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Đã giới thiệu| English | Ý nghĩa |
|---|---|
| viscosity/vɪˈskɒsɪti/ | A measure of a fluid's resistance to flow or to relative motion between adjacent layers. |
| continuity/kɒntɪˈnjuːɪti/ | continuity |
| Reynolds number/ˈrenəldz ˈnʌmbə/ | The dimensionless ratio of inertial to viscous effects in a flow. |
Same flow, different pressure
- Water flows from a wide pipe into a narrow one. The flow rate is the same in both, the speed rises fourfold, and the pressure drops — if the flow is ideal. Add viscosity 黏度 and the story changes.
- GRE fluids questions test continuity 连续性, hydrostatics, Bernoulli, Poiseuille flow and the Reynolds number 雷诺数.
Continuity and hydrostatics
- Steady incompressible flow: $Q = Av$ is constant along a pipe. Halving the radius quarters the area and quadruples the speed.
- At rest: $p = p_0 + \rho gh$ at depth $h$. Use absolute or gauge pressure consistently.
A pipe narrows to half its radius. The flow speed becomes
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Continuity keeps $Av$ constant; half the radius is a quarter of the area, so the speed becomes four times as large.
Bernoulli
- Along a streamline of steady, incompressible, inviscid flow with no pump: $p + \tfrac12\rho v^2 + \rho gz =$ const. Each term is energy per volume.
- Faster at the same height means lower pressure. A stagnation point converts $\tfrac12\rho v^2$ into a pressure rise.
Water from 4 cm² at 1 m/s into 1 cm²: $v_2 = 4$ m/s, $Q = 4\times10^{-4}$ m³/s, and $p_1 - p_2 = \tfrac12 \cdot 1000 \cdot (16 - 1) = 7500$ Pa.
Water ($\rho = 1000$) flows horizontally from 6 cm² at 2 m/s into 2 cm². Pressure drop in Pa?
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v₂ = 6 m/s; 500 × (36 − 4).
Viscous flow: Poiseuille
- Laminar flow in a tube: $Q = \dfrac{\pi R^4\Delta p}{8\eta L}$. Doubling $R$ at fixed $\Delta p$ multiplies $Q$ by 16.
- This is a different comparison from continuity, which holds $Q$ fixed within one pipe.
Bernoulli or a loss model?
Decide whether lossless Bernoulli applies.
At fixed pressure drop, a tube's radius is multiplied by 1.5. Factor change in laminar $Q$ (four decimal places)?
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Poiseuille's law gives $Q \propto r^4$, so $1.5^4 = 5.0625$.
Bernoulli's equation applies unchanged to viscous honey flowing slowly through a thin straw.
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Viscous losses dominate; use Poiseuille.
Reynolds number
- $\text{Re} = \dfrac{\rho vD}{\eta}$ (diameter, mean speed) is dimensionless. Low Re: viscous, laminar. High Re: inertia dominates, turbulence possible — the threshold depends on geometry.
- Poiseuille's $R^4$ law fails once flow is turbulent.
Water ($\rho = 1000$, $\eta = 0.001$ Pa·s) at 0.5 m/s in a 2 cm diameter pipe. Reynolds number?
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1000 × 0.5 × 0.02 / 0.001.
Recap
- Continuity: $Av$ constant; hydrostatics $\rho gh$.
- Bernoulli only for lossless steady flow along a streamline.
- Poiseuille $Q \propto R^4\Delta p$ for laminar flow; check Re.
- Reynolds number: The dimensionless ratio of inertial to viscous effects in a flow.