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 $m^{3}$ .
Another common unit is the liter (L), which is $10^{- 3} m^{3}$ .
Flow rate and velocity are related by $Q = A \bar{v}$ where $A$ is the cross-sectional area of the flow and $\bar{v}$ is its average velocity.
For incompressible fluids, flow rate at various points is constant. That is,
$\begin{matrix} Q_{1} = Q_{2} \\ A_{1} {\bar{v}}_{1} = A_{2} {\bar{v}}_{2} \\ n_{1} A_{1} {\bar{v}}_{1} = n_{2} A_{2} {\bar{v}}_{2} \end{matrix}\} .$

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:
$P_{1} + \frac{1}{2} {ρv}_{1}^{2} + ρ {gh}_{1} = P_{2} + \frac{1}{2} {ρv}_{2}^{2} + ρ {gh}_{2} .$
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
$P_{1} + \frac{1}{2} {ρv}_{1}^{2} = P_{2} + \frac{1}{2} {ρv}_{2}^{2} .$
Bernoulli's principle has many applications, including entrainment, wings and sails, and velocity measurement.

Power in fluid flow is given by the equation $(P_{1} + \frac{1}{2} {ρv}^{2} + ρ gh) Q = power,$ where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

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 $η$ is due to friction within a fluid. Representative values are given in Table 12.1. Viscosity has units of $({N/m}^{2}) s$ or $Pa \cdot s$ .
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
$R = \frac{8 η l}{{πr}^{4}} .$
Poiseuille's law for flow in a tube is
$Q = \frac{(P_{2} - P_{1}) π r^{4}}{8 η l} .$
The pressure drop caused by flow and resistance is given by
$P_{2} - P_{1} = R Q .$

The Reynolds number $N_{R}$ can reveal whether flow is laminar or turbulent. It is
$N_{R} = \frac{2 ρ vr}{η} .$
For $N_{R}$ below about 2000, flow is laminar. For $N_{R}$ above about 3000, flow is turbulent. For values of $N_{R}$ between 2000 and 3000, it may be either or both.

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

Diffusion is the movement of substances due to random thermal molecular motion.
The average distance $x_{rms}$ a molecule travels by diffusion in a given amount of time is given by
$x_{rms} = \sqrt{2 D t},$
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.