Foundations — Viscosity — dynamic μ, kinematic ν = μ - ρ; Newtonian vs non-Newtonian

Before you can read the parent note, you need to own every symbol it throws at you. Below, each item is built from nothing: plain meaning → the picture → why the topic needs it. They are ordered so each one leans only on the ones above it.

1 — Layers of fluid (the deck of cards)

Plain words: Imagine a liquid as a stack of very thin sheets, like a deck of cards lying flat. Each sheet can move sideways at its own speed.

The picture: Look at Figure 1. The bottom card is glued to the table (speed $0$). The top card is dragged to the right. The cards in between get dragged along by their neighbours, each a little faster than the one below it. This staircase of arrows is the whole story of viscosity.

Why the topic needs it: Viscosity is defined by what happens between these layers. If you don't picture the layers, none of the symbols mean anything.

2 — $u$ : the speed of a layer

Plain words: $u$ is simply the sideways speed of a fluid layer, measured in metres per second (m/s).

The picture: In Figure 1, $u$ is the length of the blue arrow on each card. Long arrow = fast layer; no arrow = frozen layer.

3 — $y$ and the velocity gradient $\frac{du}{dy}$

Plain words: $y$ is the height measured across the gap, from the bottom plate ($y=0$) up to the top plate ($y=h$). The velocity gradient $\frac{du}{dy}$ asks: "as I climb up by a tiny amount, how much faster does the layer move?"

The picture: In Figure 2, draw the speed $u$ sideways and the height $y$ upward. The layers trace a straight slanted line. The steepness of that line is $\frac{du}{dy}$. A gentle slope = layers barely differ = weak shearing. A steep slope = neighbours race past each other = strong shearing.

Why the topic needs it: $\frac{du}{dy}$ is the single input to Newton's law of viscosity. Every viscous force on the parent page starts here.

4 — Shear rate $\dot{\gamma }$ (a nickname for the gradient)

Plain words: $\dot{\gamma }$ (Greek "gamma", with a dot on top) is just another name for the velocity gradient: $\dot{\gamma }=\frac{du}{dy}$. The dot means "rate" — how fast the shape is being sheared per second.

The picture: Same slanted line as Figure 2. For the simple straight-line profile, $\dot{\gamma }$ = (top speed $U$) ÷ (gap $h$), because a straight line has constant steepness everywhere.

Why the topic needs it: When the parent draws $\tau$-versus-$\dot{\gamma }$ graphs to classify Newtonian vs non-Newtonian fluids, the horizontal axis is $\dot{\gamma }$.

5 — Shear stress $\tau$ (the sideways rub, per area)

Plain words: $\tau$ (Greek "tau") is the sideways dragging force spread over each square metre of a layer's face. It is a pressure that pushes sideways instead of straight-on.

The picture: In Figure 1, $\tau$ is the red arrow acting along the face between two cards — it points along the sliding direction, resisting it, on both the layer above and the layer below.

Why the topic needs it: $\tau$ is the output of Newton's law — the thing viscosity produces in response to a gradient.

6 — Dynamic viscosity $\mu$ (the muscle)

Plain words: $\mu$ (Greek "mu") is the number that says how much stress you get for a given gradient. It is the fluid's personal "stickiness rating."

The picture: On the $\tau$-vs-$\dot{\gamma }$ graph (Figure 3, blue line), $\mu$ is literally the slope. Steep line = high $\mu$ (honey). Flat line = low $\mu$ (water).

Why the topic needs it: $\mu$ is the star of the whole chapter — it is what "viscosity" usually means in a single word.

7 — Density $\rho$ (how heavy the deck is)

Plain words: $\rho$ (Greek "rho") is mass packed into each cubic metre — how heavy a chunk of the fluid is.

The picture: Two identical boxes: one filled with honey (heavy, high $\rho$), one with air (light, tiny $\rho$). Same size, wildly different mass.

Why the topic needs it: Newton's second law says acceleration = force ÷ mass. To predict how fast viscosity actually moves the fluid, you must divide the viscous force by the fluid's inertia — and inertia is set by $\rho$. That division gives the next symbol.

8 — Kinematic viscosity $\nu$ (the nimbleness)

Plain words: $\nu$ (Greek "nu" — looks like a "v", not the letter v) is dynamic viscosity divided by density. It answers a different question: "how fast does motion spread through the fluid?"

9 — Apparent viscosity ${\mu }_{app}$ and the power-law $n$, $K$

Plain words: For fluids whose stickiness changes as you shear them, there is no single $\mu$. Instead we measure the ratio at each shear rate and call it the apparent viscosity: ${\mu }_{app}=\frac{\tau }{\dot{\gamma }}$

The power-law model describes the bending curve: $\tau =K{\dot{\gamma }}^n$

• $K$ = the "consistency" — how sticky overall (like a stand-in for $\mu$).
• $n$ = the bending exponent. $n=1$ → straight line → Newtonian. $n<1$ → curve bends down → shear-thinning. $n>1$ → curve bends up → shear-thickening.

The picture: In Figure 3, the blue line ($n=1$) is straight. The green curve ($n<1$) starts steep then flattens — it gets easier to shear. The red curve ($n>1$) starts gentle then steepens — it fights back harder the faster you push.

Why the topic needs it: These three curves are exactly the "Newtonian vs non-Newtonian" classification at the heart of the parent note.

The prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.

Ready? Then head back to the parent topic — every symbol there is now yours.