--- title: "Viscosity, Poiseuille Flow and the Reynolds Number" description: "Viscosity resists relative motion between fluid layers. Steady, fully developed, pressure-driven laminar flow of an incompressible Newtonian fluid in a uniform circular tube with stationary, no-slip w" canonical: https://lightmysky.com/learn/science/viscosity-poiseuille-flow-and-the-reynolds-number-mt_-1sWgkRtMs source: https://lightmysky.com/learn/science/viscosity-poiseuille-flow-and-the-reynolds-number-mt_-1sWgkRtMs.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Viscosity, Poiseuille Flow and the Reynolds Number Viscosity resists relative motion between fluid layers. Steady, fully developed, pressure-driven laminar flow of an incompressible Newtonian fluid in a uniform circular tube with stationary, no-slip walls has a parabolic speed profile. For a horizontal tube, Poiseuille's law relates flow rate to pressure difference, radius, length and viscosity. Reynolds number indicates the likely flow regime; it does not establish that Bernoulli's loss-free model applies. Subject: Science · Area: Forces & Motion · Ages 20 to 21 Page: https://lightmysky.com/learn/science/viscosity-poiseuille-flow-and-the-reynolds-number-mt_-1sWgkRtMs ## Ready when they can - Explains why halving a horizontal circular tube's radius reduces Poiseuille flow to one sixteenth at fixed pressure difference, tube length and viscosity - Calculates the diameter-based Reynolds number in consistent units and describes the likely pipe-flow regime, including transition uncertainty - Checks steady, incompressible flow and mechanical energy losses before applying the usual Bernoulli equation along a streamline ## Lesson: Sticky flow, narrow pipes, rough flow Viscosity resists relative motion between neighbouring fluid layers. For a Newtonian fluid, shear stress is proportional to the velocity gradient; dynamic viscosity mu is the proportionality constant. At a stationary wall, no slip means zero fluid speed. Steady, fully developed, pressure-driven laminar flow of a Newtonian fluid through a straight circular tube has a parabolic speed profile, fastest at the centre. *(drawing: Relative speeds at five positions across a diameter. These sample a smooth parabolic profile, not separate fluid streams.)* For a horizontal tube, Poiseuille's law is Q = pi*delta_p*r^4/(8*mu*L). Q is volume per time; delta_p is inlet minus outlet pressure, r radius, L length and mu dynamic viscosity. It requires steady, fully developed laminar flow of an incompressible Newtonian fluid in a straight, rigid, uniform circular tube with stationary, no-slip walls. At fixed delta_p, L and mu, multiplying r by k multiplies Q by k^4. At fixed Q, L and mu, halving r needs 16 times delta_p. For pipe flow, Re = rho*v*D/mu. Use density rho in kg/m³, mean speed v in m/s, pipe diameter D in m, and dynamic viscosity mu in Pa s. The units cancel. Multiply rho, v and D, then divide by mu. For rho = 800, v = 0.2, D = 0.01 and mu = 0.04 in these units, Re = (800*0.2*0.01)/0.04 = 40. Other geometries need a stated characteristic length instead of D. Re compares inertial effects with viscous effects. For ordinary flow in a straight circular pipe, values well below 2000 favour laminar flow. Values far above a few thousand usually indicate turbulence. Transition depends on inlet disturbances and wall roughness, so Re is not a universal switch. Turbulent eddies mix fluid and usually cause greater pressure loss than laminar flow at the same mean speed in the same pipe. You have used Bernoulli's equation as a mechanical energy budget. Its usual form assumes steady, incompressible flow with negligible viscous losses along the chosen streamline and no pump or turbine between the points. Laminar flow can still dissipate much energy in a long, narrow tube. Reynolds number alone does not decide whether losses are negligible. With losses, use an energy balance that includes conversion of mechanical energy to internal energy. **Recap.** Poiseuille flow scales as r^4 under its stated conditions. Reynolds number compares inertia with viscosity. Bernoulli needs negligible losses; low Reynolds number does not promise that. *The lesson, questions and flashcards were written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Continuity and Bernoulli's Equation for Ideal Flow](https://lightmysky.com/learn/science/continuity-and-bernoullis-equation-for-ideal-flow-mt_pKxlVYTy0Y) ## Optional connections - [Drag and Terminal Speed from the Equation of Motion](https://lightmysky.com/learn/science/drag-and-terminal-speed-from-the-equation-of-motion-mt_In6VeoADzZ)