SCIENCE · INVESTIGATE AND EXPLAIN

Derive a pipe profile and then test its admissibility

For steady fully developed flow through a straight circular tube, an axial pressure drop supplies force on a cylindrical fluid core. Balancing it with shear around the core gives shear magnitude Δp r/(2L). Newtonian stress relates that shear to the radial velocity gradient.

Learning outcome: Derive the shear and velocity profiles, integrate their flow and independently check the regime before accepting any radius extrapolation.

Starting recap

Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.

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Try

At R = 0.001 m and Δp = 100 Pa, mean speed is 0.0125 m/s, center speed 0.025 m/s and Reynolds number 25. The wall shear magnitude is 0.05 Pa.

Open the experiment

Predict

At fixed fluid, pressure drop and length, doubling tube radius multiplies the conditional Reynolds number by:

Your prediction and reason

Experiment

Derive the shear and velocity profiles, integrate their flow and independently check the regime before accepting any radius extrapolation.

Integrate a viscous profile and then audit its regime

Worked teaching model · synthetic values · Conditional steady fully developed laminar Newtonian flow in a straight circular1 m tube, viscosity.001 Pa s,density1000 kg/m³,no-slip wall. Report mathematical profile and Reynolds screen separately; Re2000 is a heuristic boundary, not observed turbulence. No clinical or apparatus application.

Tube radius: 0.002 m Pressure drop: 200 Pa

Reset keeps your prediction, captures and practice history.

Conditional parabolic profile, subject to regime screen

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: DeltaP(R²−r²)/(4eta L) (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Conditional parabolic profile, subject to regime screen

DeltaP(R²−r²)/(4eta L)

radial coordinate (m)	axial velocity (m/s)
-0.002	0
-0.00195	0.009875
-0.0019	0.0195
-0.00185	0.028875
-0.0018	0.038
-0.00175	0.046875
-0.0017	0.0555
-0.00165	0.063875
-0.0016	0.072
-0.00155	0.079875
-0.0015	0.0875
-0.00145	0.094875
-0.0014	0.102
-0.00135	0.108875
-0.0013	0.1155
-0.00125	0.121875
-0.0012	0.128
-0.00115	0.133875
-0.0011	0.1395
-0.00105	0.144875
-0.001	0.15
-0.00095	0.154875
-0.0009	0.1595
-0.00085	0.163875
-0.0008	0.168
-0.00075	0.171875
-0.0007	0.1755
-0.00065	0.178875
-0.0006	0.182
-0.00055	0.184875
-0.0005	0.1875
-0.00045	0.189875
-0.0004	0.192
-0.00035	0.193875
-0.0003	0.1955
-0.00025	0.196875
-0.0002	0.198
-0.00015	0.198875
-0.0001	0.1995
-5.00000e-5	0.199875
0	0.2
5.00000e-5	0.199875
0.0001	0.1995
0.00015	0.198875
0.0002	0.198
0.00025	0.196875
0.0003	0.1955
0.00035	0.193875
0.0004	0.192
0.00045	0.189875
0.0005	0.1875
0.00055	0.184875
0.0006	0.182
0.00065	0.178875
0.0007	0.1755
0.00075	0.171875
0.0008	0.168
0.00085	0.163875
0.0009	0.1595
0.00095	0.154875
0.001	0.15
0.00105	0.144875
0.0011	0.1395
0.00115	0.133875
0.0012	0.128
0.00125	0.121875
0.0013	0.1155
0.00135	0.108875
0.0014	0.102
0.00145	0.094875
0.0015	0.0875
0.00155	0.079875
0.0016	0.072
0.00165	0.063875
0.0017	0.0555
0.00175	0.046875
0.0018	0.038
0.00185	0.028875
0.0019	0.0195
0.00195	0.009875
0.002	0

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Conditional mean speed	0.1	m/s
Conditional maximum speed	0.2	m/s
Conditional volume flow	1.25664e-6	m³/s
Conditional Reynolds number	400	ratio
Laminar heuristic screen below2000	1	flag
Wall shear-stress magnitude	0.2	Pa

The supplied laminar profile passes the stated Reynolds screen, not an empirical regime certification.

Axial force balance on a cylindrical core gives shear magnitude DeltaP r/(2L). Newtonian stress with no-slip wall integrates to v(r)=DeltaP(R²−r²)/(4eta L). Integrate2pi r v dr for Q=pi DeltaP R⁴/(8eta L), yielding vmax=2vmean. At fixed pressure drop, Reynolds scales R³, so radius extrapolation can invalidate the starting model.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

The supplied laminar profile passes the stated Reynolds screen, not an empirical regime certification.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Derive a pipe profile and then test its admissibility

Integrate the gradient and impose zero velocity at the stationary wall r = R. The conditional laminar profile is v(r) = Δp(R² − r²)/(4ηL). Integrating annuli, Q = integral(2πr v dr), gives πΔpR⁴/(8ηL), so mean velocity is ΔpR²/(8ηL) and center velocity is twice the mean.

With R = 0.002 m, Δp = 200 Pa, η = 0.001 Pa s and L = 1 m, mean speed is 0.1 m/s and flow is about 1.25664×10⁻⁶ m³/s. With density 1000 kg/m³, Reynolds number is 2ρR vmean/η = 400, below the stated heuristic 2000 screen.

Doubling radius at the same pressure drop multiplies conditional flow by 16, mean speed by 4 and Reynolds number by 8, reaching 3200. The same laminar formula has now helped identify its own unsupported extrapolation. The mathematical parabola can still be displayed as a conditional candidate, but it must not be called the accepted measured flow. The threshold is approximate and does not by itself prove a turbulent state.

Changed-case check

Predict before reading the comparison. At R = 0.001 m and Δp = 100 Pa, mean speed is 0.0125 m/s, center speed 0.025 m/s and Reynolds number 25. The wall shear magnitude is 0.05 Pa.

Retained investigation

Derive the shear and velocity profiles, integrate their flow and independently check the regime before accepting any radius extrapolation. Why can a formula produce a finite number even when its assumptions fail?

Source and boundary

Primary conceptual reference checked 30 October 2026: OpenStax University Physics Volume 1. All prose, functions and evidence packets are original. Bounded mathematical models do not establish measured apparatus, supervised practical skill, independent learner mastery or institutional credit. No physical loading, liquids, heat or sound exposure is requested.

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Why can a formula produce a finite number even when its assumptions fail?

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