9. Poiseuille's Method — Coefficient of Viscosity of a Liquid¶
Module
Fluid Properties: Viscosity & Surface Tension · Module 3
Try it live
Explore this on the interactive simulator with real-time sliders and graphs: 🔬 Launch in Simulator
Aim¶
To determine the coefficient of viscosity of a liquid by studying its steady, streamline flow through a narrow capillary tube under a constant pressure head.
Theoretical Background¶
When a liquid flows slowly and smoothly (in streamline, or laminar, flow) through a narrow horizontal capillary tube under a small, steady pressure difference between its two ends, adjacent layers of the liquid slide over one another at different speeds — fastest along the central axis and effectively zero at the tube wall (the no-slip condition) — because of the liquid's internal friction, or viscosity.
Poiseuille analysed this velocity profile and showed that the volume of liquid flowing per second depends extremely sensitively on the tube's radius (to the fourth power), linearly on the pressure gradient driving the flow, and inversely on both the liquid's viscosity and the tube's length — a result now called Poiseuille's law, foundational to understanding flow in blood vessels, industrial pipelines, and microfluidic devices alike.
In this experiment, the pressure head is supplied by a column of the liquid itself, of height h above the capillary, giving a driving pressure P = ρgh; a fixed volume V of liquid is collected at the outlet and the time t taken is measured, from which the coefficient of viscosity η can be calculated directly.
Because the r⁴ dependence makes the flow rate so sensitive to the capillary's bore, accurately knowing (or independently calibrating) the capillary radius is usually the single most important practical requirement of this experiment.
Governing Formulas¶
Poiseuille's equation (volume V collected in time t):
where \(P = \rho g h\) is the driving pressure (liquid density \(\rho\), head \(h\)), \(r\) the capillary radius, and \(l\) its length.
Simulator Controls¶
| Control | Symbol | Range | Unit |
|---|---|---|---|
| Pressure head | h | 5 – 40 | cm |
| Capillary radius | r | 0.20 – 0.60 | mm |
| Capillary length | l | 5 – 20 | cm |
| Liquid selection (density) | ρ | 800 – 1300 | kg/m³ |
Readouts:
| Quantity | Symbol | Unit |
|---|---|---|
| Time to collect fixed volume V = 10 cm³ | t | s |
Procedure¶
- Launch the Poiseuille's Method panel; select a liquid and note its density, and set the capillary radius and length to their default values.
- Vary the pressure head h in equal steps and record the time t (to collect the fixed volume V) shown on the readout at each step.
- Plot 1/t against h and verify that the points lie on a straight line through the origin (since Q = V/t ∝ h at fixed r, l).
- Repeat with a different liquid (different density/viscosity) and compare the resulting graphs.
- Calculate the coefficient of viscosity η from the slope of the 1/t vs h graph, using the known r, l, ρ and V.
Observation Table¶
Table: Time to collect a fixed volume of liquid, with increasing pressure head
| S. No. | h (cm) | Volume V (cm³) | Time t (s) | 1/t (s⁻¹) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 |
Graph¶
Plot 1/t (y-axis) against Pressure head, h (x-axis).
Expected trend: a straight line through the origin (1/t ∝ h).
1/t vs h, expected to be a straight line through the origin; slope = πρgr⁴/(8ηlV)
Calculation¶
- Compute the slope of the 1/t vs h graph: slope = πρgr⁴/(8ηlV).
- Rearrange to find η = πρgr⁴/(8lV × slope), using the known r, l, ρ and V.
- Repeat for a second liquid and compare the two computed viscosities against their expected relative order.
Result¶
Table: Coefficient of viscosity of the liquid(s) studied
| Liquid | ρ (kg/m³) | r (mm) | l (cm) | Slope (s⁻¹/cm) | η (poise) |
|---|---|---|---|---|---|
| 1 | |||||
| 2 |
Maximum Permissible Error¶
η depends on the capillary radius to the fourth power — even a very small uncertainty in r, typically measured with a travelling microscope on a mercury thread, dominates the total error.
Precautions¶
- Maintain a constant pressure head during collection — use a constant-level (overflow) arrangement rather than letting the head fall as liquid drains.
- Ensure the flow remains streamline (laminar) throughout — very high heads or very wide capillaries risk turbulent flow, which invalidates Poiseuille's law.
- Measure the capillary's bore radius carefully, ideally by weighing a mercury thread of known length inside it, since this is the dominant source of error.
Self-Check Quiz¶
1. If the capillary radius r is doubled (all else fixed), the flow rate Q for the same pressure head:
- A) Doubles
- B) Increases fourfold
- C) Increases eightfold
- D) Increases sixteenfold
Answer: D — Poiseuille's law gives Q ∝ r⁴, so doubling r increases the flow rate sixteenfold — this extreme sensitivity is why capillary bore must be known very precisely.
2. Poiseuille's law is strictly valid only for:
- A) Turbulent flow
- B) Streamline (laminar) flow
- C) Flow of a gas only
- D) Flow with zero viscosity
Answer: B — The derivation assumes smooth, layered (laminar) flow with no radial mixing; at high flow rates the motion becomes turbulent and the simple r⁴ law no longer applies.
3. Doubling the capillary length l (all else fixed) changes the flow rate by a factor of:
- A) 2
- B) 1/2
- C) 4
- D) 1/4
Answer: B — Q ∝ 1/l, so doubling the length halves the flow rate for the same driving pressure.
Viva-Voce Questions¶
- Why is the flow rate through a capillary so extremely sensitive to its radius, compared to its length?
- What physical assumption distinguishes streamline (laminar) flow from turbulent flow, and how would you recognise experimentally that the flow has become turbulent?
- Why must the pressure head be kept constant during the timed collection, rather than allowed to fall as the reservoir empties?
- How would the required collection time change if a more viscous liquid (e.g. glycerine instead of water) were used, all else equal?
- Derive, in outline, why the velocity profile across the capillary is parabolic in laminar flow.
- What everyday or industrial situations depend critically on the r⁴ law demonstrated in this experiment?