FLUID DYNAMICS

Air Resistance If you drop a feather and a cannonball at the

same time, from the same height, which will hit the ground first? The cannonball of course. Why does the feather take longer? For it’s mass it has much more surface area are bouncing against air molecules than the cannonball. Air resistance slows it down. If we did the same experiment on the moon (no air) they would hit at the same time. Parachutes are one way in using air resistance, by dramatically increasing your area with the parachute, you can slow yourself down to a speed where you can land without injury.

Fluid Flow Fluids in motion often behave in complex and

unpredictable ways. However, we can understand many aspects of fluid flow on the basis of a simple model that in many cases is reasonably realistic. He liquids in this model are supposed to be incompressible and have no viscosity. Viscosity is the term used to describe internal frictional resistances in a fluid.

The volume of liquid that flows through a pipe per unit is easy to figure out. The rate of flow R of a liquid through the pipe is

R = vtA/t = vA

The rate of flow is the product of the liquid speed and the cross sectional area.

If the pipe size varies, the speed of a liquid also varies so as to keep R constant, so that

v1A = v2A2

This is the equation of continuity. Hence a liquid flows faster through a constriction in a pipe and slower through a dilation.

Bernoulli’s Equation

When a liquid is flowing in a pipe in a region where the pipe diameter gets smaller, it’s speed increases, A change in speed is acceleration, which means that a net force must be acting on the liquid. This force can only come from a difference in pressure. Evidently, the pressure must be higher in the large diameter part of the pipe, because the liquid increases I speed on its way to the constriction.

The liquid speed is also affected by changes in the height of the pipe. If the liquid rises, it slows down, and if the liquid falls, its speeds up. Thus we expect the pressure, the speed, and the height of a moving liquid to be related in some way. The relationship turns out to be

P1 – pgh1 + ½ p v1² = P2 + pgh2 + ½ p v2²

Example: Water flows upward through a pipe at 15 1/s. If water enters the lower end of the pipe at 3.0 m/s, what is the difference in pressure between the two ends?