M096 · Fluid mechanics

Couette–Poiseuille Viscous Flow

Set a plate speed, pressure gradient and gap. Move a probe through the velocity and stress profiles, then infer viscosity and pressure drive from acquired readings.

Probe velocity $\text{Ready}$

Probe signed shear $\text{Ready}$

Signed flow per width $\text{Ready}$

Upper wall power per footprint $\text{Ready}$

Pressure power per footprint $\text{Ready}$

Viscous heating per footprint $\text{Ready}$

Peak-speed Reynolds number $\text{Ready}$

Power closure residual $\text{Ready}$

Viscosity inferred from raw readings $\text{Ready}$

Pressure drive inferred from raw readings $\text{Ready}$

Acquired stations $\text{Ready}$

Raw-coordinate fit residual $\text{Ready}$

Inference status $\text{Ready}$

Model domain $\text{Ready}$

GEOMETRY / PROBE / INFERENCE

What does the shear reveal?

Plate separation

H

4.00\,mm

Upper wall speed

U

20.00\,mm/s

Pressure drive (negative pressure slope)

G

30.00\,Pa/m

Dynamic viscosity

\mu

20.00\,mPa\,s

Density

\rho

1000.00\,kg/m^3

Probe gap fraction

\eta_p

0.50\,

Sensor perturbation gain

\epsilon

0.00\,

The lower wall is fixed. Flow and plate motion are signed along the conduit; height runs upward across the gap. Stress is the fluid stress on an upward-facing plane, so the upper wall receives the opposite traction. Wall power can be negative while total dissipation stays positive.

Steady, incompressible, Newtonian, no-slip, fully developed planar flow. The displayed cell is twenty gaps long and ten gaps wide; edges, entrance and sidewall effects are omitted. A peak-speed Reynolds number above 200 blocks sensor acquisition; this teaching screen is not a universal transition criterion.

Acquire velocity and shear, fit a quadratic with a free velocity offset, then fit stress against its inferred derivative through zero. Both viscosity and pressure drive come from raw readings, assuming zero stress offset and known gap. Stationary fluid cannot identify viscosity. Perturbations are deterministic and bounded, without statistical confidence claims.

Fully developed Newtonian laminar reference between infinite plates; finite hardware depicts a representative cell. Acquisition is screened by a conservative teaching Reynolds limit, without a stability claim.

Start with the essentials

Focus question: Can shear reveal a hidden viscosity?
One-sentence intuition: Infer parameters from independent raw channels and check the approximation domain first.

Core mathematical model

Signed velocity and shear

\begin{aligned}\eta&=y/H,\quad G=-dp/dx\\u&=U\eta+\frac{GH^2}{2\mu}\eta(1-\eta)\\\tau&=\mu U/H+GH(1/2-\eta)\end{aligned}

The lower wall is fixed; shear is the stress on an upward-facing fluid plane.

Flux per unit width

q=\int_0^H u\,dy=\frac{UH}{2}+\frac{GH^3}{12\mu}

Positive and negative layers can cancel even while the fluid dissipates energy.

Power closure per footprint

\begin{aligned}Gq+U\tau(H)&=\mathcal D\\\mathcal D&=\int_0^H\mu(u_y)^2\,dy\end{aligned}

The wall can extract mechanical energy; pressure plus wall work equals heating.

Infer from acquired profiles

\begin{aligned}\widehat u&=c+a\eta+b\eta(1-\eta)\\\widehat\mu&=\frac{\sum_i\widehat u_{y,i}\tau_i}{\sum_i\widehat u_{y,i}^2}\\\widehat G&=2\widehat\mu b/H^2\end{aligned}

A free velocity offset is fitted. The stress fit assumes zero sensor bias; the derivative must be nonzero.

Domain diagnostic

\mathrm{Re}_{\mathrm{peak}}=\rho H\max_y|u|/\mu

A teaching threshold of 200 blocks acquisition outside the selected low-inertia domain, without proving stability.

Common difficulties

Reference versus inference

Typical misconceptionThe estimate can be read without recording data.

Better mental modelClear the record and the estimate disappears. Sensor perturbations can produce nonphysical fits; estimates are not clipped or independently calibrated.

Run the experiment

01
Separate the two drives
Compare Wall only, Pressure only and Combined drive.
What to observe: Linear and parabolic contributions add; both meet no-slip wall values.
02
Find counterflow
Select Counterflow and move the gold probe across the gap.
What to observe: A moving upper wall can coexist with negative interior velocity.
03
Close the energy ledger
Change wall speed and inspect signed wall power.
What to observe: Pressure and wall work sum to nonnegative viscous heating.
04
Acquire and infer
Acquire the gap profile, increase perturbations and export CSV.
What to observe: The fit uses recorded velocity and shear, without reference viscosity or gradient.