5.3.2. Momentum Equations

And if you look at this momentum equations what we derived earlier and that what will be expressions like this. So what you supposed to do it whenever write these type of big equations, try to write this all these dimensions and check it the dimensions of each components whether they are the equal.

Momentum equations are derived from fundamental principles of fluid mechanics. They describe how the momentum of fluid changes as it flows through a control volume. In practical applications, it is essential to ensure that the dimensions of each component in the equation are consistent; this is termed 'dimensional analysis.' For example, if you have an equation that expresses force, you need to confirm that all components involved contribute consistently to this force unit (Newtons in SI).

Imagine measuring how fast you can push a car down a hill. You need to consider how heavy the car is (mass), how steep the hill is (angle), and how much force you can exert. If your measurements are not consistent, like mixing pounds with kilograms, the results won't make sense. Similarly, in momentum equations, every term must align dimensionally.

Linear momentum relation in the stream wise direction.

The linear momentum relation defines the change in momentum for a unit mass over time due to external forces applied in the direction of fluid flow. It highlights how the velocity of the fluid changes along the streamline, influenced by factors such as pressure and friction. When analyzing flow, it's crucial to account for all forces acting on the fluid, which can significantly affect its momentum.

Think about riding a bike downhill. As you go down, gravity pulls the bike faster, and the momentum increases. If there's a strong wind (a force acting against you), it slows down your speed. This scenario represents how different forces interact to change momentum in fluid systems.

And if you look at that basic equations of Bernoulli’s equations which has the pressure head component, the velocity head component and the elevations head those what if you include it as the flow energy for weight, the kinetic energy for weight and the potential energy for weight.

Bernoulli’s equation captures the conservation of energy in a moving fluid, outlining how pressure head, velocity head, and elevation head are interconnected along a streamline. Pressure head corresponds to the stored energy due to fluid pressure, velocity head indicates kinetic energy due to fluid velocity, and elevation head represents potential energy due to gravitational height. Each of these components is crucial for understanding how fluids move and behave under different conditions.

Consider a water slide at a water park. At the top of the slide, the water has maximum potential energy (height) and minimal kinetic energy (motion). As it slides down, this potential energy converts to kinetic energy, increasing the speed of the water. Bernoulli’s principles help explain this energy conversion through pressure, height, and velocity relationships.

Let us solve the five questions from GATE question set. The first examples one what I consider is the circular water pipes shown in the sketch are flowing full condition.

Applying momentum equations through practical problems such as fluid flow in pipes offers insights into real-world situations. By setting up equations that equate inflow and outflow, one can calculate important quantities like flow rates and velocities. For example, if you know the diameter of pipes and the velocities at different sections, you can use mass conservation and momentum principles to find unknown speeds.

Think of a water hose where you pinch one end. The pressure builds up, and when released, the water shoots out faster. By analyzing how water flows in the hose, we can predict how much pressure is needed or how fast the water will come out when comparing different hose diameters. These same principles apply to momentum equations in various engineering problems.

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