9.4.1. Speed of Efflux: Torricelli’s Law

The word efflux means fluid outflow. Torricelli discovered that the speed of efflux from an open tank is given by a formula identical to that of a freely falling body.

The term 'efflux' refers to the process whereby a fluid flows out from a container through an opening, which is essentially what happens when water leaves a tap. Torricelli's discovery highlights that this outflow can be explained using principles similar to those governing the motion of falling objects under gravity. Therefore, when fluid exits an opening, its speed can be modeled as if it were falling freely due to the force of gravity.

Consider a bucket filled with water. If you poke a small hole at the bottom, water will start flowing out due to gravity. This process—just like a rock falling to the ground—can be understood using the same mathematical concepts that describe falling bodies.

Consider a tank containing a liquid of density ρ with a small hole in its side at a height y₁ from the bottom (see Fig. 9.10). The air above the liquid, whose surface is at height y₂, is at pressure P.

In this setup, the tank holds a liquid, and there's a hole at a certain height. The liquid has a density, which means it has weight, and this weight creates pressure in the fluid due to gravity. The height y₁ indicates where the hole is compared to the total height of the liquid column. The pressure at different heights within the liquid will vary, and this pressure influences the speed at which the liquid exits through the hole.

Imagine a tall glass filled with water and a straw sticking into it. If you put your finger over the straw's top and release it, the water will rush out when you remove your finger. The same concept applies here; the pressure from the water above the hole forces the water out just like gravity pulls down an object.

From the equation of continuity [Eq. (9.10)] we have v₁ A₁ = v₂ A₂.

This is a crucial part of fluid dynamics where we relate the speed of fluid flow to the cross-sectional areas at different points in the fluid. The equation states that the product of the area and velocity of the fluid at one point (A₁, v₁) equals that at another point in the system (A₂, v₂). This principle of continuity helps us understand how fluid velocity changes when it moves through different sized openings. If one area is smaller, the fluid must move faster to keep the flow rate constant.

Think of a garden hose. If you place your thumb partially over the end of the hose, the water speed increases at the opening due to the reduced area, similar to how it behaves in our tank scenario.

If the cross-sectional area of the tank A₂ is much larger than that of the hole (A₂ >> A₁), then we may take the fluid to be approximately at rest at the top, i.e., v₂ = 0.

When the top opening of the tank is large compared to the hole from which the water is flowing, we can simplify our calculations. The fluid above the hole is considered stationary (or nearly so) because the size of the tank allows the fluid pressure to remain constant. Hence, we primarily deal with the water flowing out of the hole while neglecting its movement at the top. This simplification leads to the conclusion that the speed of efflux can be derived using just the height difference and gravity.

Picture a swimming pool where water is being poured out through a narrow drain at the bottom. While the water at the bottom is flowing quickly, the large volume of water at the top remains still enough that it doesn't significantly affect the flow rate of the water going down the drain.

Now, applying the Bernoulli equation at points 1 and 2 and noting that at the hole P₁ = Pₐ, the atmospheric pressure, we have from Eq. (9.12).

By applying Bernoulli's equation between the two points—a point in the liquid column (point 1) and the point at the hole (point 2)—we can derive a formula that gives us the speed of the outflowing liquid. The equation considers the pressures at both points and includes the effects of gravitational potential energy. The result tells us that the speed of the liquid coming out of the hole depends on the height of the liquid column above the hole.

Think of a soda can. When you poke a hole in the side, the liquid sprays out due to the pressure inside pushing against atmospheric pressure. Here, the height of the liquid in the can directly affects how fast it sprays out—just like in our experiment with the tank of water.

When P >> Pₐ and 2g h may be ignored, the speed of efflux is determined by the container pressure.

In situations where the pressure inside the tank is much greater than the atmospheric pressure, we can simplify Torricelli's law further. The main takeaway is that the speed of the fluid flowing out is primarily dictated by the pressure exerted by the liquid above the hole and not significantly influenced by gravity when considering fast flows among liquids. This simplifies our calculations and understanding of efflux speed.

This is similar to how a soda bottle with a lot of carbonation behaves. When you open it, the pressure inside pushes the liquid out very quickly, much faster than if you simply tilted the bottle partway without removing the cap. In high-pressure scenarios, there is a comparable effect in various fluid systems.

--