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Sandbox Physics
Active matter · A course with experiments

L3 · Chapter 7

Guide defects by choosing where rods prefer to point

A patterned preference for rod orientation changes the local order field, which changes its active stress and therefore the flow. To test control, compare the same initial state with and without the pattern. The browser slice uses permeable orientational anchoring; the fluid can cross the highlighted region.

Sandbox Physics ·

An orientation pattern is a different intervention from a wall

A solid wall constrains velocity: fluid cannot pass through it, and a no-slip wall fixes the tangential velocity too. An anchoring pattern instead tells nearby rods which axis they prefer. It can influence fluid indirectly through their stress while allowing fluid to cross.

Represent the preference by a target order tensor and a nonnegative strength field. A stripe can favour one fixed axis; an annular island can favour the local tangent around its centre, inside an otherwise periodic domain. The restoring term brings the current tensor toward that target. It must act inside the order equation before the stress is recomputed.

A wall with rod anchoring would require both conditions at once. Do not infer a pressure drop around a solid obstacle from a permeable anchoring disk. The present slice isolates orientation control so that the cause of a trajectory change remains identifiable.

Fanchor=12∫W(r) Tr⁡ ⁣[(Q−Qtarget)2] d2r,Hanchor=−W(r)(Q−Qtarget)F_{\rm anchor}=\frac12\int W(\mathbf r)\,\operatorname{Tr}\!\left[(Q-Q_{\rm target})^2\right]\,d^2r,\qquad H_{\rm anchor}=-W(\mathbf r)(Q-Q_{\rm target})

Use the same initial field twice

For an exact paired computation, save the initial order field, numerical resolution, activity, friction and random seed. Run once with anchoring disabled, then restore that same field and enable anchoring. In the browser, use Reset, pause promptly, and change only Orientational anchoring and Anchoring pattern. Demo presets also change friction or prepare a differently warmed field, so compare control settings directly.

Choose an observable before looking at the result. One option is the signed displacement of a tracked defect across a specified line. Another is the time-integrated fluid flux across that line. A third is the fraction of initially marked tracers that crosses a region. Each answers a different question: a defect is a feature of orientation, a tracer follows material flow.

For a vertical line, integrate horizontal velocity along the chosen segment and then over time. Incompressibility constrains the signed total through a complete periodic cross-section, which is zero when mean flow is removed. A local segment can still have nonzero flux; specify its endpoints and count the compensating return flow elsewhere.

Φ[y1,y2](T)=∫0T∫y1y2ux(x∗,y,t) dy dt\Phi_{[y_1,y_2]}(T)=\int_0^T\int_{y_1}^{y_2}u_x(x_*,y,t)\,dy\,dt

Worked example

A comparison can change sign when the box changes

Suppose a defect moves three length units to the right in a patterned run and one unit to the right in its matched control. The pattern-associated difference is two units for that initial condition and time interval. Repeat with different initial defect positions; a result caused by one close encounter may not survive.

Viscosity and friction define a screening length: beyond it, friction suppresses the flow from a local stress disturbance. If viscosity is one and friction is four in consistent units, the screening length is one half. A pattern much narrower than this length can drive a flow extending beyond its own footprint.

For an offline finite-size study, increase the domain while keeping the pattern’s physical radius, grid spacing and local coefficients fixed. Add charge-balanced distant defects if the initial state needs them. This checks how much periodic copies or nearby companions influence redirection. The browser’s box-size control also rescales the pattern: the annular radius is 0.16 of the side, its width is 0.055 of the side, and the stripe width is 0.065 of the side. It tests that combined change. Refining the grid tests numerical resolution.

ℓs=η/Γ,η=1, Γ=4 ⟹ ℓs=0.5\ell_{\rm s}=\sqrt{\eta/\Gamma},\qquad \eta=1,\ \Gamma=4\ \Longrightarrow\ \ell_{\rm s}=0.5

An experiment to think through

Predict, change, explain

  1. Predict where a change of preferred axis will bend the order field most strongly.
  2. Use Reset, pause, and compare only Anchoring pattern and Orientational anchoring at fixed Active stress and Friction screening. Export equal-duration records.
  3. Compare a defect displacement with a tracer displacement. Explain why these are different measurements of control.
Follow defects and fluid together Find half-turn defects

These Labs generate synthetic particle or continuum records. Their displayed model notes state the implemented dynamics and units.

Try explaining it yourself

1. Can a permeable anchoring disk impose no-slip flow?

Read an explained answer

No. It changes orientational dynamics. A no-slip disk needs a separate fluid boundary treatment that blocks normal flow and constrains tangential flow.

2. A local line segment has positive flux. Does that imply a net pump across the periodic box?

Read an explained answer

No. Return flow elsewhere can balance it. With the mean velocity removed, the signed integral over the full periodic cross-section is zero.

Sources to read alongside

  1. Giomi, Geometry and topology of turbulence in active nematics, arXiv:1409.1555v2 (2015), Physical Review X 5, 031003

    Equation (1), Section I.2 and Figure 2 provide the continuum context and active-length scaling. Our screened-flow benchmark is a manufactured solution, separate from the paper’s turbulent simulations.

  2. Serra, Lemma, Giomi, Dogic & Mahadevan, Defect-mediated dynamics of coherent structures in active nematics, Nature Physics 19, 1355–1361 (2023)

    Figure 1 and equations (1)–(3) introduce the flow map and finite-time stretching; Figure 2 uses experimental microtubule–kinesin velocities. This is a guided reading, with no copied images or imported experimental dataset.

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