1.6. Assumptions for Laminar Flow in Circular Pipes

So, now laminar flow in circular pipes, so, after dealing telling you the basics of how to define the, you know, how to define and find what laminar and turbulent flow is, we are going to see 188 some of the properties of laminar flow in circular pipes. So, for deriving anything there are certain assumptions first that we have to take.

This chunk introduces the concept of laminar flow in circular pipes and highlights the importance of certain assumptions needed for deriving the properties of this flow. It sets the stage for a detailed discussion by emphasizing that understanding laminar flow is crucial for hydraulic engineering.

Consider a calm stream of water flowing gently in a narrow channel – this is like laminar flow. Just as we need to note the shape of the stream and its surroundings to understand its behavior, we need to outline certain assumptions to accurately study laminar flow.

• We have to assume steady flow, what does this assumption of steady flow mean, that the situation or the condition is not dependent on time.
• Second assumption is, it is laminar flow, this means the Reynolds number is fairly less than 2300, as we have seen in the last slide.
• And the flow is incompressible which means density is constant, the density does not changes either in space or with time.
• And we also have to assume the flow is fully developed, that means there is no, you know, intermittent phenomenon that is happening, a flow has happened over a long period of time.

In this chunk, various key assumptions for analyzing laminar flow in circular pipes are discussed. The assumptions include:

1. Steady Flow: The flow parameters do not change with time. This means that if you were to measure the flow velocity at a given point, it would be the same if you measured it a minute later.
2. Laminar Flow: The flow remains laminar if the Reynolds number is less than 2300, indicating smooth and orderly motion.
3. Incompressibility: The density of the fluid remains constant regardless of its flow state, simplifying calculations.
4. Fully Developed Flow: The flow pattern remains the same along the length of the pipe, meaning any initial disturbances have settled.

Imagine a calm, steady river – the water flows uniformly at a constant rate without changes due to the time or weather. This is similar to steady flow in laminar conditions, where the properties remain constant along the flow direction.

Now, after going through the assumptions, we will consider a coaxial ring shaped fluid element of radius ‘r’ whose thickness is ‘dr’ and length is ‘dx’ and the flow is from left to right.

This chunk introduces the concept of a coaxial ring-shaped fluid element which is a theoretical tool used to analyze fluid flow within pipes. The fluid element is defined by its radius (r), a small thickness (dr), and a length (dx). Understanding this element allows for the application of principles such as force balance and shear stress analysis within the laminar flow framework.

Think of slicing a cylindrical cake – each slice has a certain radius, thickness, and height. Just like we could analyze the properties of that slice, we analyze the fluid element to understand how it behaves within the pipe.

If you see, there will be pressure forces acting P x from the left, P x + P pressure force are at x + dx from the right and then there will be shear forces acting here, in this direction and there will be shear forces acting at Tr + dr.

This chunk describes how to analyze forces acting on the coaxial fluid element. Pressure forces act on either side of the fluid element, while shear forces are acting due to the viscous nature of the fluid. It elaborates on how these forces are utilized to establish equations for analyzing the flow through the pipe.

Imagine pressing your hand against a stream of water. The pressure from behind pushes the water forward while the side of your hand creates shear forces that influence the flow. This balance of forces is crucial to understanding how fluids behave in pipes.

Equation number 2 can be written as simply you see, so this is pressure at x + dx - P x by dx, so it can be written as, dP dx and this can be written as, d r into tau dr.

Here, the chunk focuses on converting the previously discussed balance of forces into differential equations. It shows how to express the rate of change in pressure (dP/dx) in relation to shear stress, a fundamental part of fluid dynamics. This is an important step for deriving the velocity profile for laminar flow in circular pipes.

Consider the increase in speed of a car as it goes downhill—this change in speed represents a rate of change, similar to how fluid pressure changes along the pipe.

Now, if we use the standard tau is equal to minus mu du dr, why do we do this? Actually this is an assumption for laminar flow. So, if we have a laminar flow we can assume shear stress as a function of minus, you know, as a function of du dr or in other terms tau is equal to minus mu du dr.

In this chunk, the equation for shear stress (tau) in laminar flow is derived. Shear stress is directly proportional to the velocity gradient (du/dr), with the proportionality constant being the fluid's dynamic viscosity (mu). Such recognition is critical for understanding how internal friction in the fluid affects its motion.

Think about spreading butter on bread. As you push the knife through the butter, you're applying a force that changes its shape – that’s similar to how shear stress works in fluids, influencing their motion depending on the viscosity and the speed of flow.

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