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.
What a learner can do afterwards
- 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
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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.
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.
In this horizontal tube, with all other conditions fixed, tripling the pressure difference raises Q from 2 to 6 mL/s.
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.
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.
The readings and flashcards were written for this dataset. Video references credit Khan Academy: "Viscosity and Poiseuille flow" and "Turbulence at high velocities and Reynold's number".
More to read
Changing a tube changes its flow
Consider steady flow of an incompressible Newtonian liquid through a rigid, horizontal circular tube with stationary walls and no slip. A pressure difference drives the flow. Once the flow is fully developed, its speed pattern no longer changes along the tube. In laminar flow, speed is zero at the wall and greatest at the centre. The profile across a diameter is a parabola.
For this straight tube of uniform radius, Poiseuille's law gives Q = pi*delta_p*r^4/(8*mu*L). Here Q is volume per time, delta_p is inlet minus outlet pressure, r is tube radius, L is tube length and mu is dynamic viscosity. The horizontal condition removes any elevation change along the tube. An inclined tube needs gravity included in the driving term.
To compare two such flows, first list what stays fixed. With pressure difference, length and viscosity unchanged, multiply Q by the fourth power of the radius ratio. With geometry and viscosity unchanged, tripling the pressure difference triples Q. With pressure difference and geometry fixed, doubling viscosity halves Q.
A predicted increase in flow can change the flow regime. Check that the new flow still satisfies the assumptions before using the prediction.
The force balance at terminal speed
A falling particle pushes through fluid while the fluid exerts drag against its motion. For a sufficiently slow, small sphere in a Newtonian fluid, far from walls, Stokes' law gives drag magnitude F_drag = 6*pi*mu*a*v. Here a is sphere radius, v is speed relative to the fluid and mu is dynamic viscosity. This model requires a Reynolds number much smaller than one.
A falling sphere also experiences an upward buoyant force. At downward terminal speed, drag plus buoyancy balances weight. The net force is zero, so acceleration is zero even though the sphere continues moving.
Call weight minus buoyancy F_drive. In the Stokes regime, terminal speed is F_drive/(6*pi*mu*a). Holding the sphere and viscosity fixed, doubling F_drive doubles terminal speed only if the new motion remains in that regime.
Drag converts mechanical energy into internal energy. An energy calculation can include this transfer. A model that assumes no dissipation cannot be used unchanged.
Two different checks before choosing a flow model
Reynolds number compares inertial effects with viscous effects. For a circular pipe, calculate Re = rho*v*D/mu using mean speed and pipe diameter. Choosing radius instead would halve the number, so a stated length convention matters when comparing results with a transition range.
Low pipe Reynolds numbers favour laminar flow. Values far above a few thousand usually indicate turbulence, with eddies that carry fluid across the main flow. Near transition, inlet disturbances and wall roughness affect what happens. Thresholds for pipes must not be applied unchanged to flow around objects.
Choosing an energy equation is a separate decision. The usual Bernoulli relation treats pressure, kinetic and gravitational terms as a conserved mechanical energy budget along a streamline. Its assumptions include steady, incompressible flow and negligible losses, with no pump or turbine between the points.
A slowly moving liquid can lose substantial mechanical energy along a long narrow tube despite remaining laminar. Include that loss in an extended energy balance. Total energy is conserved as some mechanical energy becomes internal energy.
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