2.6. Linear Momentum Equation

So, now the linear momentum equation. So, an example here is, if you see this figure here, this is so, suppose if you shoot water jet onto a wall for example, this is one of this example where, you know, linear momentum is associated, there is a velocity with which water is approaching and when it touches this wall it comes to rest.

In this section, we introduce the concept of linear momentum using a practical example of a water jet hitting a wall. Linear momentum is the product of an object's mass and its velocity. When the water jet (having momentum) strikes the wall, it comes to a stop, demonstrating the principle of conservation of momentum. This change shows that there is a force acting on the water as it hits the wall.

Think of a basketball thrown at a hoop. When it hits the backboard, it either rebounds or stops, showing that momentum depends on the mass of the ball and its speed. Just like the water jet, the ball experiences a change in momentum when it collides with another object.

So, here we can actually apply the momentum conservation linear momentum. So, what happens is, as I said we are going to use the same concept of Reynolds transport theorem for the linear momentum equation too.

The Reynolds Transport Theorem (RTT) allows us to relate the change of a property within a control volume to the flows of that property across the control surface. In the context of linear momentum, we utilize RTT to derive the linear momentum equation, which is critical for understanding fluid motion. This equation can help us relate the forces acting on a fluid and the changes in momentum that occur due to the motion and interaction of the fluid with its surroundings.

Imagine a river flowing through a bend and carrying leaves. The RTT helps explain how the leaves' momentum is affected when the water bends and pushes the leaves toward the shore.

Here, what we do is so, this is the control volume equation, as I told you from the Reynolds transport theorem. So, what we do we say B in our case when we want to derive the linear momentum equation B should be mass into velocity or called momentum.

To derive the linear momentum equation, we define B as the product of mass and velocity (or momentum). We also consider a control volume where we can apply the principles of conservation. In mathematical terms, the change in momentum over time within a control volume can be expressed by the forces acting on it, allowing us to relate the mass flow rates at the inlet and outlet to the momentum exchange.

Think of a car driving on a highway. The car's momentum is determined by its mass and speed. If it brakes suddenly, its momentum changes because of the force applied, similar to how we derive the relationship in fluids.

So, now, we similar to the mass we assume cross sections here and we use for the steady state equation. DmV / Dt = ρ V dv dot n cap dA.

In a steady state condition, we relate the rate of change of momentum (DmV/Dt) to the net momentum flux across the control surface. This allows us to set up an equation that connects the forces acting on the fluid (such as pressure and weight) to the changes in momentum at different cross-sections of the fluid stream. By establishing these relationships, we can derive the governing equations for fluid motion.

Consider a large pipe carrying water; the water flowing in has a certain momentum. If the pipe narrows, the speed increases, changing the momentum. We can visualize this as water squeezing through a garden hose; the force applied relates to how fast the water exits.

So, now the question is, what are the forces acting on fluid in the control volume. So, one is gravity, other is shear at solid surface or pressure at solid surface, pressure on the flow surfaces.

For a fluid in control volume, several forces need to be considered, including gravitational force acting downwards, pressure forces from the fluid's internal interactions, and shear forces acting at the solid surfaces. These forces contribute to the overall momentum balance within the fluid, allowing us to write meaningful equations that capture the dynamics of fluid systems.

Imagine a tall building. The building exerts gravitational force downwards due to its weight, similarly, the fluids in a container exert force depending on their densities and the pressure at their surfaces—just like the household plumbing system, which has to withstand various pressures from water flow.

An example we are going to see is the reducing elbow using this equation, we have seen this equation already in the last slide.

The example of a reducing elbow in a piping system illustrates how the linear momentum equation is applied in real scenarios. We analyze the forces acting on a fluid flowing from a larger diameter pipe to a smaller one, which involves understanding how changes in area affect velocity and, consequently, momentum. By calculating these factors, we can derive the forces exerted on the elbow from the fluid.

A practical analogy can be drawn with traffic merging on a highway. As vehicles move from a wider lane to a narrower lane, they have to speed up to maintain their flow, just like fluid speeds up when passing through a reducing elbow, creating forces as they interact with the pipe walls.

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