2.2. Change in Momentum

The mass flow rate m is given by ρbcy, ρ is the density and we need to have volume per unit time. So, b the width, y is the depth and c is the say let us say dx / dt, for example. So, this is width, this is length and this is height.

The mass flow rate (m) can be calculated using the formula m = ρbcy. In this formula, ρ represents the density of the fluid, b is the width of the channel, y is the depth of the fluid, and c is the velocity of the flow. This relationship helps us understand how mass moves through a specific area over time, indicating how heavy the fluid is at that cross-section.

Imagine you are filling a bucket with water from a hose. The width of the hose corresponds to 'b', how deep the water is in the bucket relates to 'y', and how fast the water comes out of the hose is 'c'. The heavier the water (higher density), the more water is in the bucket in a given time.

We assume that the pressure is hydrostatic within the fluid and therefore, the pressure force on the channel in cross section 1 will be if you remember from your fluid statics class, it will be ϒy A or gamma where the height was y + delta y.

Hydrostatic pressure relates to how pressure increases with depth in a fluid. At a certain depth in the water, the pressure is calculated as ϒy, where ϒ is specific weight and y is the depth. In this case, we need to consider both y and an increment (delta y) to appropriately assess the change in force acting at different sections of the channel.

Think about diving into a swimming pool. As you go deeper, the water pressure on your body increases. This is similar to the hydrostatic pressure explained here, where the deeper you go, the more pressure you experience from the water above you.

So, we apply the change in momentum, we change, rate of change of momentum is the force. So, we obtain half gamma y square b, that is, force on the right-hand side minus force on the left-hand side is equal to rho b c y mass flow rate.

When considering momentum in fluid dynamics, the change in momentum (the product of mass and velocity) is equated to the forces acting on the fluid. The momentum change, represented as half gamma y^2 b, is derived from the difference in forces from two cross-sections of the channel, helping us understand how forces affect fluid motion.

Imagine a car accelerating on the road. The force exerted by the car's engine changes its momentum as it speeds up. In a similar way, when the fluid inside a channel accelerates, changes in pressure and flow across different sections impact its overall momentum.

If we substitute equation 2 into equation 1, we get c is equal to yg / c, and c can go this side it becomes c square is equal to gy, or the final equation c is equal to gy.

The relationships established through the previous equations can be simplified to arrive at a new equation that directly correlates the speed of water waves (c) to the fluid depth (y) and gravitational acceleration (g). This shows that wave speed is determined significantly by these factors, leading to the critical result c = sqrt(gy).

Think of the ripples that form when you throw a stone into a pond. The speed at which these ripples travel is influenced by how deep the water is (y) and the force of gravity (g). The derived formula helps us predict how quickly those ripples move across the surface.

The Froude number as we discussed is given by V under root g y. If c is greater than V, so what is going to be the Froude number, less than 1, so that means subcritical flow.

The Froude number is a dimensionless value that compares the flow speed (V) to the wave speed (c). When the wave speed exceeds the flow speed, it indicates a subcritical flow condition, which is important for analyzing fluid dynamics in channels. This concept is essential for understanding how waves interact with current flows.

Think of a river with a flowing current and a wave moving through it. If the wave travels faster than the current, those waves will move against the current, just like how a bird can fly upstream against a river's flow. This is akin to having a Froude number less than 1, indicating that the waves can travel upstream.

For finite sized waves, the wave speed actually is given by under the root of g y multiplied by one + delta y / y to the power half.

When the wave amplitude (delta y) increases, the speed of the waves also increases. This relationship is captured in the adapted wave speed formula, demonstrating that larger waves travel faster than smaller ones, which contrasts with small amplitude cases where the speed remained independent of amplitude.

Imagine a larger wave at the beach compared to a small wave. Larger waves, such as those created by big storms or swells, travel faster and have more force behind them, just as the formula suggests. When we understand this, we can predict how waves will interact when they hit the shore.

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