Viscosity

Pour honey and water side by side and the difference is obvious before either liquid even hits the plate — one flows in a slow, reluctant ribbon, the other splashes freely. That difference is viscosity, the internal friction between adjacent layers of a moving fluid. It has nothing to do with how heavy or dense a fluid is: mercury is dense yet pours almost as easily as water, while honey is far less dense yet flows sluggishly. Viscosity is about how strongly a fluid's own layers grip and drag against each other as they slide past one another. This article builds the single relation that quantifies that grip, $F = \eta A\,dv/dx$ , works through its units, and explains — physically, not just by memorising a rule — why liquids get thinner when heated while gases get thicker. Two worked problems use the formula both ways: force from viscosity, and viscosity from a measured force. The physics here sets up Stokes' law and terminal velocity, covered next.

The core insight

Viscosity is the internal friction between fluid layers, quantified by $F = \eta A\,dv/dx$ , and it falls with rising temperature in liquids but rises with rising temperature in gases because the two arise from different molecular mechanisms.

The core idea

Real fluids, unlike the "ideal" fluid of Bernoulli's theorem, resist relative sliding between their own layers. Picture laminar flow near a fixed boundary: the layer touching the boundary is at rest, and each successive layer further away moves a little faster, building up a velocity gradient $dv/dx$ perpendicular to the flow. To keep this sliding going, a tangential (shearing) force must be applied, and Newton's law of viscosity says that force is proportional to the contact area $A$ and to the velocity gradient.

F = \eta A \frac{dv}{dx}

where: $F$ is the viscous (tangential) force in newtons, $A$ is the contact area between the sliding layers in $\text{m}^2$ , $dv/dx$ is the velocity gradient in $\text{s}^{-1}$ , and $\eta$ is the coefficient of viscosity — a property of the fluid and its temperature that measures how strongly it resists shear. Rearranged, $\eta$ is read directly off a measurement of force, area and gradient:

\eta = \frac{F}{A\,(dv/dx)}

Units. With $F$ in newtons, $A$ in $\text{m}^2$ and $dv/dx$ in $\text{s}^{-1}$ , the coefficient carries an extra factor of seconds compared with ordinary stress:

[\eta] = \frac{\text{N}}{\text{m}^2\cdot\text{s}^{-1}} = \text{N}\,\cdot\,\text{s/m}^2 = \text{Pa}\,\cdot\,\text{s} = \text{kg}\,\text{m}^{-1}\text{s}^{-1}

Why layers resist sliding — two different mechanisms. In a liquid, viscosity comes mainly from intermolecular cohesive forces: a fast-moving layer tries to drag its slower neighbour along, while the slower layer holds the faster one back, through attraction between molecules straddling the boundary. Heating a liquid loosens this cohesive grip, so $\eta$ falls as temperature rises — this is why warm honey pours more easily than cold honey. In a gas, molecules are far apart and cohesion is negligible; viscosity instead comes from molecules randomly darting between layers, carrying the momentum of their home layer with them and so tending to equalise the speeds of neighbouring layers. Heating a gas speeds up this molecular traffic, so momentum is exchanged faster and $\eta$ rises with temperature — the opposite trend to a liquid.

Figure

Two adjacent fluid layers with the faster upper layer dragging the slower lower layer, arrows showing the velocity gradient and the opposing viscous force

Common traps

Confusing viscosity with density: mercury is dense but flows freely, honey is less dense than mercury but far more viscous.
Reversing the temperature trend: liquids get less viscous as they warm, gases get more viscous as they warm.
Treating $dv/dx$ as just "velocity over distance" rather than the velocity gradient measured perpendicular to the flow, between adjacent layers.
Writing the unit of $\eta$ as $\text{N/m}^2$ (which is stress or pressure) instead of $\text{N}\,\cdot\,\text{s/m}^2 = \text{Pa}\,\cdot\,\text{s}$ — the extra factor of seconds is what makes it a viscosity coefficient, not a stress.

Derivation AGetting the SI unit of viscosity from the force law

The goal is to show, starting only from the defining relation $F = \eta A\,dv/dx$ and the base units of force, area and velocity gradient, that the coefficient of viscosity must carry the composite unit $\text{N}\,\cdot\,\text{s/m}^2$ , equivalently $\text{Pa}\,\cdot\,\text{s}$ or $\text{kg}\,\text{m}^{-1}\text{s}^{-1}$ — a result students are expected to derive on demand rather than simply recall.

Step 1. Isolate $\eta$ algebraically. Rearranging $F = \eta A\,(dv/dx)$ gives $\eta = F / [A\,(dv/dx)]$ , so the unit of $\eta$ is whatever combination of units on the right-hand side reduces to.

Step 2. Fix the unit of each quantity on the right. Force $F$ is in newtons, $\text{N}$ . Area $A$ is in $\text{m}^2$ . The velocity gradient $dv/dx$ is a velocity divided by a distance, so its unit is $(\text{m/s})/\text{m} = \text{s}^{-1}$ .

Step 3. Substitute these units into the ratio. $[\eta] = \dfrac{\text{N}}{\text{m}^2 \cdot \text{s}^{-1}}$ . Dividing by $\text{s}^{-1}$ is the same as multiplying by $\text{s}$ , so this becomes $\dfrac{\text{N}\,\cdot\,\text{s}}{\text{m}^2}$ , written $\text{N}\,\cdot\,\text{s/m}^2$ .

Step 4. Convert to the named unit and to base SI units. Since $1\ \text{Pa} = 1\ \text{N/m}^2$ , $\text{N}\,\cdot\,\text{s/m}^2 = \text{Pa}\,\cdot\,\text{s}$ . Expanding $\text{N} = \text{kg}\,\cdot\,\text{m/s}^2$ gives $[\eta] = \dfrac{\text{kg}\,\cdot\,\text{m/s}^2 \cdot \text{s}}{\text{m}^2} = \text{kg}\,\text{m}^{-1}\text{s}^{-1}$ , confirming all three forms of the unit are equivalent.

Worked example

A flat plate of area $A = 0.20\ \text{m}^2$ is dragged over a layer of oil, maintaining a velocity gradient $dv/dx = 25\ \text{s}^{-1}$ in the oil, whose coefficient of viscosity is $\eta = 1.5\ \text{Pa}\,\cdot\,\text{s}$ .

Step 1. Identify the relation to use. Since $F$ , $A$ and $dv/dx$ are all wanted or given, use $F = \eta A\,(dv/dx)$ directly.

Step 2. Substitute the values. $F = (1.5)(0.20)(25) = 7.5\ \text{N}$ . So the plate needs $F = 7.5\ \text{N}$ to keep sliding at that rate against the oil's internal friction.

Competitive stretch

Now suppose the same plate is dragged so that the viscous force per unit area between two liquid layers $2.0\ \text{mm}$ apart, moving at a relative velocity of $0.10\ \text{m/s}$ , is measured to be $0.30\ \text{N/m}^2$ . Find $\eta$ for this liquid.

Step 1. Find the velocity gradient. $dv/dx = 0.10 / (2.0\times10^{-3}) = 50\ \text{s}^{-1}$ .

Step 2. Solve for $\eta$ from $F/A = \eta\,(dv/dx)$ : $\eta = (F/A)/(dv/dx) = 0.30/50 = 6.0\times10^{-3}\ \text{Pa}\,\cdot\,\text{s}$ . This is $\eta = 6.0\times10^{-3}\ \text{Pa}\,\cdot\,\text{s}$ , a few times more viscous than water at room temperature ( $\approx 1\times10^{-3}\ \text{Pa}\,\cdot\,\text{s}$ ) — consistent with a moderately viscous liquid.

Ask the tutor: Want to see how this viscosity coefficient feeds into Stokes' law and terminal velocity, or try a few more force/gradient calculations?

Why does honey get runnier when you warm it but air get thicker?

Because the two liquids and gases owe their viscosity to different mechanisms. In honey (a liquid), molecules are packed close and held together by cohesive attraction; heating gives them more thermal energy to break free of that grip, so the internal friction between layers drops and it flows more easily. In air (a gas), molecules are far apart with negligible cohesion, and viscosity instead comes from molecules crossing between layers and carrying momentum with them. Heating speeds up this molecular traffic, so more momentum is exchanged per second between layers, and the gas's viscosity rises. Same word, opposite trend, because the underlying physics is different.

Is viscosity the same thing as thickness or density?

No, and this is a very common mix-up. Density is mass per unit volume; viscosity is resistance to shearing flow. Mercury is roughly 13 times denser than water yet pours almost as easily, so it is dense but not very viscous. Honey is less dense than mercury yet flows far more sluggishly, so it is viscous but not especially dense. "Thickness" in everyday speech usually means viscosity, but keep the physical quantities separate: one is $\rho = m/V$ , the other is $\eta$ from $F = \eta A\,dv/dx$ , and they can vary independently.

Why is honey more viscous than water in the first place?

Viscosity in a liquid comes from how strongly neighbouring molecules attract and interlock as layers try to slide past each other. Honey's large sugar molecules are long, sticky, and heavily hydrogen-bonded to each other, so adjacent layers grip strongly and resist sliding — a large $\eta$ . Water's small, simpler molecules have weaker mutual attraction, so its layers slide past each other with much less resistance, giving a much smaller $\eta$ . The formula $F = \eta A\,dv/dx$ does not explain why $\eta$ differs between fluids — it only measures the effect — but the molecular picture of stronger versus weaker intermolecular grip explains the size of $\eta$ itself.

Viscosity

The core idea

Common traps

Worked example

Competitive stretch

Before this

Where this leads