lilernoEngineeringOpening Lilerno
Chapter 15 · Guided self-study
Intro + external course studyEstimate flow and pressure loss while stating where the model applies.
Build the idea
An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.
This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.
Start with these prerequisites
Shared science foundations
Fluid mechanics starts with conservation applied to a chosen control volume. Pressure, flow rate and velocity are related, but they are not the same quantity.
Module lens
For steady incompressible flow, a fixed volumetric rate means average speed changes with area.
For steady incompressible flow through one inlet and one outlet, Q = Av is constant when v is the area-average speed. A smaller area increases average speed at the same volume flow rate.
The simple Bernoulli relation assumes steady inviscid incompressible flow along a streamline without added shaft work. Real pipes have losses. Pressure may be absolute or gauge; never mix those conventions in a single calculation.
See the relationship
Change one quantity, watch the graph respond, then explain the result in your own words.
Predict → change → explain
Keep flow rate constant. What changes when the cross-section halves?
Axes: Area (cm²) → Average speed (m/s). Bounds may rescale when inputs change.
Q = 0.002 m³/s; A = 0.001 m²; v = 2 m/s.
Steady incompressible flow, one inlet and outlet. Q = Av describes area-average speed, not pressure loss or a velocity profile.
Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.
| Series | Area (cm²) | Average speed (m/s) |
|---|---|---|
| Continuity | 5 | 4 |
| Continuity | 5.438 | 3.678 |
| Continuity | 5.875 | 3.404 |
| Continuity | 6.313 | 3.168 |
| Continuity | 6.75 | 2.963 |
| Continuity | 7.188 | 2.783 |
| Continuity | 7.625 | 2.623 |
| Continuity | 8.063 | 2.481 |
| Continuity | 8.5 | 2.353 |
| Continuity | 8.938 | 2.238 |
| Continuity | 9.375 | 2.133 |
| Continuity | 9.813 | 2.038 |
| Continuity | 10.25 | 1.951 |
| Continuity | 10.69 | 1.871 |
| Continuity | 11.13 | 1.798 |
| Continuity | 11.56 | 1.73 |
| Continuity | 12 | 1.667 |
| Continuity | 12.44 | 1.608 |
| Continuity | 12.88 | 1.553 |
| Continuity | 13.31 | 1.502 |
| Continuity | 13.75 | 1.455 |
| Continuity | 14.19 | 1.41 |
| Continuity | 14.63 | 1.368 |
| Continuity | 15.06 | 1.328 |
| Continuity | 15.5 | 1.29 |
| Continuity | 15.94 | 1.255 |
| Continuity | 16.38 | 1.221 |
| Continuity | 16.81 | 1.19 |
| Continuity | 17.25 | 1.159 |
| Continuity | 17.69 | 1.131 |
| Continuity | 18.13 | 1.103 |
| Continuity | 18.56 | 1.077 |
| Continuity | 19 | 1.053 |
| Continuity | 19.44 | 1.029 |
| Continuity | 19.88 | 1.006 |
| Continuity | 20.31 | 0.9846 |
| Continuity | 20.75 | 0.9639 |
| Continuity | 21.19 | 0.944 |
| Continuity | 21.63 | 0.9249 |
| Continuity | 22.06 | 0.9065 |
| Continuity | 22.5 | 0.8889 |
| Continuity | 22.94 | 0.8719 |
| Continuity | 23.38 | 0.8556 |
| Continuity | 23.81 | 0.8399 |
| Continuity | 24.25 | 0.8247 |
| Continuity | 24.69 | 0.8101 |
| Continuity | 25.13 | 0.796 |
| Continuity | 25.56 | 0.7824 |
| Continuity | 26 | 0.7692 |
| Continuity | 26.44 | 0.7565 |
| Continuity | 26.88 | 0.7442 |
| Continuity | 27.31 | 0.7323 |
| Continuity | 27.75 | 0.7207 |
| Continuity | 28.19 | 0.7095 |
| Continuity | 28.63 | 0.6987 |
| Continuity | 29.06 | 0.6882 |
| Continuity | 29.5 | 0.678 |
| Continuity | 29.94 | 0.6681 |
| Continuity | 30.38 | 0.6584 |
| Continuity | 30.81 | 0.6491 |
| Continuity | 31.25 | 0.64 |
| Continuity | 31.69 | 0.6312 |
| Continuity | 32.13 | 0.6226 |
| Continuity | 32.56 | 0.6142 |
| Continuity | 33 | 0.6061 |
| Continuity | 33.44 | 0.5981 |
| Continuity | 33.88 | 0.5904 |
| Continuity | 34.31 | 0.5829 |
| Continuity | 34.75 | 0.5755 |
| Continuity | 35.19 | 0.5684 |
| Continuity | 35.63 | 0.5614 |
| Continuity | 36.06 | 0.5546 |
| Continuity | 36.5 | 0.5479 |
| Continuity | 36.94 | 0.5415 |
| Continuity | 37.38 | 0.5351 |
| Continuity | 37.81 | 0.5289 |
| Continuity | 38.25 | 0.5229 |
| Continuity | 38.69 | 0.517 |
| Continuity | 39.13 | 0.5112 |
| Continuity | 39.56 | 0.5055 |
| Continuity | 40 | 0.5 |
| Chosen section | 10 | 2 |
Save your place when you finish reading.
Original Lilerno example
Water flows at Q = 0.002 m³/s through an area of 0.001 m². Find average speed.
Use Q = Av.
Rearrange v = Q/A.
Divide: 0.002/0.001 = 2 m/s.
Test the model
Open learning, traceable sources
Work in this order. These links open the publisher’s material; free access does not always permit republication.
MIT OpenCourseWare
Start with: Hydrostatics → control volumes → viscous flows
Solve mass and momentum balances before moving to pipe-loss models.
CFD Direct / OpenFOAM Foundation
Start with: Tutorials and boundary conditions (later extension)
Inspect the case documentation only after you can explain the governing equations.
Recall, then record
Close the explanation and answer these in your own words. Return tomorrow, then again later in the week.
Is this flow steady and incompressible?
Average or local velocity?
Which energy losses were neglected?
Sketch a control volume with all inlets and outlets. Write its mass balance and list what more you need to estimate pressure loss.
Self-reported tasks, not an assessment of mastery or university credit. Reading a page does not complete a chapter.
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Open learning, traceable sources
Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.
Pressure, hydrostatics, control volumes, conservation laws and viscous flows.
CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.
Open source ↗Rights / publisher record ↗Start with the tutorial and learn case structure, boundary conditions and mesh setup. A versioned guide, not a claim that v13 is the newest release.
Open-source solver; guide and software have separate notices. Linked only; verify the relevant documentation and code licence before redistribution.
Open source ↗Rights / publisher record ↗