Viscosity is a fluid’s resistance to flow, caused by internal friction as layers of the fluid slide past one another. It equals the shear stress divided by the velocity gradient, and its SI unit is the pascal second (Pa·s). Honey has high viscosity; water and air have low viscosity.
Tip a jar of honey and a glass of water over at the same moment. The water is gone in a second; the honey is still deciding. Same gravity, same tilt — utterly different behaviour.
That difference has a name, a formula and a unit, and it governs far more than breakfast. It sets how thick your engine oil needs to be in January, why blood struggles through narrowed arteries, and why a spoonful of custard turns solid when you stir it fast.
What Is Viscosity?
Viscosity is a measure of how strongly a fluid resists being sheared — that is, how hard it is to make one layer of the fluid slide over the layer beneath it. High viscosity means the fluid fights the motion; low viscosity means it gives way easily.
Think of a deck of cards lying flat. Push the top card sideways and the cards below drag along a little, held back by friction between the surfaces. A fluid does the same thing — except the “cards” are molecular layers, and the resistance between them is viscosity.
Two details make this work. First, fluid touching a solid surface sticks to it and moves with it — the no-slip condition. Second, because the wall holds one edge still and something else drags the other edge along, the speed must change gradually across the gap.
That change in speed with distance is the velocity gradient. Viscosity is simply the constant that links it to the force required.
The Viscosity Formula
The viscosity formula states that shear stress equals dynamic viscosity multiplied by the velocity gradient. This is Newton’s law of viscosity, and it is the definition from which every viscosity unit follows.
Each symbol carries a specific SI unit:
- τ (tau) — shear stress: the tangential force per unit area on a fluid layer, in pascals (Pa), equal to N/m2.
- μ (mu) — dynamic viscosity, the fluid property itself, in pascal seconds (Pa·s).
- du/dy — velocity gradient or shear rate: how quickly flow speed changes across the gap, in reciprocal seconds (s−1).
- u — local flow speed, in metres per second (m/s).
- y — distance measured perpendicular to the flow, in metres (m).
Rearranged, μ = τ ÷ (du/dy). So dynamic viscosity is the shear stress needed per unit of shear rate — and that is exactly what a Pa·s means: one pascal of stress producing a shear rate of one per second.
Fluid sheared between two plates. The no-slip condition pins the fluid to each plate, producing a steady velocity gradient across the gap.
Units of Viscosity: Pa·s, Poise and Stokes
The SI unit of dynamic viscosity is the pascal second (Pa·s), also written N·s/m2 or kg/(m·s). Older CGS units survive stubbornly in industry, which is why oil datasheets still speak in centipoise and centistokes.
One poise (P) equals 0.1 Pa·s. One centipoise (cP) is a hundredth of that — exactly 10−3 Pa·s, or one millipascal second. Water sits almost precisely at 1 cP at 20 °C, which is no accident: that convenient coincidence is why the unit stuck.
| Quantity | SI unit | CGS unit | Conversion | Water at 20 °C |
|---|---|---|---|---|
| Dynamic viscosity (μ) | pascal second, Pa·s | poise, P | 1 P = 0.1 Pa·s 1 cP = 10−3 Pa·s |
1.002 × 10−3 Pa·s (1.002 cP) |
| Kinematic viscosity (ν) | square metre per second, m2/s | stokes, St | 1 St = 10−4 m2/s 1 cSt = 10−6 m2/s = 1 mm2/s |
1.004 × 10−6 m2/s (1.004 cSt) |
| Shear stress (τ) | pascal, Pa | dyne/cm2 | 1 dyne/cm2 = 0.1 Pa | depends on shear rate |
| Shear rate (du/dy) | per second, s−1 | per second, s−1 | identical | set by the flow |
The value of 1.002 cP for water at 20 °C is not a textbook rounding. It was fixed by a decade-long capillary-flow determination at the US National Bureau of Standards, now NIST, and adopted in 1952 as the primary reference standard against which other viscometers are still calibrated.
Dynamic vs Kinematic Viscosity
Dynamic viscosity measures resistance to shear on its own; kinematic viscosity measures that resistance relative to the fluid’s density. Divide one by the other and you get the second.
- ν (nu) — kinematic viscosity, in m2/s.
- μ (mu) — dynamic viscosity, in Pa·s.
- ρ (rho) — density of the fluid, in kg/m3.
Why bother with two? Because they answer different questions. Dynamic viscosity tells you the force you must supply to shear the fluid; kinematic viscosity tells you how readily the fluid’s own momentum spreads sideways compared with how much inertia it carries.
Lubricant and fuel datasheets almost always quote kinematic viscosity in centistokes, because that is what a gravity-fed capillary viscometer measures directly. Converting to Pa·s means multiplying by density — a step it is easy to forget under exam pressure, and one you can check against our Viscosity Converter when a datasheet gives you cSt and the question wants Pa·s.
How Viscosity Works Inside a Fluid
Viscosity arises from momentum being carried sideways between fluid layers, but the mechanism differs completely in liquids and gases. That difference produces one of the most counter-intuitive results in fluid mechanics.
Liquids: molecules that cling
In a liquid, molecules sit close together and attract one another. To shear the liquid you must repeatedly break and reform those bonds, and that costs force.
Heat the liquid and the molecules jiggle harder, escaping each other’s pull more easily. So liquid viscosity falls as temperature rises — steeply. Water is about 3.6 times less viscous at 100 °C than at 20 °C, which is why hot oil pours like water and cold treacle barely moves.
Gases: molecules that trade places
Gas molecules are far apart and barely attract each other, so bonding is irrelevant. Instead, fast molecules from a quick-moving layer wander into a slower layer and speed it up, while slow molecules drift the other way and drag the fast layer back.
This exchange of momentum is the gas’s viscosity. Heat the gas and the molecules cross between layers faster, so gas viscosity rises as temperature rises — the exact opposite of a liquid.
Temperature dependence of viscosity. Liquids thin dramatically on heating; gases thicken slightly. Curves are schematic, not to scale.
The no-slip condition that anchors all of this has a visible consequence: a thin layer where flow speed climbs from zero at a surface to the free-stream value. Engineers call it the boundary layer, and NASA’s aeronautics guide explains how viscosity creates it around a wing.
Real-World Examples of Viscosity
Viscosity spans an enormous range — roughly a million-fold between air and honey at room temperature. The table below lists measured values at 20 °C, with kinematic viscosity worked out from each fluid’s density.
| Fluid (20 °C) | Dynamic μ (Pa·s) | In cP | Density ρ (kg/m3) | Kinematic ν (m2/s) |
|---|---|---|---|---|
| Air | 1.81 × 10−5 | 0.018 | 1.20 | 1.50 × 10−5 |
| Water | 1.00 × 10−3 | 1.00 | 998 | 1.00 × 10−6 |
| Mercury | 1.55 × 10−3 | 1.55 | 13 534 | 1.15 × 10−7 |
| Olive oil | 8.4 × 10−2 | 84 | 915 | 9.2 × 10−5 |
| Glycerine | 1.41 | 1 410 | 1 261 | 1.12 × 10−3 |
| Honey (typical) | about 10 | about 10 000 | 1 420 | 7.0 × 10−3 |
Honey values vary widely with water content and floral source, so treat that row as a typical figure rather than a constant. Everything else in the table is a standard measured value at 20 °C.
Six places this actually matters:
- Engine oil grades. A “5W-30” label is a viscosity specification — the oil must stay thin enough to pump when cold and thick enough to keep a film between bearing surfaces when hot.
- Blood flow. Blood is roughly three to four times more viscous than water. Raised viscosity forces the heart to generate more pressure for the same flow.
- Honey off a spoon. High viscosity means the shear stress from gravity produces only a tiny shear rate, so the strand thins slowly instead of breaking.
- Paint and printing inks. Formulated to flow under the brush or roller, then stiffen fast enough not to run down the wall.
- Volcanic lava. Silica-rich lava is thousands of times more viscous than basaltic lava, which is why some volcanoes ooze and others explode.
- Aircraft skin friction. Air’s tiny viscosity still produces a large drag force, because aircraft skin area is huge and shear rates near the surface are enormous.
Newtonian vs Non-Newtonian Fluids
A Newtonian fluid has a viscosity that stays constant no matter how fast you shear it. Water, air, glycerine and most thin oils behave this way, so a single value of μ describes them completely at a given temperature.
Non-Newtonian fluids break that rule — their apparent viscosity changes with the shear rate itself. Two families cover most everyday cases.
- Shear-thinning fluids get runnier the harder you work them. Ketchup, blood, paint and shampoo all thin under stress; this is why shaking the bottle works.
- Shear-thickening fluids get stiffer. Cornflour-and-water paste is the classic demonstration — stir it slowly and it flows, punch it and it resists like a solid.
A practical consequence: quoting “the viscosity of ketchup” without stating the shear rate is meaningless. For non-Newtonian fluids the number only exists alongside the conditions it was measured under.
Common Misconceptions About Viscosity
1. “Viscosity is the same as density”
These are entirely independent properties, and the table above proves it. Mercury is 13.6 times denser than water yet only about 1.5 times as viscous — pour it and it splashes like water.
Honey does the reverse: barely 1.4 times the density of water, but around 10 000 times the viscosity. Density is how much mass is packed in; viscosity is how hard the fluid resists shearing.
2. “Air is basically inviscid”
Air’s viscosity is small in absolute terms but never zero, and the consequences are not small. Skin-friction drag on an airliner comes entirely from air viscosity.
Stranger still, air’s kinematic viscosity is about 15 times larger than water’s, because air’s density is so low. By that measure, air is the “thicker” fluid.
3. “Heating always lowers viscosity”
True for liquids, false for gases. Warm a gas and its viscosity climbs, because momentum transfer between layers speeds up. Sutherland’s law is built on exactly this behaviour.
4. “Thicker oil is always better lubrication”
Too viscous and the oil will not reach the bearing surfaces quickly on a cold start, and it wastes power as heat once running. Problem 7 below puts a number on that loss: 324 watts dissipated by a single sliding plate.
How Viscosity Relates to Drag, Flow and the Reynolds Number
Viscosity determines whether a flow is smooth or chaotic, and how strongly a fluid resists an object moving through it. The bridge between these ideas is the Reynolds number, a dimensionless ratio of inertial forces to viscous forces.
- Re — Reynolds number, dimensionless.
- v — flow speed, in m/s; L — characteristic length such as pipe diameter, in m.
- ρ, μ, ν — density (kg/m3), dynamic viscosity (Pa·s) and kinematic viscosity (m2/s).
In pipes, flow below roughly Re = 2000 is laminar and above about 4000 is turbulent. High viscosity keeps Re low and the flow orderly; that is why thick oil moves in smooth layers while water of the same speed tumbles.
Viscosity also feeds directly into drag force. At very low Reynolds numbers, Stokes’ law gives drag on a sphere as F = 6πμrv, and setting that against gravity yields the terminal velocity of a falling ball bearing — the basis of the falling-sphere viscometer.
For pipe flow, viscosity sets how much pressure you lose along the length. Poiseuille’s law, Q = πΔP r4 / (8μL), shows flow rate falling inversely with viscosity and rising with the fourth power of radius.
Compare that with Bernoulli’s principle, which assumes zero viscosity. Bernoulli predicts no pressure loss along a level pipe of constant width — real pipes always lose pressure, and viscosity is the reason.
Worked Problems
Show Solution
Solution:
Step 1: With no-slip at both surfaces the profile is linear, so du/dy = U / h.
Step 2: du/dy = 2.0 m/s ÷ (4.0 × 10−3 m) = 500 s−1
Step 3: τ = μ(du/dy) = 0.50 Pa·s × 500 s−1 = 250 Pa
Step 4: F = τA = 250 Pa × 0.20 m2 = 50 N
Answer: du/dy = 500 s−1, τ = 250 Pa, F = 50 N
Show Solution
Solution:
Step 1: By definition, 1 P = 0.1 Pa·s and 1 cP = 0.01 P = 10−3 Pa·s.
Step 2: μ = 250 cP × 10−3 Pa·s/cP = 0.250 Pa·s
Step 3: μ = 250 cP ÷ 100 cP/P = 2.50 P
Answer: 0.250 Pa·s, which is 2.50 poise
Show Solution
Solution:
Step 1: ν = μ / ρ, so rearranging gives μ = νρ.
Step 2: Convert the unit: 1 cSt = 10−6 m2/s, so ν = 46 × 10−6 m2/s.
Step 3: μ = (46 × 10−6 m2/s)(870 kg/m3) = 0.0400 kg/(m·s)
Step 4: kg/(m·s) is identical to Pa·s, so no further conversion is needed.
Answer: μ = 0.040 Pa·s (40 mPa·s, or 40 cP)
Show Solution
Solution:
Step 1: Re = ρvd / μ, taking the characteristic length as the pipe diameter.
Step 2: Re = (998 kg/m3)(0.60 m/s)(0.025 m) ÷ (1.00 × 10−3 Pa·s)
Step 3: Numerator = 14.97 kg/(m·s), so Re = 14.97 ÷ 1.00 × 10−3 = 1.497 × 104
Step 4: Re is well above 4000, so the flow is turbulent.
Answer: Re ≈ 1.5 × 104 — turbulent flow
Show Solution
Solution:
Step 1: At terminal velocity, weight = buoyancy + viscous drag, using Stokes’ law F = 6πμrv.
Step 2: (4/3)πr3ρ_s g = (4/3)πr3ρ_f g + 6πμrv
Step 3: Rearranging gives v = 2r2(ρ_s − ρ_f)g / (9μ)
Step 4: Numerator = 2(2.0 × 10−3 m)2(7800 − 1260 kg/m3)(9.81 m/s2) = 0.5133
Step 5: Denominator = 9 × 1.41 Pa·s = 12.69, so v = 0.5133 ÷ 12.69 = 0.0404 m/s
Step 6: Check Stokes’ law applies: Re = ρ_f v d / μ = 0.14, comfortably below 1.
Answer: v ≈ 0.040 m/s (about 4.0 cm/s)
Show Solution
Solution:
Step 1: Poiseuille’s law for laminar flow gives Q = πΔP r4 / (8μL).
Step 2: Numerator = π(4000 Pa)(1.0 × 10−3 m)4 = 1.257 × 10−8
Step 3: Denominator = 8(1.00 × 10−3 Pa·s)(0.50 m) = 4.00 × 10−3
Step 4: Q = 1.257 × 10−8 ÷ 4.00 × 10−3 = 3.14 × 10−6 m3/s = 3.1 mL/s
Step 5: Q depends on r4, so halving r divides the flow by 24 = 16, giving 0.20 mL/s.
Answer: Q ≈ 3.1 × 10−6 m3/s (3.1 mL/s); halving the radius drops it to ≈ 0.20 mL/s
Show Solution
Solution:
Step 1: du/dy = U / h = 1.5 m/s ÷ (0.50 × 10−3 m) = 3000 s−1
Step 2: τ = μ(du/dy) = 0.80 Pa·s × 3000 s−1 = 2400 Pa
Step 3: A = 0.30 m × 0.30 m = 0.090 m2, so F = τA = 2400 Pa × 0.090 m2 = 216 N
Step 4: P = Fv = 216 N × 1.5 m/s = 324 W
Answer: F = 216 N, and 324 W is dissipated as heat in the oil film
Show Solution
Solution:
Step 1: ν = μ / ρ for both fluids.
Step 2: ν_water = 1.00 × 10−3 ÷ 998 = 1.00 × 10−6 m2/s
Step 3: ν_air = 1.81 × 10−5 ÷ 1.20 = 1.51 × 10−5 m2/s
Step 4: Dynamic comparison: 1.00 × 10−3 ÷ 1.81 × 10−5 = 55, so water wins by 55 times.
Step 5: Kinematic comparison: 1.51 × 10−5 ÷ 1.00 × 10−6 = 15, so air wins by 15 times.
Answer: ν_water ≈ 1.0 × 10−6 m2/s and ν_air ≈ 1.5 × 10−5 m2/s. Water is 55 times more viscous dynamically; air is 15 times more viscous kinematically. The question is incomplete without stating which viscosity is meant.
