### Summary

## 14.1 Fluids, Density, and Pressure

- A fluid is a state of matter that yields to sideways or shearing forces. Liquids and gases are both fluids. Fluid statics is the physics of stationary fluids.
- Density is the mass per unit volume of a substance or object, defined as $\rho =m\text{/}V.$ The SI unit of density is ${\text{kg/m}}^{3}.$
- Pressure is the force per unit perpendicular area over which the force is applied, $p=F\text{/}A.$ The SI unit of pressure is the pascal: $1\phantom{\rule{0.2em}{0ex}}\text{Pa}=1\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$.
- Pressure due to the weight of a liquid of constant density is given by $p=\rho \phantom{\rule{0.2em}{0ex}}gh$, where
*p*is the pressure,*h*is the depth of the liquid, $\rho $ is the density of the liquid, and*g*is the acceleration due to gravity.

## 14.2 Measuring Pressure

- Gauge pressure is the pressure relative to atmospheric pressure.
- Absolute pressure is the sum of gauge pressure and atmospheric pressure.
- Open-tube manometers have U-shaped tubes and one end is always open. They are used to measure pressure. A mercury barometer is a device that measures atmospheric pressure.
- The SI unit of pressure is the pascal (Pa), but several other units are commonly used.

## 14.3 Pascal's Principle and Hydraulics

- Pressure is force per unit area.
- A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
- A hydraulic system is an enclosed fluid system used to exert forces.

## 14.4 Archimedes’ Principle and Buoyancy

- Buoyant force is the net upward force on any object in any fluid. If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object can remain suspended at its present depth. The buoyant force is always present and acting on any object immersed either partially or entirely in a fluid.
- Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid it displaces.

## 14.5 Fluid Dynamics

- Flow rate
*Q*is defined as the volume*V*flowing past a point in time*t*, or $Q=\frac{dV}{dt}$ where*V*is volume and*t*is time. The SI unit of flow rate is ${\text{m}}^{3}\text{/s,}$ but other rates can be used, such as L/min. - Flow rate and velocity are related by $Q=Av$ where
*A*is the cross-sectional area of the flow and*v*is its average velocity. - The equation of continuity states that for an incompressible fluid, the mass flowing into a pipe must equal the mass flowing out of the pipe.

## 14.6 Bernoulli’s Equation

- Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
$${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}+\rho g{h}_{1}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}+\rho g{h}_{2}.$$
- Bernoulli’s principle is Bernoulli’s equation applied to situations in which the height of the fluid is constant. The terms involving depth (or height
*h*) subtract out, yielding$${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}.$$ - Bernoulli’s principle has many applications, including entrainment and velocity measurement.

## 14.7 Viscosity and Turbulence

- Laminar flow is characterized by smooth flow of the fluid in layers that do not mix.
- Turbulence is characterized by eddies and swirls that mix layers of fluid together.
- Fluid viscosity $\eta $ is due to friction within a fluid.
- Flow is proportional to pressure difference and inversely proportional to resistance:
$$Q=\frac{p-2{p}_{1}}{R}.$$
- The pressure drop caused by flow and resistance is given by ${p}_{2}\u2013{p}_{1}=RQ$.
- The Reynolds number ${N}_{\text{R}}$ can reveal whether flow is laminar or turbulent. It is ${N}_{\text{R}}=\frac{2\rho vr}{\eta}$.
- For ${N}_{\text{R}}$ below about 2000, flow is laminar. For ${N}_{\text{R}}$ above about 3000, flow is turbulent. For values of ${N}_{\text{R}}$ between 2000 and 3000, it may be either or both.