Fluids

## Big Picture

A fluid is any substance that takes the form of its container. This includes both liquids and gases, but we will focus on liquids in this study guide. Since fluids are amorphous (shapeless) and made of molecules that are allowed to move relative to each other, we cannot treat them the same as solids. Unlike solids, forces cannot be exerted on a single point of a fluid. Therefore, we must measure quantities such as force and energy in terms of their density. In the study of fluids, volume and mass density are more important quantities than mass.

## Key Terms

Pressure: Measure of force per unit area. SI units: Pa

Pascal’s Law: The pressure on a fluid is distributed equally throughout the whole fluid.

Archimedes’ Principle: Describes the upward buoyant force on an object in a fluid. The buoyant force is proportional to the weight of the displaced volume of the fluid. SI units: N

Bernoulli’s Equation: Equation for the conservation of energy of fluids.

## Pressure of a Fluid

The pressure of a fluid represents the force exerted by the fluid’s molecules colliding with each other. Pressure can be measured in two ways. Absolute pressure is measured with 0 pressure being a vacuum. Gauge pressure is measured where 0 is the ambient air pressure at sea level.

### Pascal’s Law

A small force applied over a small surface on one end of a fluid's container will produce a large force on a large surface on the other end of the container - the pressure ratio is equal. This is known as Pascal's law and is the physical principle behind all hydraulic machinery. To the right is a diagram that illustrates this concept.

Image Credit: Lidingo, CC-BY-SA 3.0

### Buoyancy

If there is an object within the fluid, the fluid exerts a pressure on the object.

• The pressure exerted by the fluid depends on the depth of the object because the weight of the fluid above the object exerts a downward pressure on the object.
• The upward force (called a bouyant force) is described by Archimedes’ Principle. There is an upward bouyant force because the bottom of the object is at lower depth and higher pressure than the top of the object.
• If the object has a density equal or lower than the density of the fluid, this buoyant force will cause the object to float so that the volume of displaced water
• If the object has a density greater than the density of the fulid, the weight of the displaced volume is always less than the object’s weight, so the object will sink.

Image Credit: Yupi666, GNU-FDL 1.2

Fluids cont.

## Properties of Fluids

• Continuity of fluid flow: The amount of liquid entering a system in any period of time is equal to the amount of liquid leaving. The continuity of fluid flow can be demonstrated by putting your finger over the end of a running hose. By making the area of the hose’s end smaller, the water must come out faster to maintain continuity.
• Conservation of energy is applied to fluids using Bernoulli’s equation, which substitutes energy density for actual energy values.
• Surface tension is the result of the cohesive forces of the molecules in a liquid. The molecules resist any force pushing down on the surface that would split the molecules apart. Surface tension allows some very light objects that would normally sink to float on the surface.
• Viscosity in fluids is analogous to friction for solid objects. A fluid’s viscosity measures its resistance to flow or change its shape under stress. Laminar flow is when kinetic energy is lost by a flowing liquid due to its viscosity.
• Turbulence describes how the flow of a liquid in a certain direction becomes irregular as the liquid flows in some random directions, dissipating some of the kinetic energy of the liquid

## Important Equations

$$p = \frac{m}{v}$$

ρ - mass density

m - mass

V - volume

$$p = \frac{F}{A}$$

P - pressure

F - force

A - area

$$u_g = pgh$$

$$u_g$$ - gravitational potential energy density

g - acceleration due to gravity

h - height

$$ke = \frac{1}{2}pv^2$$

ke - kinetic energy density

$$p = \frac{m}{v}$$

Continuity of fluid flow: $$\Delta p + \Delta ke + \Delta U_g = 0$$

Bernoulli’s equation: $$A_iV_i = A_fV_f$$