Converting Velocity-Delta Products to Divergences of Dyads

The Problem

When we compute (the time derivative of momentum density), we get:

The first term is easy (acceleration). The second term involves , which looks messy. This note shows how to turn it into a clean divergence of a dyadic tensor.

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The Goal

Show that:

where is the outer product (dyad): in components, .

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The Derivation — Three Steps

Step 1: Use the moving delta identity

From The moving delta identity, we have:

Substitute:

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Step 2: The Key Insight — Does Not Act on

This is crucial: When we write , we mean the gradient with respect to the field coordinate (the spatial point where we observe the field).

• is the Lagrangian velocity of particle . It depends on the particle index and time , but not on the field coordinate .
• When we take of something, we differentiate only w.r.t. .
• Therefore: .

TL;DR

Treat as a constant vector when we’re doing spatial derivatives. The only spatial dependence is in the delta spikes.

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Step 3: Convert to Divergence of Dyad

Use the vector identity: For any constant vector and any scalar field :

Why? Use the product rule on the RHS:

The first term vanishes (the divergence of a constant dyad is zero), so:

(Here denotes the double-dot product: summed over .)

Apply with and :

Substitute back:

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Component View (for those who prefer indices)

In index notation: Let . Then:

Use :

Define the kinetic momentum-flux tensor (as a matrix):

Then:

Or in vector form:

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Interpretation: What is the Kinetic Momentum Flux?

Physical meaning: This tensor describes the transport of momentum due to particle motion.

• Diagonal component = flux of -momentum in the -direction (normal stress)
• Off-diagonal component = flux of -momentum in the -direction (shear)

When particles move with velocity , they carry momentum with them. If the velocity varies from point to point (e.g., shear flow), then momentum fluxes arise—and those are captured by .

Example (1D, shear flow): If (linear shear), then:

• Particles at height move faster than particles at height .
• Fast particles carry momentum downward and slow particles carry momentum upward.
• The net transport of -momentum in the -direction is .
• This is exactly where viscosity comes from! (See 2-What-is-Viscosity.)

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Common Confusion: Why Is This a Dyad, Not a Vector?

Question: is a vector. Why does (the outer product) give a tensor?

Answer: When we write in bold, it’s shorthand for the outer product. The component form makes it clear:

This is a 3×3 matrix (or 2×2 in 2D). When we take the divergence , we get:

which is a vector (summing over the second index).

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Summary

Term	Type	Meaning
	Vector	Particle velocity (doesn’t depend on field point )
	Operator	Gradient w.r.t. field coordinate
	Dyad (tensor)	Outer product; components
	Tensor field	Kinetic momentum-flux density
	Vector field	Momentum transport due to bulk and thermal motion

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The Takeaway

Summary

The key algebraic move is recognizing that can be rewritten as because is constant w.r.t. the field gradient .

This gives the microscopic momentum-flux tensor , which—when averaged—contains both thermal and bulk contributions to stress.

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