20.4. Steady Flow Across Missions

Now let us come to today's lecture, will have a steady flow across missions for the linear momentum equations. Many of the times, we have a linear momentum equations we solve for the one inlet or one outlet and in one directions and some of the problems we have solved with no external forces, then how the momentum equations can be simplified.

In this section, we will discuss the fundamentals of steady flow in fluid mechanics, focusing on linear momentum equations. A steady flow means that the flow characteristics (like velocity and pressure) do not change with time at any point in the flow field. When solving linear momentum equations, we often deal with scenarios where there is only one inlet or one outlet or where external forces acting on the control volume are negligible, making the equations simpler to manage.

Imagine water flowing steadily out of a hose. If you keep the hose at a fixed position and only change the nozzle type (one inlet, one outlet), the water characteristics become predictable, allowing you to focus on the pressure and flow rate without worrying about changes over time.

Then fourth what we will talk about, when you apply the linear momentum equations, what are the hints and tips, what should we consider when you apply that linear momentum equations.

When applying the linear momentum equations to steady flow situations, key tips include identifying all inflow and outflow momentum, recognizing that momentum is a vector quantity (affected by direction), and ensuring proper use of momentum flux terms. Understanding how mass and velocity distributions impact momentum is crucial, especially when the flow is non-uniform; hence the use of momentum flux correction factors becomes necessary.

Think of a traffic system at an intersection. The flow of cars represents fluid flow. Each road (inlet or outlet) has a certain number of cars (momentum) entering or exiting. If it’s steady, you can analyze the car counts without worrying about the time it takes for each car to arrive; you just need to focus on managing the traffic flow for safety and efficiency.

Now let us go for a simple problems, where you have a linear momentum equations like you have a fixed control volume, just to look at, you have a fixed control volume. There are the in are there, okay, and there are the outs are three, and this is a fixed control volume. So there is inflows, this is outflows.

The fixed control volume concept sets boundaries for analyzing fluid flow. In our example, we consider a control volume that has various inflows and outflows. For a system to be steady, the sum of amplitudes entering must equal the sum of those leaving over time. By applying momentum equations, we can analyze how forces act on the fluid moving in and out of this control volume, leading to insights on how to manipulate flow dynamics in practical applications.

Consider a bathtub as a control volume. When you fill the tub with water (inflow), you need to account for water leaking out (outflow) from the drain to understand how much water remains. Similarly, analyzing the inflows and outflows helps manage the system efficiently.

Now, go for a simple problems, where is come it, very interestingly that we can apply the momentum equations as you can go it, this momentum equation is a vector equation, it has a three components in scalar directions, X component, Y component, Z component.

In steady flow problems, we often only focus on one direction (usually the x-direction) to simplify our calculations. By analyzing for just that one component, we can derive force components related specifically to that direction. This step allows us to ignore other dimensions temporarily and focus on the primary flow direction, easing the calculations needed for momentum interactions.

Using a football as an analogy, if the player passes only along the field's x-axis (forward), they can ignore movements in the y-axis (side to side) during that specific moment. This focus simplifies their analysis of the ball's momentum and how to adjust when it gets to another player.

Now if you look it there another set of the problems we solve it, where there is no external forces is there, there is no external forces is there, very simple cases.

In scenarios devoid of external forces, the linear momentum equations indicate that the net force on the system is zero. This means the rate of change of momentum within the control volume equals the difference between incoming and outgoing momentum. Such a situation simplifies our analytical efforts as we can focus on momentum exchanges within the controls, therefore making calculations and results clearer and more manageable.

Think about a closed train car moving at a constant speed on tracks. Without other forces acting on it (like wind or friction changes), any package inside that doesn’t alter its movement doesn’t experience any outside forces ensuring that the train's constant motion provides a consistent frame for analyzing any movement inside.

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