College Physics for AP® Courses

# Section Summary

### 12.1Flow Rate and Its Relation to Velocity

• Flow rate $QQ size 12{Q} {}$ is defined to be the volume $VV size 12{V} {}$ flowing past a point in time $tt size 12{t} {}$, or $Q=VtQ=Vt size 12{Q= { {V} over {t} } } {}$ where $VV size 12{V} {}$ is volume and $tt size 12{t} {}$ is time.
• The SI unit of volume is $m3m3 size 12{m rSup { size 8{3} } } {}$.
• Another common unit is the liter (L), which is $10−3m310−3m3 size 12{"10" rSup { size 8{ - 3} } m rSup { size 8{3} } } {}$.
• Flow rate and velocity are related by $Q=A v ¯ Q=A v ¯ size 12{Q=A {overline {v}} } {}$ where $AA size 12{A} {}$ is the cross-sectional area of the flow and $v ¯ v ¯ size 12{ {overline {v}} } {}$ is its average velocity.
• For incompressible fluids, flow rate at various points is constant. That is,
$Q 1 = Q 2 A 1 v ¯ 1 = A 2 v ¯ 2 n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 . Q 1 = Q 2 A 1 v ¯ 1 = A 2 v ¯ 2 n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 . size 12{ left none matrix { Q rSub { size 8{1} } =Q rSub { size 8{2} } {} ## A rSub { size 8{1} } {overline {v}} rSub { size 8{1} } =A rSub { size 8{2} } {overline {v}} rSub { size 8{2} } {} ## n rSub { size 8{1} } A rSub { size 8{1} } {overline {v}} rSub { size 8{1} } =n rSub { size 8{2} } A rSub { size 8{2} } {overline {v}} rSub { size 8{2} } } right rbrace "." } {}$

### 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:
$P1+12ρv12+ρgh1=P2+12ρv22+ρgh2.P1+12ρv12+ρgh1=P2+12ρv22+ρgh2. size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } } {}$
• 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
$P1+12ρv12=P2+12ρv22.P1+12ρv12=P2+12ρv22. size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } } {}$
• 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 $P1+12ρv2+ρghQ=power,P1+12ρv2+ρghQ=power, size 12{ left (P rSub { size 8{1} } + { {1} over {2} } ρv rSup { size 8{2} } +ρ ital "gh" right )Q="power"} {}$ where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

### 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 $ηη size 12{η} {}$ 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 size 12{ $$"N/m" rSup { size 8{2} }$$ s} {}$ or $Pa ⋅ s Pa ⋅ s size 12{"Pa" cdot s} {}$.
• Flow is proportional to pressure difference and inversely proportional to resistance:
$Q=P2−P1R.Q=P2−P1R. size 12{Q= { {P rSub { size 8{2} } - P rSub { size 8{1} } } over {R} } } {}$
• For laminar flow in a tube, Poiseuille's law for resistance states that
$R=8ηlπr4.R=8ηlπr4. size 12{R= { {8ηl} over {πr rSup { size 8{4} } } } } {}$
• Poiseuille's law for flow in a tube is
$Q=(P2−P1)πr48ηl.Q=(P2−P1)πr48ηl. size 12{Q= { { $$P rSub { size 8{2} } - P rSub { size 8{1} }$$ πr rSup { size 8{4} } } over {8ηl} } } {}$
• The pressure drop caused by flow and resistance is given by
$P2−P1=RQ.P2−P1=RQ. size 12{P rSub { size 8{2} } - P rSub { size 8{1} } =R`Q} {}$

### 12.5The Onset of Turbulence

• The Reynolds number $NRNR size 12{N rSub { size 8{R} } } {}$ can reveal whether flow is laminar or turbulent. It is
$NR=2ρvrη.NR=2ρvrη. size 12{N rSub { size 8{R} } = { {2ρ ital "vr"} over {η} } } {}$
• For $NRNR size 12{N rSub { size 8{R} } } {}$ below about 2000, flow is laminar. For $NRNR size 12{N rSub { size 8{R} } } {}$ above about 3000, flow is turbulent. For values of $NRNR size 12{N rSub { size 8{R} } } {}$ 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 $N′R=ρvLη(object in fluid),N′R=ρvLη(object in fluid), size 12{ { {N}} sup { ' } rSub { size 8{R} } = { {ρ ital "vL"} over {η} } } {}$ which indicates whether flow is laminar or turbulent.
• For $N′RN′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ less than about one, flow is laminar.
• For $N′RN′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ greater than $106106 size 12{"10" rSup { size 8{6} } } {}$, 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 $xrmsxrms size 12{x rSub { size 8{"rms"} } } {}$ a molecule travels by diffusion in a given amount of time is given by
$xrms=2Dt,xrms=2Dt, size 12{x rSub { size 8{"rms"} } = sqrt {2 ital "Dt"} } {}$

where $DD size 12{D} {}$ 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.