FR22 · Orientations push a fluid

Active Nematic Flow

Why do differently shaped defects stir a fluid differently? Active rods exert stress along their local orientation. Where that orientation bends, the stress pushes the fluid unevenly. A positive half defect has a polar shape; a negative half defect has threefold symmetry, so their surrounding flows differ.

Watch a field of rods drive a moving sea of tracers. Reverse the active stress, add an orientational anchoring island or stripe, and follow the defects detected in the evolving field.

Model assumptions

A small periodic two-dimensional Q-tensor model evolves with relaxation, advection, rotation and alignment. The same field drives an incompressible screened Stokes flow through its active stress. The anchoring pattern is permeable to fluid. Passive elastic stress, thermal noise and material calibration are omitted.

Positive half defectNegative half defectPassive fluid tracers

Lengths and times use model units; hue shows direction. Anchoring lets fluid through.

Positive half defects

2

Negative half defects

2

RMS fluid speed

0.2

Half-section flux

0.951

A flow that changes with its texture

Cyan is RMS speed; amber is signed horizontal flux through the lower half of the central vertical section. A complete periodic section has zero net flux. Both curves come from the evolving velocity field.

Equations, approximations and sources

The two-component alignment tensor relaxes, moves and rotates with a periodic incompressible flow. A spectral projection solves the centered-difference screened Stokes equation with active stress from the same tensor. Local anchoring is an orientation potential, not a solid boundary.

The coarse browser model isolates coupling and sampling. It omits passive elastic stress and material calibration; its trajectories are numerical examples. Export includes raw tensor samples, velocity samples, seeds, parameters and tracked charges.

Giomi (2015), active nematodynamics, equations 1–2 and Figure 1

Continue through the active-matter course

Read the active matter course: from persistent motion to defects and flow

Start with the essentials

Focus question: Why do differently shaped defects stir a fluid differently?
One-sentence intuition: The divergence of the active stress drives the flow; the flow then transports and rotates the same orientation field.

Core mathematical model

Active stress drives screened Stokes flow

0=-\nabla p+\nabla^2\mathbf v-\xi\mathbf v-\zeta\nabla\cdot Q,\qquad\nabla\cdot\mathbf v=0

Viscosity is one in the chosen units. Positive activity is extensile in this sign convention. Pressure removes the compressible part of the force.

Relaxation and anchoring

H=0.12\nabla^2Q+(1-2\operatorname{tr}Q^2)Q+Ww(\mathbf r)(Q_{\rm a}-Q)

Relaxation smooths the field and restores order. The last term turns rods toward a prescribed orientation near a permeable island or stripe.

Common difficulties

Two meanings of motion

Typical misconceptionA bright marker is a particle with a prescribed speed.

Better mental modelMarkers follow zeros detected in the evolving tensor field. Tracers are separate passive points carried by the computed fluid velocity.

Model scale

Typical misconceptionThe grid is a calibrated sample of a biological material.

Better mental modelA small periodic two-dimensional Q-tensor model evolves with relaxation, advection, rotation and alignment. The same field drives an incompressible screened Stokes flow through its active stress. The anchoring pattern is permeable to fluid. Passive elastic stress, thermal noise and material calibration are omitted.

Run the experiment

01
Reverse the stress
Compare Defects drive the flow and Reverse the stress immediately after preparation.
What to observe: The initial tensor field is the same. Changing the stress sign reverses its instantaneous velocity; later the fields evolve differently.
02
Add an anchoring pattern
Choose An anchoring island or Guide with a stripe. Drag the pattern across the field.
What to observe: The rods turn near the pattern, which changes the active force and the detected defect paths. The pattern is permeable.
03
Compare screening and size
Increase Friction screening. In Numerical settings, change the box size and repeat the same preset.
What to observe: Friction weakens long-range flow. Changing the periodic box also changes the neighboring images of every defect.

Active Nematic Flow

A flow that changes with its texture