Fluid continuity, pressure energy and viscous flow
| English | Español |
|---|---|
| volume flow rate | volume flow rate |
| dynamic viscosity | dynamic viscosity |
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:
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:
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.
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.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.