8. Bernoulli's equation normal to streamlines

So, we have seen this in the previous, you know, when we are dealing with the Bernoulli’s equation along the stream line. So, similarly, we can write minus the force equation we can write, - delta p / delta n + rho a n + rho g dz dn. Here, a n is V square / R, where, R is the local radius of curvature, n is towards the center of the radius of curvature and this is equal to 0, s is constant along the stream line. So, therefore, proceeding in a similar way, we obtain dp = del p del s into ds + delta p delta n into dn, you remember, we had the same along the stream line ds = 0 because and we can write dp dn = del p del n. So, what happens is we get - dp dn = rho V square / R + rho g dz dn because we can write delta p delta n is equal to and we obtained this from this, so, this equation becomes this, the same procedure, no changes whatsoever.

In this section, we begin with the general form of Bernoulli's equation as it relates to the flow of fluids normal to streamlines. We start by noting that the pressure differential can be described by the negative change in pressure over a normal direction (delta p / delta n). This is influenced by the acceleration (a_n), which is derived from the velocity squared divided by the radius of curvature (V²/R). The derivation essentially parallels what was discussed for flow along the streamline, except it accounts for the aspects affecting the flow in the perpendicular direction to the streamline. The result is the equation that relates changes in pressure to velocity and height in a fluid under flow conditions.

Imagine standing next to a fast-flowing river. If you were to throw a small rock into the water, you would notice that the water not only moves downstream but also pushes outward. The pressure changes around the rock as it creates disturbances both directly downstream (along the streamline) and also sideways (normal to the streamline). Just like the analysis of flow normal to the streamlines considers these pressure changes, valuing how the water interacts with the environment around it.

So, this is the equation we are not going into too much detail about this, but this is how you can have the Bernoulli equation normal to the streamlined. So, you see when you integrate this, this will have a different consequence. So, now the Bernoulli’s equation applications; 1 is stagnation tube that you have already seen, the other was pitot tube which we discussed, the free jets equation we have seen. We have not seen orifice, but orifice is one of the other areas where Bernoulli’s equation is there. Venturi meter, the Bernoulli’s equation is applied across sluice gate and sharp crested weir.

This section addresses the practical applications of Bernoulli’s equation when considering flow normal to streamlines. It lists several fluid mechanics devices that utilize this principle, including stagnation tubes (used in various fluid measurements), pitot tubes (used in aviation for measuring airspeed), and orifices (used for measuring flow rates). The application of the Bernoulli equation in devices like venturi meters demonstrates how engineers can apply theoretical principles to real-world scenarios effectively, providing critical measurements in fluid systems.

Think of a large water slide at a theme park. As you sit at the top with water pooling around you, you are at a point of high pressure but little movement. As you slide down, the angle of the slide changes, creating both forward motion (along the streamline) and pressure that pushes in different directions (normal to the streamlines). The water powering the slide’s flow is similar to applications of Bernoulli’s equation, demonstrating how speed and pressure constantly adjust based on the shape and direction of the slide.

Now, proceeding forward, so, the Bernoulli nation normal to the streamlines. So, can you find the flow Q given the pressure drop between point 1 and 2 and the diameter of the 2 section? You may assume the head loss is negligible. Draw the EGL and the HGL over the contracting section of the venturi metre?

This segment prompts students to apply Bernoulli’s equation in a practical context—calculating flow (Q) using pressure differences measured at two points along a streamline, considering changes in diameter in a venturi meter. By utilizing Bernoulli's principles, students can visualize the hydraulic grade line (HGL) and energy grade line (EGL) over different points in the system. This understanding helps students comprehend how fluids behave under varying pressures and velocities in a fundamental way.

Picture the process of siphoning water from a bucket with a tube. When you suck the air out to start the flow, the initial pressure drop creates a siphon effect, moving water from one bucket to another based on the height difference. The adjustments in speed and pressure along the tube mirror the behavior examined in this section. Calculating flow becomes similar to understanding how the siphon effectively works, reinforcing the principles of fluid mechanics.

--