6.1. Assumptions for Velocity Potential Derivation

We started this chapter by saying that it is irrotational flow. So, following are the assumptions of deriving the expression for the velocity potential phi that has much to do with hydraulic engineering or fluid mechanics.

Irrotational flow means that the flow is smooth and does not have any rotation at any point within the fluid. It is a key condition because it simplifies our analysis. If the fluid flows in an irrotational manner, we can derive and use potential functions that describe the flow behaviour without dealing with complex rotational dynamics.

Imagine riding a bike smoothly along a straight path without any sudden turns. The way the bike moves forward without wobbles or rotations is similar to irrotational flow in fluids—where the movement is steady and predictable.

This ideal fluid condition assumes that the fluid is inviscid, which means it has no viscosity, and there are no effects of friction within the fluid.

An ideal fluid is a hypothetical fluid that experiences no resistance to shear stress (i.e., it's frictionless). This assumption helps us in simplifying the equations governing fluid flow. In reality, all fluids have some viscosity, but for mathematical models, we often treat fluids as ideal to focus on the primary effects without the complicating factor of viscosity.

Think of an ideal fluid as a perfectly smooth surface, like an ice rink. When you glide over it, there's little to no friction compared to a rough, ragged surface where every bump slows you down. This smoothness allows for simpler calculations of motion.

Surface tension is also neglected in this analysis, meaning we assume that it does not significantly affect the flow dynamics.

Neglecting surface tension means we consider the fluid as having flat and uniform properties at small scales. In practice, surface tension can affect how fluids interact at their boundaries and with solid objects, but for our derivation, we focus only on the larger-scale motion of the fluid.

Imagine pouring a cup of water. On a very small scale, the surface tension might make the water bead slightly at the edge. However, when you look at the full cup, the effect is negligible compared to the volume of water contained in it.

The pressure at the free surfaces is assumed to be uniform and constant.

Assuming that the pressure is uniform means that at every point on the surface of the fluid, the pressure is the same. This simplifies the calculations significantly as we can use the same pressure value throughout without considering variations caused by wind or other factors.

Think about a balloon filled with air. If you squeeze the balloon gently, the air pressure feels uniform across the surface of the balloon. Similarly, for our fluid, we assume that the 'skin' or surface interacts consistently with its environment.

We also assume that the seabed is rigid, horizontal, and impervious.

This means that the seabed does not change shape or shift. It stays flat across the area we are studying. This helps us to focus on how the water moves above, rather than dealing with potential bed deformation or changes in depth that would complicate our analysis.

Picture a smooth, solid foundation of a building that does not shift or settle unevenly over time. Just as this stable base makes construction easier, a rigid and horizontal seabed allows us to apply our assumptions without considering ground movement.

We assume that wave height is small compared to the length.

This assumption is essential as it allows us to approximate the flow behaviour without considering drastic changes in wave shape. If the wave heights are significantly smaller than their lengths, the flow remains primarily uniform and manageable. This is particularly important in wave mechanics where we use sinusoidal approximations.

Imagine a gentle wave rolling into the shore at the beach. The height of the wave is much smaller compared to the distance it travels from one point to another before breaking. This ‘gentle’ nature makes it easier to predict where and how the wave will act as it lands.

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