22.5.3. Flow Classifications

Now let us look at the flow classifications or the simplifications that in this case we have considered one-dimensional steady incompressible or compressible that is what we will discuss just a bit here. We have a stream tube, fixed control volume, okay.

This chunk introduces the concept of flow classifications in fluid mechanics. It emphasizes the importance of various types of flow, notably one-dimensional flow which can be either steady or unsteady, and compressible or incompressible. A 'stream tube' is mentioned, which refers to a series of streamlines defining the flow of fluid, and a 'fixed control volume' is a chosen region in which we analyze the fluid's behavior.

Think of a garden hose. When you turn on the hose and let the water flow out of it, the water flows in a specific direction, and you can imagine the flow lines as a series of connected paths inside the hose. This is like the 'stream tube' concept. If you pinch the hose to make the water flow faster, you're creating a scenario that can be classified as one-dimensional steady flow.

The advantage of the stream tube is that there is no flow across this stream tube, because that is what the definitions of the streamlines, there will be no flow component to normal to this part, because all the flow components will be tangential to the spot.

Stream tubes have a critical property: there is no flow of fluid across them. This means the fluid moves solely along the direction of the streamlines without any perpendicular flow. The streamlines represent the path of individual fluid elements, ensuring that fluid either flows into or out of the stream tube without mixing with other flows.

Imagine a narrow river surrounded by banks. The shape of the river can be thought of as a stream tube, where the water flows straight along its path without jumping over the banks. Any debris in the water follows the same flow and cannot cross over to different areas, similar to how fluid elements in a stream tube behave.

Because of that, we do not have any flow pass through perpendicular to the surface. The flow comes from this direction and goes from this.

The concept of a control volume in fluid mechanics is linked closely with stream tubes. By defining a 'control volume', we identify a specific region through which we analyze the mass inflow and outflow without considering flow across its surface. This simplification helps in applying conservation laws such as mass and momentum more easily.

Consider a balloon filled with air. The surface of the balloon acts as a control volume. Air molecules can move in and out of the balloon, but they cannot pass through the balloon's rubber surface. When you squeeze the balloon, you see the air inside being compressed, demonstrating how we can analyze changes within our defined control volume without worrying about external airflow.

Now if you look it, since there is no friction is there, okay. So, we can easily consider is that shear stress is equal to zero.

When analyzing fluid flow in a stream tube, some assumptions simplify the calculations. One key assumption is that there is no friction, meaning shear stress can be considered zero. This allows us to analyze the ideal conditions under which the fluid flows, making it easier to derive equations of motion and energy conservation.

Think about a perfectly smooth slide at a playground. Without any friction, a child would glide down effortlessly, illustrating how idealized conditions help clarify the underlying physics. In real-world situations, friction exists but considering it to be zero simplifies our initial understanding of fluid behavior.

So, whenever you have a streamtube or the steamlines, any point you consider 1, 2, 3, 4. If it is a fictional loss, there is no energy losses in these 1, 2, 3, 4 points, we will have a total energy becomes constant.

In applications of fluid mechanics, particularly when analyzing flow through stream tubes, we often assume that energy is conserved throughout the flow (unless there are losses due to friction or turbulence). This means that the total energy—composed of potential energy, kinetic energy, and flow energy—remains constant along a streamline under ideal conditions.

Using the slide analogy again, if a child goes down without any obstacles or friction, they reach the bottom with a certain energy that has converted from potential to kinetic. Similarly, in fluid mechanics, as fluid moves through a streamline, its energy transformations can be predicted based on these conservation principles.

--