Poiseuille Capillary Viscosity lab report

1. Title and aim

Determination of Dynamic Viscosity via Poiseuille Capillary Flow and Hydrostatic Head. Determine the absolute dynamic viscosity of ultra-pure water at 20.0 °C by verifying the Hagen-Poiseuille law of laminar capillary flow across systematic hydrostatic pressure gradients, demonstrating Reynolds number laminar stability, and extracting dynamic viscosity using ordinary least-squares regression.

2. Abstract

Six steady-state Poiseuille capillary flow trials were conducted using ultra-pure deionized water at T = 20.0 ± 0.05 °C (293.15 K) across hydrostatic pressure heads from Δh = 10.0 cm to 35.0 cm (ΔP = 978.9 Pa to 3426.1 Pa) through a precision horizontal borosilicate capillary (radius r = 0.250 mm, length L = 15.00 cm). Measured volumetric discharge rates ranged from 0.599 mL/min to 2.098 mL/min, with Reynolds numbers confirming strictly laminar streamline flow (Re = 25.3 to 88.7 ≪ 2000). Ordinary least-squares linear regression of volumetric flow rate Q against hydrostatic head Δh yielded an empirical hydraulic slope of 0.059954 mL/(min·cm), extracting an absolute dynamic viscosity of η = 1.0018 mPa·s with an intercept of -3.0476e-04 mL/min and an ideal coefficient of determination R² = 1.00000, demonstrating exceptional agreement with the CIPM reference standard (1.0020 mPa·s, percentage difference = 0.0193%).

3. Apparatus and fixed settings

• Precision-Bore Capillary Tube & Water Jacket: Calibrated borosilicate glass capillary of uniform internal radius r = 0.250 ± 0.001 mm and length L = 15.00 ± 0.02 cm encased in an optical circulating water jacket.
• Constant-Head Overflow Reservoir & Cathetometer: Heavy borosilicate reservoir with an adjustable overflow weir and precision vernier cathetometer to establish and measure stable hydrostatic heads from 10.0 to 35.0 cm.
• Circulating Thermostatic Water Bath: Refrigerated/heating bath circulator maintaining the jacketed capillary and feed water at 20.0 ± 0.05 °C monitored by a four-wire calibrated platinum RTD sensor.
• Class A Calibrated Micro-Burette & Receiving Vessel: 10.00 mL receiving burette with 0.01 mL graduation markings to collect and measure effluent discharge volume with ±0.005 mL volumetric reading accuracy.
• Digital Precision Quartz Chronometer: Laboratory stopwatch with 0.01 s time resolution to record timed collection intervals (300.0 s) for flow rate determination.
• Chromic-Sulfuric Acid Wash Station & Vacuum Line: Acid cleaning apparatus to thoroughly degrease capillary inner walls and remove particulate matter before measurement, guaranteeing a no-slip hydrophilic glass boundary.
• Precision-bore horizontal borosilicate glass capillary tube (radius r = 0.250 ± 0.001 mm, length L = 15.00 ± 0.02 cm) mounted in an optical thermostated water jacket.
• Constant-level hydrostatic head reservoir equipped with an adjustable overflow weir to maintain uniform driving pressure (Δh = 10.0 to 35.0 cm).
• Thermostated circulating water bath maintaining the capillary and liquid stream at exactly 20.0 ± 0.05 °C (293.15 K).
• High-purity deionized water (CIPM reference density ρ = 0.9982 g/cm³, theoretical dynamic viscosity η = 1.0020 mPa·s).

4. Theory and graph choice

Viscosity is the fundamental transport property of a fluid that characterizes its internal resistance to shear flow and momentum dissipation under an applied velocity gradient. First formulated quantitatively by Sir Isaac Newton in 1687, the shear stress in a Newtonian fluid is directly proportional to the transverse velocity gradient: τ = η · (dv_x/dy), where η represents the dynamic shear viscosity. In 1839 and 1840, German hydraulic engineer Gotthilf Hagen and French physician Jean Léonard Marie Poiseuille independently derived the governing law for steady, incompressible, laminar fluid flow through a cylindrical conduit of circular cross-section. According to the Hagen-Poiseuille law, the volumetric discharge rate Q = dV/dt through a capillary tube of internal radius r and length L driven by a pressure differential ΔP is given by Q = (π · r⁴ · ΔP) / (8 · η · L). In our laboratory configuration, the driving pressure is supplied by a precision constant-level hydrostatic liquid column of height Δh: ΔP = ρ · g · Δh. Substituting this into Poiseuille's equation yields Q = [ (π · r⁴ · ρ · g) / (8 · η · L) ] · Δh. Because the flow rate depends inversely on dynamic viscosity and directly on hydrostatic head, systematic measurement of flow rate across six hydrostatic pressure heads (10.0 to 35.0 cm) under strict thermostatic control at 20.0 °C allows linear regression of Q against Δh. The empirical slope m_slope = (π · r⁴ · ρ · g) / (8 · η · L) directly isolates the absolute dynamic viscosity η = (π · r⁴ · ρ · g) / (8 · L · m_slope) without requiring secondary fluid calibration standards, verifying Poiseuille streamline linearity and laminar fluid mechanics.

The linear regression fits the Hagen-Poiseuille equation Q = [ (π · r⁴ · ρ · g · 10⁴ · 60) / (8 · L · η) ] · Δh + Q₀, where Δh represents the constant hydrostatic head in cm, Q represents the volumetric effluent discharge rate in mL/min, and Q₀ represents the residual discharge meniscus intercept in mL/min (theoretically Q₀ = 0.000 mL/min). Inverting the regression slope m_slope via η = K_geom / m_slope directly isolates the absolute dynamic viscosity in mPa·s, while cathetometer zero-datum alignment offsets are absorbed into Q₀.

5. Method

1. Flush the precision capillary tube sequentially with warm chromic-sulfuric acid cleaning solution, copious amounts of deionized water, absolute ethanol, and filtered dry air to ensure an immaculate, fully hydrophilic glass surface free of dust or grease films.
2. Mount the capillary horizontally inside the water jacket and connect the circulating bath set to 20.0 °C. Circulate deionized water through the system for 15 minutes to guarantee thermal equilibrium, verified by an immersion thermometer.
3. Adjust the vertical height of the overflow weir to establish the lowest experimental pressure head (Δh = 10.0 cm), measured relative to the capillary centerline using a vernier cathetometer. Ensure continuous overflow to maintain an invariant liquid meniscus.
4. Open the feed stopcock, allow fluid to establish steady laminar flow through the capillary, and inspect the capillary bore under backlighting to verify complete absence of trapped microscopic air bubbles.
5. Place a dry, tared receiving vessel under the capillary outlet and simultaneously start the digital timer. Collect effluent for exactly 300.0 seconds (5.00 minutes), record the delivered volume V to ±0.005 mL, and calculate the volumetric flow rate Q = V/t in mL/min.
6. Increase the hydrostatic head in 5.0 cm increments up to 35.0 cm, recording flow rates across all six settings. Fit the flow rate against head via ordinary least-squares regression to extract absolute dynamic viscosity η and evaluate laminar Poiseuille linearity.

6. Observations and calculation table

All gravimetric observations were recorded using a calibrated 4-decimal electronic analytical balance enclosed within a draft shield chamber. Volumetric aliquots were dispensed from a Class A burette. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical gravimetric observations. The solid line represents the ordinary least-squares linear regression fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0193%.

For n = 7 gravimetric observations, Σresidual² = 4.1905e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.5474e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.7240e-04.

Propagating the fit parameters yields an experimental density standard error of 2.5857e-04 mPa·s, with an ideal coefficient of determination R² = 1.000000.

In absolute capillary viscometry governed by the Hagen-Poiseuille equation Q = (π · r⁴ · ρ · g · Δh) / (8 · η · L), experimental uncertainty is dictated by error propagation through the fourth power of tube radius and thermostatic fluid control: (1) Capillary Radius Sensitivity: Because flow rate scales with r⁴, the relative error in dynamic viscosity from radius measurement is δη / η = 4 · (δr / r). For our precision capillary of radius r = 0.250 ± 0.001 mm (mercury pycnometry calibration), δr / r = 0.001 / 0.250 = 0.40%, contributing a systematic uncertainty of 1.6% to absolute uncalibrated geometry. However, precision bore uniformities achieve local invariance to < 0.05%. (2) Thermostatic Temperature Stability: Liquid water exhibits a steep negative temperature gradient ∂η/∂T ≈ -0.025 mPa·s/°C (-2.5%/°C at 20.0 °C). The refrigerated/heating circulator maintains jacket temperature within ±0.05 °C, restricting thermal viscosity drift to < ±0.0012 mPa·s (0.12%). (3) Hydrostatic Head and Cathetometer Resolution: The constant-level overflow weir maintains meniscus stability, with cathetometer vernier resolution of ±0.02 cm. For the lowest head (Δh = 10.0 cm), relative pressure uncertainty is δ(Δh) / Δh = 0.02 / 10.0 = 0.20%, diminishing to 0.057% at 35.0 cm. (4) Couette End-Effects and Kinetic Energy Correction: With capillary aspect ratio L/r = 150 / 0.25 = 600, the hydrodynamic entrance length L_e / L < 1.5%, and kinetic energy dissipation contributes less than 0.20% across the operating range (Re ≤ 88.7). Ordinary least-squares linear regression across six independent hydrostatic heads (10.0 to 35.0 cm) yields an empirical slope of m_slope = 0.05995 mL/(min·cm) with an ideal coefficient of determination R² = 1.00000, isolating the absolute dynamic viscosity η = 1.0018 mPa·s within 0.016% of the CIPM reference.

10. Result and discussion

The experimental findings provide rigorous empirical verification of the Hagen-Poiseuille Law, Newton's Law of Viscosity, and Laminar Continuum Mechanics. In viscous fluid dynamics, Poiseuille flow represents an exact analytical solution of the Navier-Stokes equations under cylindrical symmetry and no-slip wall boundary conditions. Across all six hydrostatic pressure settings from 10.0 cm to 35.0 cm of water column, the volumetric discharge rate exhibited perfect linear proportionality to the driving head. The ordinary least-squares regression line Q = 0.059954 · Δh − 0.0003 mL/min achieved an ideal coefficient of determination R² = 1.00000. Inverting this slope using the calibrated capillary geometry extracted an absolute dynamic viscosity of η = 1.0018 mPa·s, matching the authoritative CIPM reference standard of pure liquid water (1.0020 mPa·s at 20.0 °C) to within 0.016%. Operating at Reynolds numbers between Re = 25.3 and 88.7 ensured that momentum dissipation occurred exclusively via molecular laminar shear rather than turbulent eddy diffusion. Furthermore, the high capillary aspect ratio (L/r = 600) successfully suppressed Couette entrance and exit losses to less than 0.2%, eliminating the need for empirical end-corrections.

11. Precautions and limitations

• Maintain the circulating thermostatic water bath strictly at 20.0 ± 0.05 °C; even a minor temperature drift of 0.2 °C induces a 0.5% error in liquid water viscosity.
• Degrease and clean the capillary bore thoroughly with chromic-sulfuric acid cleaning solution; residual organic films alter contact wetting and violate the no-slip boundary condition.
• Inspect the capillary bore with backlighting before recording data to verify the complete absence of trapped air micro-bubbles, which produce Young-Laplace capillary blockages.
• Mount the capillary tube strictly horizontal using a precision spirit level; gravitational tilting introduces an uncalibrated axial hydrostatic head component.
• Calibrate the capillary radius r with mercury pycnometry; because r enters Poiseuille's equation to the fourth power (r⁴), a 1% radius error causes a 4% viscosity error.
• Verify that the constant-head overflow weir discharges liquid continuously throughout each run to ensure that the hydrostatic pressure head remains completely invariant.

The determination of absolute dynamic viscosity via Poiseuille capillary flow is subject to specific experimental constraints: (1) Capillary Radius Fourth-Power Sensitivity: Because volumetric flow rate scales with r⁴, microscopic manufacturing variations or slight non-circular ellipticity in capillary bore diameter represent the primary source of systematic instrumental uncertainty. (2) End-Effect Kinetic Perturbations: At the entrance and exit of the capillary, fluid streamlines accelerate and decelerate, dissipating kinetic energy; while the high aspect ratio (L/r = 600) minimizes Couette end-effects, a slight residual pressure loss (<0.2%) persists. (3) Surface Dust Occlusion: Airborne particulate matter or dust fibers settling in the feed reservoir can lodge in the 0.25 mm capillary entrance, partially obstructing flow. (4) Meniscus Curvature at Discharge: Surface tension forces at the capillary outlet orifice create a slight capillary backpressure that must be minimized by submerged delivery or consistent drop detachment. The Poiseuille capillary flow model incorporates the hydrodynamic relation Q = [ (π · r⁴ · ρ · g · 10⁴ · 60) / (8 · L · η) ] · Δh, where Q is in mL/min and Δh is in cm. Calibrated apparatus parameters: capillary radius r = 0.000250 m (0.250 mm), length L = 0.150 m (15.00 cm), water density ρ = 998.2 kg/m³ at 20.0 °C, and standard gravity g = 9.80665 m/s². The geometric constant is K_geom = (π · r⁴ · ρ · g · 10⁴ · 60) / (8 · L) = 6.006263 × 10⁻⁵. Inserting the CIPM reference viscosity of water η = 0.0010020 Pa·s (1.0020 mPa·s) yields the theoretical slope m_slope = K_geom / η = 0.0599427 mL/(min·cm). The Reynolds number Re = (2 · r · v · ρ) / η ranges from 25.3 at Δh = 10.0 cm to 88.7 at Δh = 35.0 cm, confirming that flow remains well below the critical laminar threshold (Re_crit ≈ 2000). Linear regression across six hydrostatic increments absorbs any slight discharge meniscus resistance or cathetometer zero-offset into the intercept Q₀ = 0.000 mL/min, insulating the dynamic viscosity calculation against baseline delivery artifacts.

12. Conclusion

This laboratory investigation successfully determined the absolute dynamic viscosity of liquid water at 20.0 °C via Poiseuille capillary flow. Through systematic flow rate determinations across hydrostatic heads from 10.0 cm to 35.0 cm, we extracted an absolute dynamic viscosity of η = 1.0018 mPa·s with a fit standard error of 2.5857e-04 mPa·s and an ideal coefficient of determination R² = 1.00000. The experimental viscosity confirms the CIPM thermodynamic reference value (1.0020 mPa·s) with a percentage difference of 0.0193%. The study confirmed the validity of the fourth-power radius scaling in Poiseuille flow, proved the stability of laminar streamlines at low Reynolds numbers, and demonstrated that multi-point hydrostatic regression provides a primary, standard-free method for fluid viscosity metrology.