### 12.1 Flow Rate and Its Relation to Velocity

- Flow rate
*$Q$*is defined to be the volume*$V$*flowing past a point in time*$t$*, or $Q=\frac{V}{t}$ where $V$ is volume and $t$ is time. - The SI unit of volume is ${\text{m}}^{3}$.
- Another common unit is the liter (L), which is ${\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$.
- Flow rate and velocity are related by $Q=A\overline{v}$ where $A$ is the cross-sectional area of the flow and $\overline{v}$ is its average velocity.
- For incompressible fluids, flow rate at various points is constant. That is,$$\left.\begin{array}{c}{Q}_{1}={Q}_{2}\\ {A}_{1}{\overline{v}}_{1}={A}_{2}{\overline{v}}_{2}\\ {n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\end{array}\right\}\text{.}$$

### 12.2 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:
$${P}_{1}+\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}+\rho {\mathrm{gh}}_{1}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}+\rho {\text{gh}}_{2}.$$
- Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height
*h*) subtract out, yielding$${P}_{1}+\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}.$$ - Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

### 12.3 The Most General Applications of Bernoulli’s Equation

- Power in fluid flow is given by the equation $\left({P}_{1}+\frac{1}{2}{\mathrm{\rho v}}^{2}+\rho \text{gh}\right)Q=\text{power}\text{,}$ where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

### 12.4 Viscosity and Laminar Flow; Poiseuille’s Law

- 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. Representative values are given in Table 12.1. Viscosity has units of $({\text{N/m}}^{2})\text{s}$ or $\text{Pa}\cdot \text{s}$. - Flow is proportional to pressure difference and inversely proportional to resistance:
$$Q=\frac{{P}_{2}-{P}_{1}}{R}.$$
- For laminar flow in a tube, Poiseuille’s law for resistance states that
$$R=\frac{8\eta l}{{\mathrm{\pi r}}^{4}}.$$
- Poiseuille’s law for flow in a tube is
$$Q=\frac{({P}_{2}-{P}_{1})\pi {r}^{4}}{8\eta l}.$$
- The pressure drop caused by flow and resistance is given by
$${P}_{2}-{P}_{1}=RQ.$$

### 12.5 The Onset of Turbulence

- The Reynolds number ${N}_{\text{R}}$ can reveal whether flow is laminar or turbulent. It is$${N}_{\text{R}}=\frac{2\rho \text{vr}}{\eta}.$$
- 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.

### 12.6 Motion of an Object in a Viscous Fluid

- When an object moves in a fluid, there is a different form of the Reynolds number ${N}_{\text{R}}^{\prime}=\frac{\rho \text{vL}}{\eta}\text{(object in fluid),}$ which indicates whether flow is laminar or turbulent.
- For ${N}_{\text{R}}^{\prime}$ less than about one, flow is laminar.
- For ${N}_{\text{R}}^{\prime}$ greater than ${\text{10}}^{6}$, flow is entirely turbulent.

### 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Diffusion is the movement of substances due to random thermal molecular motion.
- The average distance ${x}_{\text{rms}}$ a molecule travels by diffusion in a given amount of time is given by$${x}_{\text{rms}}=\sqrt{2D\text{t}},$$
where $D$ is the diffusion constant, representative values of which are found in Table 12.2.

- Osmosis is the transport of water through a semipermeable membrane from a region of high concentration to a region of low concentration.
- Dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference.
- Both processes can be reversed by back pressure.
- Active transport is a process in which a living membrane expends energy to move substances across it.