Continuity equation

The continuity equation relates the area available for a fluid to flow and its velocity.

When we cover the water outlet of a hose, the fluid flow velocity increases
When we are pouring water on the plants in the garden or washing a car with the help of a hose, it is common to use the thumb to close the water outlet a little and then increase the speed of the liquid exit. The demonstration of the explanation for this fact is made from the continuity equation .

This equation relates the flow velocity of a fluid and the area available for such flow. From the image below, notice that the path made by the fluid has two different areas: A 1 > A 2. Imagine, therefore, that, in a time interval (Δt), a volume (ΔV) of the fluid enters through area A 1 . Adopting the fluid as incompressible, we must assume that the same volume (ΔV) must leave the edge of area A 2 .

During the time interval considered, the space covered by the fluid can be given, from the average velocity equation , by Δs = v.Δt, where v is the flow velocity. Taking the gray markings on the figure as the volumes occupied by the moving fluid and knowing that they are equal, we have:

V 1 = V 2

1. Δs = A 2. Δs

1. v 1 .Δt = A 2. v 2 .Δt

1 . 1 = A 2. v 2

Thus, we can see that the smaller the flow area available for a fluid, the greater its velocity and vice versa. As a final example, we can imagine the “thread” of water formed by a half-open faucet. Note that the lower you look, the thinner the water stream will be, because, with the action of gravity acceleration , the fluid’s velocity increases, decreasing its flow area.

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