Concept

Deformation gradient — where it appears

The matrix mapping an initial separation between two fluid particles to its current value, obtained by integrating the velocity gradient along a trajectory. Its singular values are the principal stretches and its determinant is the volume ratio, so every question about how far a flow has pulled fluid apart is a question about it.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

kinematics · Material lines

A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

kinematics · Dilatation $One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.$

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

kinematics · Material lines

One pair of strainings, two orders, two lengths. The stretch of the most-stretched material direction against time, for a simple shear followed by a pure strain and for the same two in the other order. The two curves are identical until the swap and separate afterwards, ending a factor of 2.16 apart.

Two strainings, and the order they came in

A material line is stretched by a shear and then by a pure strain, and then by the same two in the other order. Every instantaneous measure of how hard the fluid was being worked is identical in the two cases. The lengths at the end differ by a factor of 2.16.

kinematics · Material lines

Named alongside it

The objects these essays reach for when they reach for this one.

Mixing Model limit The Lyapunov exponent Material line Strain rate Stretching ABC flow Averaging Chaotic advection Commutator Compressibility Density

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