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Section Summary

12.1 Flow 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 m3.m3. size 12{m rSup { size 8{3} } } {}
  • Another common unit is the liter (L), which is 10−3m3.10−3m3. 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 "." } {}

12.2 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.
    P1+12ρv12+ρgh1=P2+12ρv22+ρgh2P1+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.

12.3 The 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.

12.4 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 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−P1RQ=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= { { \( P rSub { size 8{2} } - P rSub { size 8{1} } \) π`r 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} {}

12.5 The 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.

12.6 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′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 106,106, size 12{"10" rSup { size 8{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 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.