1.5. Terminal Fall Velocity

Terminal velocity is the maximum velocity that can be attained by an object as it falls through a fluid. It occurs when the sum of the drag force and the buoyant force on the particle is equal to the weight of the particle.

Terminal velocity occurs when an object moves through a fluid (like air or water) and reaches a constant speed. At this point, two forces acting on the object balance each other: the force of gravity pulling it down and the upward forces of drag and buoyancy. When these forces are balanced, the net force on the object is zero, resulting in no acceleration – it falls at a steady speed.

Imagine a skydiver reaching a steady speed during freefall. Initially, they accelerate due to gravity, but as they speed up, air resistance (drag) increases. Eventually, the drag force equals the weight of the skydiver, and they stop accelerating, falling at a constant speed – this is their terminal velocity.

The downward force is the weight of the particle, calculated as W =   / 6  3    . The upward forces include the drag force ( F_D) and buoyant force ( F_b).

When considering a sphere falling through a fluid, the main forces are:

1. Weight (W): This is the force due to gravity acting on the sphere and is calculated by the volume of the sphere multiplied by the density of the sphere and gravitational acceleration.
2. Drag Force (F_D): This results from the fluid resistance and depends on the sphere's speed, its radius, and other fluid characteristics.
3. Buoyancy Force (F_b): This is equal to the weight of the fluid displaced by the sphere. At terminal velocity, the sum of the drag and buoyant forces equals the weight of the sphere.

Think of a marble falling in a glass of water. Initially, it sinks faster, but as it moves through the water, it pushes some water out of the way (buoyancy) and encounters water resistance (drag). Eventually, it reaches a speed where the upward forces of water resistance and buoyancy equal its weight, resulting in a constant falling speed.

For creeping flow (Reynolds number < 1), terminal velocity can be expressed as V = D^2 / 18   (_p - _f) / g.

The formula for terminal velocity shows how different factors are intertwined. D represents the diameter of the particle, _p is the density of the particle, _f is the density of the fluid, and g is the acceleration due to gravity. In a regime where the flow is slow and viscous (creeping flow), the equation helps predict how fast an object will fall in a fluid based on these physical properties. It illustrates that smaller particles will fall more slowly compared to larger ones due to less gravitational force acting on them, relative to drag forces.

Consider a small feather and a large rock. When both are released in air, the feather drifts down slowly while the rock falls rapidly. This is because the feather experiences significant drag relative to its weight, while the rock's greater weight allows it to overcome drag more effectively, both demonstrating principles from the terminal velocity equation.

To determine the fall velocity of a 0.06 millimeters sand particle (specific gravity 2.65) in water and viscosity of 10^-3 Pascal second, we calculate using the terminal velocity formula.

To solve this problem, first, we need to convert the particle diameter to meters and calculate the density of the particle based on its specific gravity. The formula for terminal velocity in creeping flow is then applied:

• First, calculate the density of the particle using its specific gravity.
• Next, substitute the values into the equation for terminal velocity to find out how fast the particle falls through water. After the calculation, we also need to check if our assumption (Reynolds number < 1) is satisfied by using the resultant velocity and diameter.

This situation is like determining how quickly a grain of sand will settle at the bottom of a still pond. Understanding the velocity of the grain helps in predicting sedimentation patterns, essential for engineers when designing structures or for environmental studies when measuring sediment transport.

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