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Graduate · Fluid mathematics

Animate an exact channel-flow solution

Compare stationary walls, a parabolic velocity profile, and particles moving fastest at the center of a pressure-driven channel.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Animate an exact channel-flow solution. Elapsed model time: 0 s. Center speed: 1 m/s. Flux per unit width: 1.333 m²/s. Mean speed: 0.667 m/sPressure drives. Viscosity balances.u(y)0 → u maxNo-slip wall10 m window →t = 0 s · Center particles move fastest
An exact special solution for a Newtonian, incompressible fluid with constant viscosity; infinite parallel stationary plates; steady, fully developed unidirectional flow; no extra body force. The display shows a 10 m streamwise window, with tracers re-entering for visualization and time from 0 to 8 s. The right-hand profile rescales horizontally to the current maximum speed. This model makes no claim about arbitrary three-dimensional flows or their regularity.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Elapsed model time
0 s
Center speed
1 m/s
Flux per unit width
1.333 m²/s
Mean speed
0.667 m/s

HD animation studio

From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

For steady, fully developed flow between stationary parallel plates, pressure pushes fluid along the channel while viscosity balances that push. The Navier–Stokes equations reduce to a second-order equation in the cross-channel coordinate. Integrating and imposing zero wall velocity produces a parabola.

A relationship to keep

u(y) = G y(h−y)/(2μ); Q per unit width = Gh³/(12μ)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Balance the equation

    With dp/dx = −G and velocity depending only on y, the steady momentum equation becomes 0 = G + μu″. The time derivative and the advective acceleration are both zero for this particular flow.

  2. STEP 2

    Apply the walls

    Integrating twice and imposing u(0) = u(h) = 0 gives the displayed parabolic profile. The center speed is Gh²/(8μ); the wall particles remain still because the no-slip condition fixes their velocity at zero.

  3. STEP 3

    Compare speeds and flux

    Particles travel horizontally at their row's velocity. Integrating the profile from 0 to h gives flux per unit width, which has units m²/s. Doubling h multiplies that flux by eight when G and μ stay fixed.

Your turn to explain

Make a prediction. Test your reasoning.

For G = 2 Pa/m, μ = 1 Pa·s, and h = 2 m, find the center speed and flux per unit width.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The center speed is 1 m/s. The flux per unit width is 4/3 m²/s; the mean velocity is (4/3)/2 = 2/3 m/s.

Work through a full lesson

Connect the animation to a worked example and practice questions.

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