Longer, with nothing pulling it

Take two neighbouring particles and the vector between them. It obeys

$$\frac{\mathrm d(\delta\mathbf x)}{\mathrm dt} = (\delta\mathbf x\cdot\nabla)\mathbf u,$$

which is the velocity gradient acting on it and nothing else. Integrating that along a trajectory gives the deformation gradient $\mathsf F$, with $\dot{\mathsf F} = \nabla\mathbf u\,\mathsf F$ and $\mathsf F(0) = \mathsf I$, and every question about stretching is a question about $\mathsf F$: a line that started as $\hat{\boldsymbol\ell}_0$ has length $|\mathsf F\hat{\boldsymbol\ell}_0|$, an area element grows by $\det\mathsf F$, and the most any direction can achieve is the largest singular value.

The rate of change of a line’s length is

$$\frac{\mathrm d}{\mathrm dt}\ln|\delta\mathbf x| = \hat{\boldsymbol\ell}\cdot\mathsf D\cdot \hat{\boldsymbol\ell},$$

with $\mathsf D$ the symmetric part of the velocity gradient. The rotation does not appear. So it is the strain that stretches lines and the vorticity contributes exactly nothing — which is the second half of the split deformation made and this collection had never used.

The reading that is wrong #

It is very tempting to read that identity as lines grow at the strain rate, and it is false, because $\hat{\boldsymbol\ell}$ moves too.

Two flows, chosen so that the Frobenius norm of $\mathsf D$ is identical in both — $\alpha\sqrt2$ for the first and $\gamma/\sqrt2$ for the second, so the matching is $\gamma = 2\alpha$:

$$\text{pure strain}\quad \mathbf u = (\alpha x, -\alpha y),\qquad \text{simple shear}\quad \mathbf u = (\gamma y, 0).$$

Both are incompressible. Both have the same rate of strain at every point and every instant. And a material line in the first grows as $e^{\alpha t}$ for ever, while in the second it grows as $\gamma t$ and no faster.

At $\alpha t = 5$ that is a factor of 148 against a factor of 10.1 — measured from integrated deformation gradients and checked against $e^{\alpha t}$ and against $(\gamma t + \sqrt{\gamma^2t^2+4})/2$ respectively, both to two parts in a thousand. The ratio is fifteen and it grows without bound.

Why the exponential happens at all #

The exponential in the first flow deserves a sentence, because it is not obvious that a line released at a generic angle should reach it.

A line at forty-five degrees to the extensional axis has half its length along each principal direction. The component along the extensional axis grows as $e^{\alpha t}$ and the one along the compressional axis decays as $e^{-\alpha t}$, so their ratio moves at $e^{2\alpha t}$ and the line turns onto the extensional axis at twice the strain rate.

That is a measurement, not a hand-wave: the angle to the axis decays with a fitted exponent of $-1.996$ against a predicted $-2$, and falls from $0.314$ radians at $t = 0.5$ to $4.5\times10^{-5}$ at $t = 5$.

So in a steady pure strain almost every line ends up aligned and almost every line ends up growing at $\alpha$. The exceptional set — a line released exactly along the compressional axis — has measure zero and is not a physical case.

What simple shear does instead #

Simple shear has the same strain rate and also has vorticity, and the vorticity is what carries the line away from the extensional direction as fast as the strain pulls it along.

The result is that the alignment never completes. The line ends up asymptotically along the shear direction, where the extensional rate is zero, and its growth degenerates from exponential to algebraic — the classic $\gamma t$ of a shear layer.

This is why a shear layer is a poor mixer for its strain rate, and it is the quantitative form of an observation this collection made from a different direction: a steady two-dimensional flow cannot mix, and the reason a stirred one can is that its straining directions keep changing relative to the material.

Reading the deformation gradient #

The object doing the work here has not appeared on this site before and is worth a paragraph on its own, because it is the natural home of several results already scattered across the collection.

$\mathsf F$ maps an initial separation to a current one. Its singular values are the principal stretches — the largest and smallest factors any direction achieves — and its determinant is the volume ratio, which is one for an incompressible flow and is the exponential of the integrated divergence otherwise. Its polar decomposition splits it into a rotation and a stretch, and that is the finite-deformation counterpart of the instantaneous split deformation makes into vorticity and rate of strain.

The relationship between the two splits is where the confusion lives. The instantaneous one is a statement about $\nabla\mathbf u$ at one point at one time; the finite one is a statement about the whole history. They are not the time integral of one another, because $\mathsf F$ is a time-ordered product of matrices that do not commute, and this essay’s entire result is a consequence of that non-commutation. A flow whose rate of strain is constant and whose rate of rotation is constant can have a deformation gradient that grows exponentially, algebraically, or not at all, depending only on the ratio of the two.

The blob is not getting bigger #

There is a picture that goes wrong here often enough to be worth stating plainly.

A circle of dye stretched into a long thin ellipse looks, in a photograph, like dye that has spread. Nothing has spread. The area is unchanged to fifteen decimal places, the amount of dye is unchanged because a material region keeps what it started with, and the concentration at the centre of the filament is what it always was. What has happened is that the interface between dye and clear fluid has grown by the same factor the perimeter has.

That distinction is the whole of why stretching matters. Molecular diffusion acts across a gradient, and the gradient is set by the filament’s thickness, which falls as the reciprocal of the stretch. So a factor of 148 in length is a factor of 148 in thinness and a factor of $148^2$ in the diffusive flux per unit area — which is why a stirred cup of tea mixes in seconds and an unstirred one takes a day, with the same molecular diffusivity in both.

The number that decides, and the number that does not #

The result can be put as a competition between two quantities that are usually quoted as one.

The rate-of-strain magnitude $|\mathsf D|$ answers how fast is the fluid being deformed, and it is the quantity that appears in the dissipation — viscous dissipation is $2\mu\mathsf D\!:\!\mathsf D$ exactly, with no contribution from the rotation, so two flows with the same $|\mathsf D|$ dissipate identically.

The stretching exponent answers how fast is the fluid being pulled apart, and it is not the same number. Pure strain and simple shear at matched $|\mathsf D|$ dissipate at exactly the same rate and mix at rates whose ratio grows without bound.

So a flow can be expensive and useless: paying the full viscous bill for its strain rate while doing almost nothing to the material. A shear layer is precisely that, and the roll-up that ends it is the flow finding a configuration in which the same strain rate buys stretching. That is a more useful way to read the instability of a shear layer than as a story about wiggles growing.

The criterion, derived #

Put the strain in a frame turning at rate $\Omega$ and the velocity gradient in that frame is constant, with eigenvalues $\pm\sqrt{\alpha^2-\Omega^2}$. That single line contains the whole subject:

• $\Omega < \alpha$: the eigenvalues are real and lines grow exponentially at $\sqrt{\alpha^2-\Omega^2}$;
• $\Omega = \alpha$: the eigenvalues collide at zero — and this case is simple shear, exactly, which is why simple shear’s growth is algebraic;
• $\Omega > \alpha$: the eigenvalues are imaginary and no line grows at all, however large the strain rate is.

The prediction is checked by fitting exponents to integrated deformation gradients over a sweep of $\Omega$, with the strain rate held at $|\mathsf D| = \alpha\sqrt2$ in every member — measured, so that the result is not about the strain having been quietly turned off.

Below the threshold a line reaches $1.2\times10^6$ over the run. Above it, the largest stretch reached at any time is 2.7. A factor of half a million, across a boundary at which the strain rate does not change.

And this is the Q-criterion #

The combination $\alpha^2 - \Omega^2$ is, up to a factor, the Q-criterion — the excess of rotation over strain that computational fluid dynamics uses to decide where the vortices are. It usually arrives as a definition with an appeal to intuition attached.

Here it arrives as a consequence: the surface $Q = 0$ is the surface on which material lines stop being pulled apart, and that is a statement about what the flow does to the fluid rather than about how a tensor is decomposed. It is the best justification the criterion has.

It is also, as the next essay shows, not enough — because the quantity is not objective and an observer who spins fast enough moves the surface. The derivation here inherits that: $\Omega$ in the analysis above is the rotation rate of the strain axes relative to the material, and an observer’s own rotation adds to it.

What is not in the velocity field #

The line’s own direction, and the history of the strain axes.

A velocity gradient at an instant says how fast a given direction is stretching. It does not say how fast the fluid is stretching, because that depends on which directions the line has been carried through since it was released. Every quantity in this essay is an integral along a trajectory and none of them can be read off a snapshot.

That is why the two flows in the first figure — identical strain rate, identical divergence, identical everything a single measurement of $\nabla\mathbf u$ contains — do different things to a blob of dye by a factor that grows without bound. A snapshot of the velocity gradient field is not enough information to say how well a flow mixes, and no amount of resolution in the snapshot repairs that.

Where this bites #

Three places, all of which this collection has met without the machinery.

Mixing in a shear layer. Free shear layers have a strain rate that looks adequate and a stretching that is algebraic, and it is the roll-up into vortices — which puts material into regions where the strain axes turn slowly relative to it — that does the mixing rather than the mean shear.

Scalar gradients. The gradient of a passive scalar obeys the adjoint equation and sharpens at exactly the rate a line stretches, so everything above transfers directly to how fast molecular diffusion is handed something to work on. A flow that cannot stretch lines cannot sharpen gradients and cannot mix, whatever its strain rate.

Vortex stretching. The spin that feeds itself is this mechanism applied to the vorticity vector, which obeys the same equation as a material line in an inviscid flow. The threshold above is therefore also a statement about when vorticity amplification happens and when it does not — and it explains why a two-dimensional flow, whose vorticity vector is always perpendicular to the plane of straining, has none of it at all, which is the mechanism behind the cascade that runs backwards.

And the drop that will not break. A drop in a shear is stretched by the strain and rotated by the vorticity, and whether it breaks is decided by exactly the competition above rather than by the strain rate alone. That is why a simple shear needs a capillary number several times larger to break a drop than a pure strain does, and why the shape at which it happens is different. The threshold in the figure is the same threshold, with surface tension supplying a restoring force the material line does not have.

What is conserved while all this goes on #

One reassurance. Every flow in this essay is incompressible, and $\det\mathsf F = 1$ throughout — checked at $4\times10^{-15}$ in the two-flow comparison and to a part in ten thousand across the rotation sweep, where the accumulated Runge–Kutta error over a stretch reaching a million is what the residual is.

So the area of a material patch is unchanged while its perimeter grows by a factor of a hundred and fifty. A patch of dye is not being made larger; it is being made thinner, and the thinning is what brings molecular diffusion into range. That is the whole mechanism by which a flow mixes, and it is why the interesting quantity is the stretch rather than the strain.

The third eigenvalue, which has no two-dimensional counterpart #

Everything above is planar, so the rate of strain has two eigenvalues and incompressibility makes them $\pm\alpha$. In three dimensions there are three, summing to zero, and the middle one is free — it may be positive, negative or zero, and what it is decides what shape a blob of fluid becomes.

Two arrangements sit at the ends of that freedom. With eigenvalues in the ratio $2:-1:-1$ the element is pulled out along one axis and squeezed equally in the other two: axial extension, and a sphere of dye becomes a filament. With $1:1:-2$ it is pulled out in two directions and squeezed in one: biaxial extension, and the sphere becomes a sheet. Both conserve volume, both can have the same rate-of-strain magnitude, and they are different mixing behaviours entirely — a filament presents a small interface and a large one respectively for the same amount of stretching.

Turbulence turns out to have a strong preference, and it is not the obvious one. Measured in experiments and in simulations, the three eigenvalues sit in a ratio near $3 : 1 : -4$ — the intermediate one is positive on average, so the typical local deformation is biaxial. Fluid in a turbulent flow is being flattened into sheets more often than drawn into threads, which is why the fine structure of a dissipation field is made of sheets, and why those sheets subsequently roll up into the tubes that the vortex-stretching picture emphasises.

There is a companion result which is stranger and even better established. The vorticity vector aligns preferentially with the intermediate eigenvector, not the most extensional one. That is thoroughly counter-intuitive — the naive picture of vortex stretching has the vorticity lying along the direction of greatest stretching — and it has been confirmed everywhere it has been looked for since it was first measured in the 1980s. The two facts together are consistent: the intermediate eigenvalue is positive on average, so vorticity aligned with it is still being stretched, just not as fast as something aligned with the leading direction would be.

For this essay the point is that the middle eigenvalue is a quantity the planar analysis has no room for at all. In two dimensions it is identically zero by construction, so nothing about the shape of a stretched element is in question. In three it is the variable that decides whether a flow makes threads or sheets, and it is decided by the flow rather than by the observer’s choice of frame.

The model limit #

Everything here is computed for spatially linear velocity fields, where the deformation gradient depends only on time and the answers are closed-form. A real flow’s velocity gradient varies along the trajectory, and the deformation gradient is then a time-ordered product that does not commute with itself — so the growth rate is not the average of the local growth rates, and can be much smaller.

That non-commutation is the honest reason a real turbulent flow stretches material lines more slowly than its strain-rate statistics suggest, and it is the same effect the rotation sweep above shows in its simplest form: a fluid element that keeps being handed a new straining direction gets less out of each one than a fluid element that is handed the same one twice.

The second limit is that everything here is two-dimensional. In three dimensions $\mathsf F$ is a three-by-three matrix with three principal stretches whose product is one, so a material element can be pulled out in one direction and squashed in two, or pulled out in two and squashed in one — and those are different mixing behaviours with the same determinant. Which of them a flow does is decided by the intermediate strain eigenvalue, a quantity with no two-dimensional counterpart at all, and one whose statistics in turbulence are still argued about.