ME3311 · Hydraulic & Pneumatic
Theme 2 · The fluid

Dynamic viscosity & shear

"Thickness" is the oil's most important property. Here is what viscosity really is — the oil's resistance to being sheared — and the simple law that measures it.

Source: Rabie, Fluid Power Engineering, Ch. 2.

Before you start

What you need first

  • The oil's jobs — lubricating and sealing depend on "thickness" (Topic 6).

What you'll be able to do

  • Say what viscosity physically is.
  • Use Newton's law \(\tau = \mu\,\dfrac{du}{dy}\).
  • Read dynamic viscosity \(\mu\) and its units.

Start here · the idea

What is viscosity?

Viscosity is the oil's resistance to flowing — its internal friction. Honey has high viscosity (it resists, pours slowly); water has low viscosity (it flows freely).

In a hydraulic machine viscosity is a balancing act: too thin and the oil leaks past clearances and the film breaks (poor sealing & lubrication); too thick and it drags, wastes power and is hard to pump. The "right" thickness is the whole game.

Shearing the oil between two surfaces

Put oil between a fixed surface and a moving one. The oil sticks to both (the "no-slip" rule), so its speed grows smoothly from zero at the fixed wall to U at the moving wall.

That change of speed across the gap is the velocity gradient, \(\dfrac{du}{dy}\). Dragging one layer of oil over the next is what viscosity resists.

A thinner gap (same speed) makes a steeper gradient — the oil is sheared harder.
moving plate U fixed plate h du/dy
Speed grows from 0 (fixed wall) to \(U\) (moving wall) across the gap \(h\); the slope is the gradient \(du/dy\).

Newton's law of viscosity

To keep the top plate moving you must keep pushing it sideways, and the oil drags back along the surface with an equal force. Spread that sideways force over the contact area and you get the shear stress \(\tau\) — the drag force per unit area, measured in pascals (Pa). It is just like pressure, except pressure pushes into a surface while shear stress acts along it.

The bigger the area you drag, or the harder the oil resists, the more total force — so we work per unit area: \(\tau = F/A\), in Pa.

Newton's result is that this shear stress is proportional to the velocity gradient, and the constant of proportionality is the viscosity:

$$\tau = \mu\,\dfrac{du}{dy}$$
where:
SymbolMeaningSI unit
\(\tau\)shear stress (drag per unit area)Pa
\(\mu\)dynamic (absolute) viscosityPa·s
\(\dfrac{du}{dy}\)velocity gradient across the gap1/s
Read it as: thicker oil (bigger \(\mu\)) → more shear stress for the same gradient. For a straight-line profile, \(\dfrac{du}{dy}=\dfrac{U}{h}\).

A note on the units of \(\mu\)

The SI unit is the pascal-second (Pa·s). The old workshop units are the poise and centipoise:

$$1~\text{Pa·s} = 10~\text{poise} = 1000~\text{cP}$$
A feel for size: water is about 1 cP (0.001 Pa·s); a typical hydraulic oil is a few tens of cP at working temperature. Always convert cP → Pa·s before using a formula (÷1000).

✏️ Try it yourself — no numbers needed

A plate slides over an oil film at a fixed speed. You then make the film half as thick (same plate, same speed, same oil). What happens to the drag (shear stress) — and why?

It doubles. With a straight profile, \(\dfrac{du}{dy}=\dfrac{U}{h}\). Halving \(h\) doubles the gradient, and \(\tau=\mu\dfrac{du}{dy}\), so the shear stress doubles. Why: the same speed change is now squeezed into half the distance — a steeper gradient means the oil is sheared harder, so it pushes back harder.

Common mistakes to avoid

MistakeFix
Using speed \(U\) where the law wants the gradient Newton's law uses \(du/dy=U/h\) — divide by the gap.
Leaving \(\mu\) in centipoise Convert to Pa·s first: \(1~\text{cP}=10^{-3}~\text{Pa·s}\).
Confusing \(\mu\) with \(\nu\) \(\mu\) is dynamic viscosity; the kinematic one \(\nu=\mu/\rho\) comes a little later (after density).

Recap — the whole topic on one screen

$$\tau = \mu\,\dfrac{du}{dy} \qquad \dfrac{du}{dy}=\dfrac{U}{h}\qquad 1~\text{Pa·s}=1000~\text{cP}$$
IdeaWhat you own now
ViscosityThe oil's resistance to being sheared (internal friction)
Newton's law\(\tau=\mu\,du/dy\) — stress ∝ velocity gradient
Dynamic viscosity \(\mu\)The proportionality constant, in Pa·s

Next topic

Density

Before we can combine \(\mu\) with density into the kinematic viscosity, we need density itself: what \(\rho\) is, its typical value for oil, and why it matters for inertia and head.

→ Density