Section Summary

Flow 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=Av¯𝑄=𝐴𝑣¯ 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,Q1=Q2A1v¯1=A2v¯2n1A1v¯1=n2A2v¯2⎫⎭⎬⎪⎪.𝑄1=𝑄2𝐴1𝑣¯1=𝐴2𝑣¯2𝑛1𝐴1𝑣¯1=𝑛2𝐴2𝑣¯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+12ρv21+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, yieldingP1+12ρv21=P2+12ρv22.𝑃1+12ρv12=𝑃2+12ρv22.
• Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

The Most General Applications of Bernoulli’s Equation

• Power in fluid flow is given by the equation (P1+12ρv2+ρgh)Q=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.

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 is due to friction within a fluid. Representative values are given in Table 12.1. Viscosity has units of (N/m2)s(N/m2)s or Pa⋅sPa⋅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 thatR=8ηlπr4.𝑅=8𝜂𝑙πr4.
• Poiseuille’s law for flow in a tube isQ=(P2−P1)πr48ηl.𝑄=(𝑃2−𝑃1)𝜋𝑟48𝜂𝑙.
• The pressure drop caused by flow and resistance is given byP2−P1=RQ.𝑃2−𝑃1=𝑅𝑄.

The Onset of Turbulence

• The Reynolds number NR𝑁R can reveal whether flow is laminar or turbulent. It isNR=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.

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),𝑁R′=𝜌vL𝜂(object in fluid), which indicates whether flow is laminar or turbulent.
• For N′R𝑁R′ less than about one, flow is laminar.
• For N′R𝑁R′ greater than 106106, flow is entirely turbulent.

Molecular 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 byxrms=2Dt−−−√,𝑥rms=2𝐷t,where D𝐷 is the diffusion constant, representative values of which are found in Table 12.2.
• Osmosis is a process by which molecules of a solvent pass through a semipermeable membrane from a less concentrated solution into a more concentrated one, thus, equalizing the solute concentrations on each side of the membrane.
• 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.