Editing Viscometer (section)

===Falling-sphere viscometers===
[[File:Terminal velocity.svg|thumb|100px|right|Creeping flow past a sphere]]
[[Stokes' law]] is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches [[terminal velocity]], which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the [[density]] of the liquid, Stokes' law can be used to calculate the [[viscosity]] of the fluid. A series of steel ball bearings of different diameter are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses [[glycerol]] as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. It includes many different oils and [[polymer]] liquids {{clarify span|such as solutions|date=January 2019}}.

In 1851, [[George Gabriel Stokes]] derived an expression for the frictional force (also called [[drag force]]) exerted on spherical objects with very small [[Reynolds number]]s (e.g., very small particles) in a continuous [[viscosity|viscous]] [[fluid]] by changing the small fluid-mass limit of the generally unsolvable [[Navier–Stokes equations]]:

: <math>F = 6 \pi r \eta v,</math>

where
: ''<math>F</math>'' is the frictional force,
: ''<math>r</math>'' is the radius of the spherical object,
: ''<math>\eta</math>'' is the fluid viscosity,
: ''<math>v</math>'' is the particle velocity.

If the particles are falling in the viscous fluid by their own weight, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the [[buoyant force]] exactly balance the [[gravitational force]]. The resulting settling velocity (or [[terminal velocity]]) is given by

: <math>V_\text{s} = \frac{2}{9} \frac{r^2 g (\rho_p - \rho_f)}{\mu},</math>

where:
: {{math|''V''<sub>s</sub>}} is the particle settling velocity (m/s), vertically downwards if {{math|''ρ''<sub>p</sub> &gt; ''ρ''<sub>f</sub>}}, upwards if {{math|''ρ''<sub>p</sub> &lt; ''ρ''<sub>f</sub>}},
: {{mvar|r}} is the [[Stokes radius]] of the particle (m),
: {{mvar|g}} is the [[gravitational acceleration]] (m/s<sup>2</sup>),
: {{math|''ρ''<sub>p</sub>}} is the [[density]] of the particles (kg/m<sup>3</sup>),
: {{math|''ρ''<sub>f</sub>}} is the [[density]] of the fluid (kg/m<sup>3</sup>),
: {{mvar|μ}} is the (dynamic) fluid [[viscosity]] (Pa·s).

Note that [[Stokes flow]] is assumed, so the [[Reynolds number]] must be small.

A limiting factor on the validity of this result is the [[surface roughness|roughness]] of the sphere being used.

A modification of the straight falling-sphere viscometer is a rolling-ball viscometer, which times a ball rolling down a slope whilst immersed in the test fluid. This can be further improved by using a patented V plate, which increases the number of rotations to distance traveled, allowing smaller, more portable devices. The controlled rolling motion of the ball avoids turbulences in the fluid, which would otherwise occur with a falling ball.<ref>{{Cite web|last=tec-science|date=2020-04-04|title=Experimental determination of viscosity (viscometer)|url=https://www.tec-science.com/mechanics/gases-and-liquids/experimental-determination-of-viscosity/|access-date=2020-06-25|website=tec-science|language=en-US}}</ref> This type of device is also suitable for ship board use.{{why?|date=January 2019}}