6.2. Deriving Velocity Profile

So, this is a snapshot of a fluid element at time t. In the fully developed laminar flow this is the velocity profile, as we have seen in our laminar and turbulent flow analysis. This velocity is only a function of radial distance r from the pipe, this is the diameter D of the pipe, this is the x dimension and the fluid element is of length l, that we have considered.

In this part, we begin with a fluid element depicted at a specific time. The focus is on the velocity profile in laminar flow, which is a crucial concept in fluid mechanics. The behavior of fluid flow is different across various radial distances from the pipe's center. In laminar flow, the fluid moves smoothly in parallel layers with the velocity being dependent purely on the distance from the center (denoted as r) and not uniform across the diameter (D) of the pipe.

Think of a smooth road with multiple lanes. The vehicles in the outer lanes (further from the center of the road) may move at a different speed compared to those in the inner lanes. Similarly, in pipe flow, fluid particles at different radial distances experience different velocities.

The assumptions is that the local acceleration is 0 since the flow is steady. We also assume that the convective acceleration is 0 since the flow is fully developed. Now, the every fluid particle flows along streamline with constant velocity. The neighboring particle have slightly different velocities, because the velocities are a function of radial distance r.

Several assumptions are vital to simplifying our understanding of fluid motion in laminar flow. First, we assume that there’s no local acceleration due to the steady flow – this means that the flow characteristics do not change over time. Second, we ignore the convective acceleration which arises in unsteady flow, thus indicating that every fluid particle flows steadily along its streamline. However, we still recognize that adjacent particles may flow at differing velocities due to their varying radial positions in the pipe.

Imagine a train where the cars in the middle are moving at a steady pace, while the ones on the edges may have slight speed differences depending on their distance from the center of the train track. In the same way, the assumptions about acceleration highlight the consistent behavior of the flow, while still acknowledging slight differences in speed across fluid layers.

The force acting from this side, if there is a pressure p1, on the left side at section 1 and this is section 2 and if we assume the delta p is the pressure drop, so p will be here, will be p2 is going to be, p1 - delta p.

This chunk introduces the concept of force balance in the context of the velocity profile. At one end of our fluid element, we have pressure p1, and at the other end, due to the pressure drop, the pressure is p2 = p1 - delta p. The pressure difference across the fluid element is crucial for determining the motion of the fluid, specifically how it contributes to the overall velocity profile and flow rate in the pipe.

Consider a water hose. The water pressure at the start (p1) and at the endpoint (p2) varies, creating a push that causes the water to flow. The difference in pressure essentially acts like a 'force' that propels the water through the hose, similar to how the pressure drop works in a fluid element.

The equation for fully developed laminar flow in pipe can be derived using 3 approaches... We are going to start the derivation of fully developed laminar flow in pipe, using Newton’s second law now.

Three approaches are available to derive the equation for fully developed laminar flow: Newton's second law, Navier-Stokes equations, and dimensional analysis. The derivation process involves applying Newton's second law, which relates the forces acting on a fluid element to its motion. As we approach this derivation, we will simplify our concepts and ensure that we consider the realities represented by fluid behaviors under flow conditions.

Consider a skateboarder going down a hill. The gravitational force acts as the primary force that causes acceleration. If we want to predict how fast they will go at the bottom, we apply principles of physics to derive their velocity. Similarly, in fluid dynamics, we use established laws like Newton's to predict various flow characteristics.

Now, we have determined u as a function of r and the flow rate across the entire pipe will be integral of 2 pi r dr, ... The integral 0 to D/2, u r was delta p D square/16 Mu l into 1 – 2r/D into 2 pi r dr.

This part focuses on the relationship between shear stress and flow velocity as a function of radial distance. Shear stress varies linearly with distance from the center of the pipe and is directly related to the pressure gradient. The flow rate can be calculated by integrating the product of the velocity profile and the differential area. This calculation leads to the final expression for flow rate, also known as Poiseuille's law, which is critical in fluid dynamics.

Think of honey pouring through a funnel. The thickness of the honey (viscosity) affects how quickly it flows, similar to how shear stress impacts flow in a pipe. Just as the honey flows more slowly near the funnel’s sides than at the center, fluid velocity also varies within a pipe, all of which is captured by the velocity profile in our equations.

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