Neutron STEM Lab
Virtual engineering lab

Transient Couette–Poiseuille Flow

Resolve viscous channel startup with an implicit momentum solver and compare against transient and steady analytical profiles.

Fluid Mechanics / Numerical MethodsintermediateValidated educational model
Learning mode

Guided laboratory

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Set up → solve → verify

Build your numerical experiment

Read the equations

Start with an example. Change one setting at a time and compare accuracy, conservation, and computation cost.

Boundary conditions. No slip at both walls. Lower wall stationary; upper wall moves at the specified speed. Fully developed along x.

Full gap between parallel walls.
Constant Newtonian viscosity.
kg/m³
Used for pressure acceleration and shear.
Pa/m
Negative gradient drives positive x flow.
The lower wall stays stationary.
The last step lands at this physical time.
unitless
Uniform cells; resolution increases both accuracy and computational cost.
unitless
Backward Euler is stable but large time steps reduce accuracy.

Choose an example or use the defaults, then run. Larger grids may take several seconds.

Keep this experiment

Valid settings are saved on this device. Project JSON preserves physical inputs and display units. Restore it and run again for fresh results.

Settings stay in this browser. Simulations run locally.

Measurements

Instrument readings

Mean axial velocity
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Recorded from the current completed simulation.
Reference RMS error
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Recorded from the current completed simulation.
Theory

Equations and model

Fully developed channel startup

Initially the interior fluid is at rest. Constant pressure forcing and upper-wall speed start at time zero. Velocity depends on wall-normal position and time. Backward Euler solves a tridiagonal momentum system and is first-order accurate in time.

∂u∂t=g+ν∂2u∂y2,g=−1ρdpdx\frac{\partial u}{\partial t}=g+\nu\frac{\partial^2u}{\partial y^2},\quad g=-\frac1\rho\frac{\mathrm dp}{\mathrm dx}u(0,t)=0,u(H,t)=U,us(y)=UyH+g2νy(H−y)u(0,t)=0,\quad u(H,t)=U,\quad u_s(y)=U\frac yH+\frac{g}{2\nu}y(H-y)

Independent sine-series reference

Subtract the steady solution and expand the initially resting interior in sine modes. The bounded reference is resolved for the allowed viscous times. Displayed shear is the fluid stress component μ du/dy; the sign of each wall traction also depends on its outward normal.

u(y,t)=us(y)+∑n=1∞bnsin⁡(nπyH)e−ν(nπ/H)2tu(y,t)=u_s(y)+\sum_{n=1}^\infty b_n\sin\left(\frac{n\pi y}{H}\right)e^{-\nu(n\pi/H)^2t}bn=2U(−1)nnπ−2gH2[1−(−1)n]ν(nπ)3b_n=\frac{2U(-1)^n}{n\pi}-\frac{2gH^2[1-(-1)^n]}{\nu(n\pi)^3}τxy=ρν∂u∂y\tau_{xy}=\rho\nu\frac{\partial u}{\partial y}
Validation

Independent checks

Numerical balance or equation residualNot run

Review conservation, boundary, entropy, or residual checks.

Expected
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Simulated
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Error
—
Tolerance
0%

Run the experiment to perform this check.

Reference, bounds, or convergenceNot run

Review the specific check and its limits; this is not a universal accuracy certificate.

Expected
—
Simulated
—
Error
—
Tolerance
0%

Run the experiment to perform this check.

Engineering interpretation

Run the experiment to generate an engineering interpretation.

Assumptions and limitations
Assumptions
  • No slip at both walls. Lower wall stationary; upper wall moves at the specified speed. Fully developed along x.
  • Constant properties; uniform one- or two-dimensional meshes.
  • Physical quantities stored in SI. Changing display units preserves the simulation.
Limitations
  • Educational numerical verification does not certify a physical design.
  • No arbitrary CAD geometry, turbulence, compressibility, multiphase flow, or free surfaces.
  • Grid resolution, bounded computation, and reference limits are reported explicitly.
Local experiment export

Save your measurements and setup

Exports are generated in your browser. No account or server upload is required.

Settings JSON
Current parameters and instrument controls, including waveform, output enable, scope coupling, timebase, trigger, and cursors when present. Data labs also include the dataset, mappings, exclusions, budget, or propagation setup.
Measurements CSV
Completed instrument readings with labels, units, and descriptions.
Plot PNG
The current canvas-based scope, transient, sweep, or statistics visualization when available.
Summary
Markdown report with objectives, setup, measurements, validation, interpretation, assumptions, and graded guided concept-check results.

Ready to export the current local experiment state.

Learning objectives

What this experiment should establish

  • Follow startup.
  • Isolate each driver.
  • Refine time.
  • Reverse the forcing.
Check your understanding

Questions to answer from the experiment

  1. Which pressure gradient drives positive flow with stationary walls?
  2. Why refine time steps in a stable backward-Euler run?
  3. What is the final Couette profile with zero pressure gradient?
  4. Which effect is omitted by this fully developed model?
Continue learning
CFD LaboratoryExplore the other numerical experiments.Fluid mechanics toolsConnect numerical results to fluid mechanics calculations.Data Analysis & Uncertainty LabAnalyze exported numerical data.

Sources and model review

Reviewed 2026-09-19. The educational model exposes its assumptions and validation; source references do not imply external certification.