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Sandbox Physics
Active matter · A course with experiments

L3 · Chapter 6

Turn an orientation field into a flow

Active rods exert a direction-dependent stress. When that stress varies across the plane, it applies a force to the fluid. A low-Reynolds-number flow follows from balancing that force against pressure, viscosity and substrate friction. The force and the displayed flow must come from the same orientation field.

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Store an axis without choosing a head

Use a symmetric, traceless two-by-two tensor. The director angle determines its two components; the scalar order measures how strongly the rods align. Reversing the director leaves both components unchanged. In this convention, perfect order has scalar amplitude one, while the diagonal component for horizontal rods is one half.

Different papers absorb the factor of two into their order variable. Compare definitions before comparing numerical activities. The formula below fixes the convention used for this derivation.

Q=S(nn−12I)=S2(cos⁡2θsin⁡2θsin⁡2θ−cos⁡2θ)Q=S\left(\mathbf n\mathbf n-\frac12I\right)=\frac S2\begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}

Only spatial changes of stress supply a bulk force

Choose the active stress as minus activity times the order tensor; positive activity is extensile in this convention. A uniform order tensor gives a uniform active stress and no bulk force in a periodic domain. Curvature and changes of order create spatial stress gradients.

Pressure adjusts to prevent compression. Viscosity resists velocity gradients and substrate friction resists velocity itself. The resulting screened Stokes equation is an instantaneous force balance: there is no fluid acceleration term. Friction is a deliberate addition to the unfrictional continuum context of Giomi’s 2015 paper.

σija=−ζQij,fi=∂jσija,−∇p+η∇2u−Γu+f=0,∇⋅u=0\sigma^{\rm a}_{ij}=-\zeta Q_{ij},\quad f_i=\partial_j\sigma^{\rm a}_{ij},\qquad -\nabla p+\eta\nabla^2\mathbf u-\Gamma\mathbf u+\mathbf f=0,\quad\nabla\cdot\mathbf u=0

Pressure removes the longitudinal part

A Fourier mode is a sinusoidal pattern with a known wave vector. Spatial derivatives become multiplications by its wave-vector components. Split the force into a part parallel to this vector and a part perpendicular to it. Pressure cancels the parallel part; only the perpendicular part drives an incompressible velocity.

Divide that transverse force by viscosity times wave number squared plus friction. This gives both the magnitude and phase of the velocity mode. The zero-wave-number velocity is set to zero in a periodic simulation with no imposed throughflow.

u^(k)=(I−kkT∣k∣2)f^(k)η∣k∣2+Γ(k≠0)\widehat{\mathbf u}(\mathbf k)=\frac{\left(I-\frac{\mathbf k\mathbf k^{\mathsf T}}{|\mathbf k|^2}\right)\widehat{\mathbf f}(\mathbf k)}{\eta|\mathbf k|^2+\Gamma}\quad(\mathbf k\ne0)

Worked example

Test one sinusoid all the way through

Set the off-diagonal order component to an amplitude times the sine of a vertical wave number times height, and set the diagonal component to zero. Its stress divergence points horizontally and varies as a cosine in height. Because the force is perpendicular to the wave vector, pressure does not remove it.

With activity 2, amplitude 0.1, viscosity 1, friction 1 and wave number 1 in consistent dimensionless units, the horizontal velocity amplitude is minus 0.1. Reverse activity and the velocity reverses. Set activity to zero and the flow vanishes. This signed example catches an incorrect stress sign or derivative as well as an incorrect amplitude.

A grid derivative approximates a continuous derivative, so compare progressively finer grids. The benchmark supplied with this course records the continuum error at resolutions 16, 32 and 64. Its decreasing error tests the force-to-velocity calculation independently of whether a moving defect looks plausible.

Qxy=Asin⁡(ky),Qxx=0,ux=−ζAkηk2+Γcos⁡(ky),uy=0Q_{xy}=A\sin(ky),\quad Q_{xx}=0,\qquad u_x=-\frac{\zeta Ak}{\eta k^2+\Gamma}\cos(ky),\quad u_y=0

Let the rods respond to the flow

A dynamic nematic field needs another equation. Local relaxation favours a finite degree of order, elasticity smooths rapid spatial changes, advection carries the pattern, and flow rotation turns it. A flow-alignment term can also orient rods along a stretching direction. These ingredients create feedback between order and flow.

The browser model evolves two order components with a stated minimal relaxation, advection, rotation and alignment law, and recomputes flow from the resulting active stress. It is a small periodic continuum model with friction. Its parameter choices are teaching choices, not measured coefficients for a particular tissue.

A defect track is extracted from this evolving order field. For a positive half defect, asymmetry permits directed motion; a negative half defect has threefold symmetry in its ideal texture. In a many-defect flow, both also respond to neighbours and advection. Compare tracks in controlled initial configurations before assigning all motion to self-propulsion.

An experiment to think through

Predict, change, explain

  1. Predict the initial velocity reversal when the sign of activity changes, holding the initial field fixed.
  2. Reset the same defect pattern, compare extensile and contractile activity, then increase friction.
  3. Trace one cause through the order pattern, its stress gradient and the transverse velocity. Use the sampled field and tracked defects together.
Follow defects and fluid together

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

Try explaining it yourself

1. Does a large uniform active stress necessarily drive a periodic bulk flow?

Read an explained answer

No. The bulk force is the stress divergence. A spatially uniform stress has zero divergence; boundary tractions are a separate issue.

2. A force Fourier mode is parallel to its wave vector. What does the incompressible velocity solver return?

Read an explained answer

Zero velocity for that mode. Pressure balances the full longitudinal force. Applying the transverse projector gives 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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