Check whether a fourth-power flow prediction remains admissible

Declare the fluid, reference and evidence

Viscosity describes resistance to relative motion between neighboring fluid layers. A Newtonian fluid has shear stress proportional to velocity gradient, with viscosity η in Pa s. Laminar flow has organized layers; turbulent flow includes irregular fluctuations and mixing. A smooth tube does not guarantee laminar flow at every speed.

Build the algebraic model

For fully developed laminar flow in a straight round tube with no slip at the wall, Poiseuille’s relation is Q=πr⁴Δp/(8ηL). It assumes a Newtonian incompressible fluid and a pressure drop allocated to that developed section. At fixed pressure drop, doubling radius would multiply conditional flow by 16 and mean speed by four. This strong geometric effect does not excuse checking the assumptions again.

Check the original numerical packet

The workbench uses density 1000 kg/m³, viscosity 0.001 Pa s and length 1 m. At radius 1 mm and pressure drop 1000 Pa, conditional Q=3.92699×10⁻⁷ m³/s, mean speed 0.125 m/s and Reynolds number 2ρvr/η=250. Doubling radius gives conditional Reynolds number 2000, at the approximate transition boundary, not a secure new laminar prediction. The plotted curve remains explicitly conditional outside the screened branch.

Challenge the assumptions and transfer

Reynolds thresholds near 2000 and 3000 are approximate tube-flow guides; entrance geometry, roughness and disturbances matter. In a separate low-Reynolds sphere model, supplied Stokes drag is 6πηrv, so doubling speed doubles drag; a quadratic high-speed model instead quadruples it. Molecular diffusion is transport down a concentration gradient without requiring bulk flow: under a supplied one-dimensional relation rms distance=√(2Dt), quadrupling time doubles distance. Osmosis concerns solvent transport across a selectively permeable boundary, not unrestricted solute diffusion. For a supplied dilute ideal osmotic law Δπ=RTΔc, doubling concentration difference at fixed temperature doubles pressure difference. These are model-comparison actions, not medical guidance or exposure experiments.

Worked original case

At radius 2 mm and pressure drop 1600 Pa, conditional speed is 0.8 m/s and Reynolds number 3200. The laminar relation fails its own screening premise; do not report its conditional flow as a verified actual value.

Bounded investigation

Predict the radius ratios for flow, speed and Reynolds number before changing radius. Retain at least one screened laminar and one rejected case, and explain which independent premises a Reynolds screen cannot establish.

Changed-case numerical practice

Try the changed case before feedback

A conditional laminar tube flow has Reynolds number 150 at radius r. At the same pressure drop, viscosity, density and length, what is the conditional Reynolds number at 2r?

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated educational evidence. No pressure rigs, liquid handling, bodily measurements or physical exposure protocols are requested. Virtual admissibility does not certify real equipment safety, practical skill or independent mastery.