College Physics for AP® Courses 2e

# Section Summary

### 12.1Flow Rate and Its Relation to Velocity

• Flow rate $QQ$ is defined to be the volume $VV$ flowing past a point in time $tt$, or $Q=VtQ=Vt$ where $VV$ is volume and $tt$ is time.
• The SI unit of volume is $m3m3$.
• Another common unit is the liter (L), which is $10−3m310−3m3$.
• Flow rate and velocity are related by $Q=A v ¯ Q=A v ¯$ where $AA$ is the cross-sectional area of the flow and $v ¯ v ¯$ is its average velocity.
• For incompressible fluids, flow rate at various points is constant. That is,
$Q 1 = Q 2 A 1 v ¯ 1 = A 2 v ¯ 2 n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 . Q 1 = Q 2 A 1 v ¯ 1 = A 2 v ¯ 2 n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 .$

### 12.2Bernoulli’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:
$P1+12ρv12+ρgh1=P2+12ρv22+ρgh2.P1+12ρv12+ρgh1=P2+12ρv22+ρgh2.$
• 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
$P1+12ρv12=P2+12ρv22.P1+12ρv12=P2+12ρv22.$
• Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

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

• Power in fluid flow is given by the equation $P1+12ρv2+ρghQ=power,P1+12ρv2+ρghQ=power,$ where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

### 12.4Viscosity 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 $ηη$ is due to friction within a fluid. Representative values are given in Table 12.1. Viscosity has units of $( N/m 2 ) s ( N/m 2 ) s$ or $Pa ⋅ s Pa ⋅ s$.
• Flow is proportional to pressure difference and inversely proportional to resistance:
$Q=P2−P1R.Q=P2−P1R.$
• For laminar flow in a tube, Poiseuille’s law for resistance states that
$R=8ηlπr4.R=8ηlπr4.$
• Poiseuille’s law for flow in a tube is
$Q=(P2−P1)πr48ηl.Q=(P2−P1)πr48ηl.$
• The pressure drop caused by flow and resistance is given by
$P2−P1=RQ.P2−P1=RQ.$

### 12.5The Onset of Turbulence

• The Reynolds number $NRNR$ can reveal whether flow is laminar or turbulent. It is
$NR=2ρvrη.NR=2ρvrη.$
• For $NRNR$ below about 2000, flow is laminar. For $NRNR$ above about 3000, flow is turbulent. For values of $NRNR$ between 2000 and 3000, it may be either or both.

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

• When an object moves in a fluid, there is a different form of the Reynolds number $NR′=ρvLη(object in fluid), NR′=ρvLη(object in fluid),$ which indicates whether flow is laminar or turbulent.
• For $NR′NR′$ less than about one, flow is laminar.
• For $NR′NR′$ greater than $106106$, flow is entirely turbulent.

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

• Diffusion is the movement of substances due to random thermal molecular motion.
• The average distance $xrmsxrms$ a molecule travels by diffusion in a given amount of time is given by
$xrms=2Dt,xrms=2Dt,$

where $DD$ 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. Do you know how you learn best?