流体连续性、圧力エネルギーおよび粘性流れ
導入済み| English | 意味 |
|---|---|
| volume flow rate/ˈvɒljuːm fləʊ reɪt/ | 体積流量 |
| dynamic viscosity/daɪˈnæmɪk vɪˈskɒsɪti/ | 動粘度 |
A decision before an answer
- A narrow pipe carries the same steady flow rate as a wide pipe, but its pressure behaviour depends on both speed and viscous losses.
- Your goal: Combine incompressible continuity with pressure, height and kinetic energy.
Conserve the flow rate
- For steady flow, the mass passing successive cross-sections per unit time is equal: ρAv is constant. An incompressible liquid has nearly constant ρ, so Q=Av is the volume flow rate. Halving pipe radius quarters area and multiplies mean speed by four for the same Q.
- Pressure does not determine Q without a model of the rest of the system. In a stationary liquid, dp/dz=−ρg with z upward; therefore pressure increases by ρgh a distance h below a surface. Use absolute or gauge pressure consistently on both sides.
At fixed steady incompressible Q, pipe radius is halved. Mean speed becomes:
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Area is proportional to radius squared, so the area quarters and speed quadruples.
Account for pressure energy
- For steady, incompressible, inviscid motion along a streamline with no pump or dissipative loss, Bernoulli gives p+ρv²/2+ρgz=constant. These three terms are energy per volume and have pressure units. At equal height, larger speed requires smaller static pressure under these assumptions.
- A higher outlet also uses pressure/kinetic energy to gain gravitational energy. Bernoulli along one streamline need not imply the same constant on different streamlines in rotational flow. A stagnation point has v=0 and converts local speed energy to a pressure rise in the ideal model.
For fully developed laminar tube flow, radius doubles while Δp, η and L stay fixed. Q becomes:
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Poiseuille flow has Q proportional to R⁴ under these stated conditions.
Include viscous loss
- Viscosity transports momentum between neighbouring fluid layers. For steady fully developed laminar flow of a Newtonian incompressible liquid in a circular tube, Q=πR⁴Δp/(8ηL). Here Δp is the pressure drop along the tube, η dynamic viscosity and L tube length.
- Doubling radius at fixed Δp, η and L multiplies Q by sixteen. Doubling Q at fixed geometry needs twice the pressure drop. This viscous pressure loss cannot be added to a loss-free Bernoulli equation as though nothing changes; include a dissipative pressure/head loss or use the viscous model.
Water with ρ=1000 kg/m³ flows horizontally from area 4 cm² at 1 m/s into 1 cm². Continuity gives v₂=4 m/s and Q=0.0004 m³/s. Ideal Bernoulli predicts p₁−p₂=1000(16−1)/2=7500 Pa. In a separate laminar tube at fixed pressure drop, increasing radius from 1 mm to 2 mm multiplies Q by 16; this is not the same fixed-Q experiment.
A stationary liquid with ρ=800 kg/m³ and g=10 has a gauge-pressure rise of ____ Pa at depth 0.5 m.
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ρgh=800·10·0.5=4000.
Check the flow regime
- The Reynolds number Re=ρvD/η compares inertial and viscous effects using characteristic speed v and length D. For a pipe use mean speed and internal diameter, not radius. Re is dimensionless: density times speed times length has the same units as dynamic viscosity.
- Small Re favours viscous dominance; transition to turbulence depends on geometry and disturbances, so a single threshold is not a universal law. Poiseuille scaling is not safe after assuming a turbulent flow. Check volume continuity, sign of pressure change and the regime before selecting a formula.
Continuity keeps Q constant across one steady pipe; Poiseuille compares Q between different systems at a specified pressure drop. State what is held fixed before comparing radius powers.
Which answer fits this case?
Combine incompressible continuity with pressure, height and kinetic energy
Loss-free Bernoulli can be used unchanged to describe a finite viscous pressure drop in a uniform horizontal tube.
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Equal speed and height would predict equal pressure; a viscous loss requires an additional term/model.
Keep the distinctions
- volume flow rate 体积流量 — Volume crossing a section per unit time, Q=Av for mean speed v.
- dynamic viscosity 动力黏度 — The coefficient relating shear stress to velocity gradient in a Newtonian fluid.
- Combine incompressible continuity with pressure, height and kinetic energy.
- Apply hydrostatic and Bernoulli relations only under their stated conditions.
- Use viscosity, Poiseuille scaling and Reynolds number to distinguish flow regimes.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.