that the mass flow rate of a fluid is constant in a steady flow. This means that the product of cross-sectional area and velocity at one point in a pipe must be the same as at any other point.

Mathematically, A_1 × v_1 = A_2 × v_2, where A_1 and A_2 are the cross-sectional areas at points 1 and 2, and v_1 and v_2 are the velocities at those points.

Now, let's consider how this relates to pressure:

3. Inverse Relationship

As the fluid flows through a constriction (a narrower section), the velocity increases due to the conservation of mass and the principle of continuity. According to the equation m_dot = A × v, if v increases, A must decrease to keep m_dot constant.

Now, when the cross-sectional area decreases, the fluid particles must move closer together. This results in more collisions between the particles and the walls of the pipe, leading to an increase in pressure.

Therefore, in a fluid flow, there is an inverse relationship between pressure and velocity: as velocity increases, pressure decreases, and vice versa. This relationship is a fundamental aspect of fluid dynamics, even before considering Bernoulli's equation. Bernoulli's equation provides a more comprehensive explanation, but the basic principles of conservation of mass and the principle of continuity offer insights into the pressure-velocity relationship.