6.1. Stagnation tube

One of the other applications of Bernoulli equation is the stagnation tube. What happens when the water starts flowing in the channel? For example, this and so, the equation along the stream line by Bernoulli equation will be can be given as p / gamma + z + V square / 2g = constant. Here, the orientation of the tube is like this, but does the orientation of the tube matter? Yes, because it can either lower up, I mean, it can change the different value of z so, yes.

In this chunk, we introduce the stagnation tube, which is a practical application of Bernoulli's equation. Bernoulli's equation relates the pressure, height, and velocity of a fluid flowing in a streamline. The equation itself states that the sum of the pressure head (p/gamma), the elevation head (z), and the velocity head (V^2/2g) is constant along a streamline. The orientation of the stagnation tube is crucial because it can affect the level of fluid (z) measured in the tube, highlighting the importance of understanding the impact of external conditions on fluid behavior.

Imagine a water fountain where water jets up into the air. The pressure and height of the water vary depending on how high the water has to rise and how fast it is flowing. If you consider how a tube positioned in different angles would show varying levels of water due to these pressure and speed changes, you start to visualize why the height of the fluid in the stagnation tube is influenced by its orientation.

How does high does the water rise in this stagnation tube that we have to calculate. How do we choose the points on this timeline, another important question that we will solve? So, we assume, points 1a, 2a,1b and 2b like this, this point, this point, this point and this point. The equation, Bernoulli equation will be the same p / gamma + z + V square / 2g = constant.

To analyze the behavior of fluid within the stagnation tube, we select specific points along the streamline. By considering points 1a, 2a, 1b, and 2b, we can apply Bernoulli's equation consistently across these points. The equation indicates that at any two points along the same streamline, changes in height, speed, and pressure are interrelated. This relationship helps us to understand how variations in fluid height occur within the stagnation tube, following Bernoulli's principle.

Think of it like measuring different heights of water in various cups placed at different levels along a slope. You can compare how the water in a cup (point) at a higher elevation behaves differently than one at a lower elevation, depending on how fast the water flows and how much pressure there is. This analogy helps visualize how we can apply Bernoulli’s equation across different points in a fluid stream.

If you go from 1a to 2a, velocity will be a function of pressure, it is the same streamline; 1a is going to 1a to 2a. If you go from 1b to 2a it crosses perpendicular stream lines. If you go from 1a to 2b, it does not cross the stream lines, going from 1a to 2a does not cross the stream line.

When calculating how high the water rises in the stagnation tube, we differentiate between points on the same streamline and those that are not. Moving from point 1a to point 2a means staying on the same streamline, where Bernoulli's equation applies directly. Conversely, moving from 1b to 2a crosses streamlines, which is not permissible under the assumptions of Bernoulli’s principles. This distinction helps us accurately calculate fluid height changes through direct application of Bernoulli’s formula for the points that are relevant and compliant with its assumptions.

Imagine trying to compare the flow of water from two different hoses. If you watch how water comes out of one hose into a bucket (same streamline), you can measure things easy. But if you tried to take water from part way up the hose (crossing streamlines) and compare it to water from a point below (not on the same streamline), your observations wouldn't make sense. This paints a clear picture of why it's important to stick to the same path when measuring fluid behavior.

In all cases, what we do not know is p1. So, the equation, Bernoulli equation p 1 / gamma + z 1 + V 1 square / 2g is constant. So, p 1 / gamma + z 1 is going to be, we can set it as 0 at 0.1 and because this is exposed to atmosphere we can assume to be 0 and because this has reached at the top what happens is, there will be no velocity because the water will have high rise until the maximum, you know, that it can and there the velocity will be 0.

In this chunk, we focus on determining the initial pressure (p1) at point 1. By applying Bernoulli's equation and setting atmospheric pressure as the reference point (0 at point 0.1), the pressure at the stagnation point can be deduced indirectly. When the fluid rises to its highest point in the tube, the flow velocity reduces to zero. This point represents the maximum height the water can reach, illustrating the dynamic interplay between pressure and velocity in fluid systems.

Picture a diver pushing water to the surface with a pool noodle. As the diver pushes upward quickly, the water splashes out, but when the diver eases off the push, the water comes to a standstill at the maximum height it can reach. This analogy helps explain how the pressure at that maximum height is zero in relation to velocity and helps anchor the principles we're learning.

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