4.2. Unsteady Bernoulli's Equation and Application

Dynamic free surface boundary condition requires that the pressure on the free surface be uniform along the waveform. To derive this, we will use unsteady Bernoulli’s equation. This means that we will add another term to the standard Bernoulli’s equation to account for time variations.

The dynamic free surface boundary condition is vital in fluid mechanics because it acknowledges that pressure varies across a surface which can deform, such as the surface of water in a wave. By using the unsteady Bernoulli’s equation, we can express the relationship between fluid velocity, pressure, and elevation at that surface. The additional term makes the equation account for changes over time, reflecting how pressure is distributed over such dynamic surfaces.

Think of the ocean surface during a storm. As waves rise and fall, we can't simply assume the pressure above the water is constant; it fluctuates based on the wave height and speed. By using the dynamic free surface boundary condition, we can better understand and predict phenomena like the force of waves on a shore.

To derive the dynamic boundary condition, we assume the gauge pressure Pn is 0. If the wavelength is very short, we have to consider the importance of surface tension, leading to equations where curvature at the water's surface affects pressure distribution.

When deriving the dynamic boundary condition using unsteady Bernoulli’s equation, we simplify by assuming that gauge pressure is zero at the surface. In scenarios where wavelengths are short, surface tension becomes significant, impacting the pressure at the water surface. This leads us to express how pressure variations occur due to curvature at the water surface, providing insights into wave dynamics.

Imagine blowing bubbles into a drink; the surface tension creates a curved shape at the surface of the bubble, which results in different pressure readings at the bubble’s edge compared to the flat surface of the drink. This principle can also explain how raindrops form; they resist deformation due to surface tension, just like waves resist flattening.

In analyzing the water surface under wave motion, we can derive that the pressure beneath the free surface (P) is affected by the curvature (represented by surface tension) leading to representation where P is expressed as Pn - σ (d²η/dx²), linking pressure variation to wave curvature.

When looking at fluid dynamics with waves, the shape of the surface impacts the underlying pressure. The formula indicates this relationship—taking into account the curvature represents how waves distort the regular flow pattern, affecting pressure distribution underneath. This is crucial for understanding how waves carry energy and momentum.

Consider how a basketball appears as it bounces on the court. As it hits the ground, the shape curvature alters and creates pressure that helps it bounce back. Similarly, in waves, the curvature at the peaks alters the pressure beneath, influencing how the wave propagates.

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