Foundations — Fluid definition — shear stress, no fixed shape

This page assumes you have seen nothing. Before you can read the parent note Fluid definition, you must own every symbol it throws at you. We build them one at a time, each from the picture it lives in.

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1. Force $\vec{F}$ — a push with a direction

Why the topic needs it. A fluid is defined by how it reacts to a push. But not every push is the same: pushing a surface straight-on is very different from scraping it sideways. So the very first thing we need is an arrow whose direction we can talk about.

Look at figure s01: the same arrow can point straight into a wall, or slide along it, or anything between.

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2. Surface and area $A$ — the patch the force lands on

Why the topic needs it. Deformation cares about intensity, not raw force. To get intensity we will divide force by area, so we must first have area as a clean, named quantity.

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3. Splitting a force: normal part $F_n$ and tangential part $F_t$

Here is the single most important picture in the whole topic. Take a force arrow hitting a surface. We break that one arrow into two arrows at right angles:

• the part pointing straight into (perpendicular to) the surface — call it $F_n$ ("n" for normal, which in geometry means "at $90^{\circ }$");
• the part pointing along (parallel/tangent to) the surface — call it $F_t$ ("t" for tangential, meaning "sliding along").

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4. Stress $\sigma$ and $\tau$ — force per unit area

Now we combine "split the force" (step 3) with "divide by the patch" (step 2).

Why the topic needs it. The parent note's headline — "a fluid cannot resist shear stress at rest" — is a sentence about $\tau$. Without $\tau$ defined, the definition is just words. The symbol $\tau$ is the topic.

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5. Shear strain $\gamma$ — how much a slab got tilted

Push the top of a soft block sideways while its base stays put. The block leans over into a slanted shape. The amount of lean is the shear strain.

Why the topic needs it. For a solid, the shear stress is proportional to this lean: $\tau =G\gamma$. This is the law the parent note contrasts fluids against. You cannot appreciate what makes a fluid special until you understand the solid's $\gamma$.

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6. Rate of change and the gradient $\frac{du}{dy}$ — the fluid's twist

Here is the leap from solids to fluids, and it needs one new idea: rate, i.e. how fast something changes.

Now the velocity gradient. Imagine fluid filling the gap between a moving top plate and a fixed bottom. Layers near the top move fast; layers near the bottom barely move. The speed $u$ changes as you climb in height $y$. How fast speed changes per unit height is written

$\frac{du}{dy}=\frac{\text{difference in layer speeds}}{\text{height between them}}.$

Why the topic needs it. The whole payoff — Newton's law $\tau =\mu \frac{du}{dy}$ — is built on this gradient. If $du/dy$ is a mystery, that boxed formula is a magic spell. Deeper detail on where the profile comes from lives in Velocity Profile and No-Slip Condition.

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7. Viscosity $\mu$ — the "thickness" constant

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8. Putting the alphabet together

Symbol	Says out loud	Picture	Job in the topic
$\vec{F}$	force arrow	arrow with length + direction	the push we apply
$A$	area	a flat patch	what the push spreads over
$F_n,F_t$	normal / tangential parts	two arrows at $90^{\circ }$	split into "into" vs "along"
$\sigma ,\tau$	normal / shear stress	force-per-patch	$\tau$ is the fluid definition
$\gamma$	shear strain	slanted block	the solid's response
$\dot{\gamma }=\frac{du}{dy}$	rate of tilt / gradient	layers sliding	the fluid's response
$\mu$	viscosity	"thickness" slope	links $\tau$ to $du/dy$

Recall Self-check: name each symbol before scrolling

$\tau$ ::: shear stress, the sliding-along force per unit area, in Pa. $\sigma$ ::: normal stress, the straight-in force per unit area, in Pa. $\gamma$ ::: shear strain, the (unitless) amount a slab is tilted. $\dot{\gamma }$ ::: rate of shear strain = how fast the tilt grows = $du/dy$. $\mu$ ::: dynamic viscosity, the fluid's "thickness" constant, in Pa·s.

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How these foundations feed the topic

Once you can read this map left-to-right, the parent note reads like plain English. Compare the two response laws in Stress and Strain in Solids and see the pressure story in Pressure in Fluids and Hydrostatics — Fluids at Rest.

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Equipment checklist

Recall Am I ready for the parent note?

I can draw a force as an arrow and say what its direction means ::: Yes — a force is a push/pull with size (arrow length) and direction. I can split a force on a surface into a perpendicular part and a parallel part ::: Yes — $F_n$ points into the surface, $F_t$ slides along it, at $90^{\circ }$. I know why we divide force by area ::: To get intensity (fierceness per patch), which is what actually deforms matter. I can state $\sigma$ and $\tau$ with their formulas and units ::: $\sigma =F_n/A$, $\tau =F_t/A$, both in pascals (${\text{N/m}}^2$). I know what shear strain $\gamma$ is and why it is a ratio ::: The tilt $\Delta x/y$; a ratio because tilt is relative to the slab's height. I understand "rate" and can read $du/dy$ ::: A rate is tiny-change ÷ tiny-time; $du/dy$ is how fast layer speed changes with height, units ${\text{s}}^{-1}$. I know why fluids use $\dot{\gamma }$ instead of $\gamma$ ::: Because a fluid never stops tilting, so there is no final $\gamma$ — only a rate. I know what viscosity $\mu$ is and its units ::: The "thickness" slope linking $\tau$ to $du/dy$; units $\text{Pa⋅s}$.