Couette-Poiseuille Channel Flow

Fluid

Drive

Top plate velocity U 0.40 m/s

Pressure gradient dp/dx 0 Pa/m

Gap height h 30 mm

Live readings

Mean velocity ū

0.20 m/s

Reynolds number

6.0k

Wall shear τ (bottom)

0.013 Pa

Wall shear τ (top)

0.013 Pa

Volumetric flow (per m depth)

6.0e-3 m²/s

How it works

Two infinite parallel plates enclose a gap of height h filled with a viscous incompressible fluid. The top plate is dragged at velocity U while a constant streamwise pressure gradient dp/dx also pushes (or resists) the flow. For steady, fully developed, laminar flow the Navier-Stokes equations reduce to a single balance between viscous shear and the pressure force:

μ d²u/dy² = dp/dx

u(y) = -(1/2μ)(dp/dx) · y(h - y)  +  U·y/h
        \_______ Poiseuille term _______/    \_ Couette term _/

The two limits are the textbook cases: with dp/dx = 0 the profile is the straight line of pure Couette flow (drag-driven shear, u ∝ y/h). With U = 0 it is the parabola of pure Poiseuille flow (pressure-driven, u ∝ y(h−y)). In between, the sliders blend the two — a favorable gradient (dp/dx < 0) accelerates flow beyond the plate speed near mid-gap, while a strong adverse gradient (dp/dx > 0) can push part of the fluid backward even though the top plate still drags forward, a real reverse-flow effect seen in journal bearings and blood-vessel segments downstream of a valve.

Wall shear stress:  τ = μ (du/dy)|_wall     (Newton's law of viscosity)
Mean velocity:      ū = (dp/dx)·h²/(12μ) + U/2
Channel Reynolds:   Re = ρ ū h / μ

Top plate velocity U — the drag-driven (Couette) contribution; negative runs the plate backward.
Pressure gradient dp/dx — the pressure-driven (Poiseuille) contribution; sign sets favorable vs. adverse.
Gap height h — sets the channel scale; shear stress grows sharply as the gap narrows.
Fluid presets — swap dynamic viscosity μ (honey ≈ 10 Pa·s down to air ≈ 1.8×10⁻⁵ Pa·s) to see how thick fluids flatten the velocity gradient while thin ones let a small force drive fast, high-Reynolds flow.

The Reynolds-number readout is a laminar-flow sanity check: this closed-form solution only holds while Re stays below roughly 1400 for channel flow — past that, real flow transitions to turbulence and the tidy parabola/line breaks down into eddies, which is why the badge above the canvas flips from "laminar" to "transitional" as you push velocity or gap height up.

LAMINAR

Slow tracer Fast tracer Velocity profile u(y)

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