Pour honey and water side by side and the difference is obvious before the liquids leave the spoon. Water races; honey crawls. Both are pulled by the same gravity and both are simple liquids, yet one resists motion far more stubbornly than the other. That resistance has a name and a number: dynamic viscosity.
Viscosity is the internal friction of a fluid — the property that resists one layer of fluid sliding past another. It sets how much pressure a pump must supply, how quickly a particle settles out of suspension, and how thick the boundary layer grows on a moving surface. This article explains what dynamic viscosity is, how it enters the physics of a settling particle, and how to compute a real settling speed from it.
Why this calculation matters
Dynamic viscosity is one of a handful of fluid properties that appear in almost every flow calculation an engineer performs. It is in the Reynolds number, which decides whether a flow is laminar or turbulent. It is in the friction factor that sizes a pump. It is in the drag law for a particle, the spreading rate of a coating, and the lubricating film inside a bearing.
Settling is one of the clearest places to see viscosity at work. Water-treatment basins rely on sediment dropping out of suspension; paint and ink must keep their pigment particles dispersed for a usable shelf life; geologists read the size of grains in a rock from how fast they once settled. In every one of these cases the settling speed is set by a competition between gravity pulling the particle down and viscous drag holding it back. Change the viscosity and you change the speed directly. Knowing how to compute that relationship turns a vague intuition — thicker fluids slow things down — into a number you can design around.
The core formula
Dynamic viscosity, written mu, is defined through the way a fluid responds to shear. If two parallel layers of fluid slide past each other, the shear stress required is proportional to the velocity gradient between them:
tau = mu * (du / dy)
Here tau is the shear stress, du/dy is the rate at which velocity changes across the gap, and the constant of proportionality mu is the dynamic viscosity, measured in pascal-seconds (Pa.s). A fluid that obeys this linear relation is called Newtonian; water, air, and most oils are close to Newtonian over ordinary conditions.
Viscosity shows its influence most cleanly in the slow settling of a small sphere. When a particle falls through a fluid slowly enough that the flow around it stays laminar, the drag force is given by Stokes' law. Balancing that drag against the net of gravity and buoyancy gives the terminal velocity:
v = (2/9) * r^2 * (rho_p - rho_f) * g / mu
In this expression r is the particle radius, rho_p is the particle density, rho_f is the fluid density, g is gravitational acceleration, and mu is the dynamic viscosity. The structure of the formula tells the whole story. Settling speed grows with the square of the radius, so larger particles drop dramatically faster. It grows with the density difference, so a particle barely denser than the fluid drifts down slowly. And it falls in direct inverse proportion to viscosity: double the viscosity and the particle settles at exactly half the speed.
A worked example
Take a small sphere of radius r = 0.001 m and density rho_p = 2500 kg/m^3 — roughly a grain of fine sand — settling through a viscous fluid of density rho_f = 1000 kg/m^3 with a dynamic viscosity of mu = 1.0 Pa.s. Use g = 9.81 m/s^2.
Step 1 — assemble the Stokes' law expression.
v = (2/9) * r^2 * (rho_p - rho_f) * g / mu
Step 2 — evaluate the leading constant and the radius term.
2/9 = 0.2222
r^2 = (0.001)^2 = 1e-6 m^2
Step 3 — evaluate the density-difference and gravity term.
(rho_p - rho_f) * g = (2500 - 1000) * 9.81 = 1500 * 9.81 = 14715
Step 4 — combine all the factors.
v = 0.2222 * 1e-6 * 14715 / 1.0
v = 0.2222 * 1e-6 * 14715
v = 0.00327 m/s
So the sphere settles at about 0.00327 m/s, or roughly 3.3 mm/s — slow enough to watch with the naked eye. The viscosity sits in the denominator, so the dependence is simple and direct: a fluid twice as viscous would slow this descent to about 1.6 mm/s, and a fluid ten times as viscous would drop it to a third of a millimetre per second. A more viscous fluid slows the descent in direct proportion.
Common mistakes
Confusing dynamic and kinematic viscosity. Dynamic viscosity mu is measured in Pa.s; kinematic viscosity nu is mu divided by density, measured in m^2/s. Stokes' law as written above uses dynamic viscosity. Substituting kinematic viscosity by mistake scrambles the answer by a factor equal to the fluid density.
Forgetting that viscosity changes sharply with temperature. The viscosity of a liquid drops steeply as it warms — engine oil can change by a large factor between a cold start and operating temperature. A settling or flow calculation done at one temperature can be badly off at another, so always pair a viscosity value with the temperature it was measured at.
Applying Stokes' law outside the creeping-flow regime. Stokes' law assumes the flow around the particle stays laminar, which holds only at a low particle Reynolds number. For larger or faster particles, inertial drag takes over and the simple formula overpredicts the settling speed. Check the Reynolds number before trusting the result.
Ignoring buoyancy. The driving term is the density difference rho_p minus rho_f, not the particle density alone. A particle only slightly denser than the fluid settles slowly, and one less dense than the fluid rises instead of falling. Dropping the buoyancy term overstates the speed.
Assuming every fluid is Newtonian. Many practical fluids — paints, gels, slurries, blood — have a viscosity that depends on shear rate. For these the single number mu is not enough, and a non-Newtonian model is needed to describe the flow correctly.
Try the interactive NovaSolver calculator
Computing one viscosity-driven result by hand is quick, but seeing how a fluid behaves across shear rates and temperatures is where understanding takes hold. The Viscosity & Rheology Simulator on NovaSolver lets you pick a fluid model — Newtonian, power-law, Bingham, or Herschel-Bulkley — set the viscosity, yield stress, temperature, and a reference velocity, and watch the Couette flow profile, apparent viscosity, Reynolds number, and flow regime update in real time, with an Arrhenius temperature correction built in.
Related calculators
- Reynolds Number calculator — viscosity is one of its inputs; use it to check whether your flow is laminar or turbulent.
- Drag & Terminal Velocity calculator — extends settling beyond the creeping-flow regime where Stokes' law no longer applies.
- Pipe Pressure Drop calculator — shows how viscosity translates into the pumping pressure a pipeline demands.
You can browse the rest in the fluid dynamics calculators hub.
Closing note
Dynamic viscosity is a small property with a long reach. It sets the internal friction of a fluid, and through that one quantity it governs settling speed, pressure drop, drag, and the character of every boundary layer. The settling example shows the dependence at its simplest: viscosity in the denominator means a fluid twice as thick halves the speed. Measure mu at the right temperature, confirm the flow regime, keep dynamic and kinematic viscosity apart, and the fluid side of an engineering problem becomes far easier to predict.
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