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1Rheology: An Historical Perspective; R.I Tanner and K.Walkers, Elsevier 19982

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6Source: Wikipedia, “Viscosity.gif” https://en.wikipedia.org/wiki/File:Viscosity.gif

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• This method requires large sample volumes and is only accurate for Newtonian fluids

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SUS (at 37.8oC) cSt

32.6 2

97.8 20

463 100

• We jumped straight from the era of Greek philosophers to the 20’th century, using the term ‘viscosity’ that was only introduced in 1929!1

• But how did the concept of viscosity develop in between?

• For a very long time…. it didn’t!

• That is to say…the underpinnings of viscosity have been developed over the last two millennia-topics such as buoyancy or density, uniformly accelerated motion and drag. Because viscosity measurement relies on many other concepts, it is difficult to concisely explain it’s history. As a result, we’ll only mention the major developments.

91Rheology: An Historical Perspective; R.I Tanner and K.Walkers, Elsevier 1998

• One of the largest early developments was Newton’s work on conservation of momentum (1687) and the concept of pressure, which was later expanded upon by Pascal. Newton calls the viscosity “the lack of slipperiness of the parts of the liquid”1 . Newton postulated on the idea of a fluid exhibiting a friction force, and how it relates to shear stress.

• The deformation and flow of materials was expanded upon Daniel Bernoulli (1743); with the Bernoulli principle.

• At around this time (1757), Leonhard Euler developed his equations governing inviscid flows (known as Euler’s equations)- leading to a complete, modern statement of mass-conservation. Interestingly enough, the earliest references to these relations date back to 2’nd century AD.

101Sir Isaac Newton, Principia, 1687

"GodfreyKneller-IsaacNewton-1689" by Sir Godfrey Kneller -http://www.newton.cam.ac.uk/art/portrait.html

• In 1822, Claude-Loise Navier and George Stokes publish the Navier-Stokes equations, which expand upon Euler’s equations and are a key underpinning of studies involving fluid behavior.

• In 1840, Gotthilf Heinrich Ludwig Hagen and Jean Léonard Marie Poiseuille published the Hagen-Poiseuilleequation, dictating the pressure drop of a incompressible, Newtonian fluid in laminar pipe flow.

• A short while later the Darcy-Weisbach equation was put in its final form, relating the pressure loss due to friction to the average velocity of a fluid.

• Many more developments occurred since then, but we will not focus on them at this time.

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Licensed under Public Domain via Commons -https://commons.wikimedia.org/wiki/File:Poiseuille.jpg#/media/File:Poiseuille.jpg

These viscometers are known by many names, such as U-tube viscometers, Ostwald viscometers (after Wilhelm Ostwald, 1853-1932, one of the major founders of physical chemistry), and Ubbelohde viscometers (after Leo Ubbelohde, 1877-1964, another notable German chemist)

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"Ostwaldscher Zähigkeitsmesser" by Wilhelm Ostwaldhttps://commons.wikimedia.org/wiki/File:Ostwaldscher_Z%C3%A4higkeitsmesser.jpg#/media/File:Ostwaldscher_Z%C3%A4higkeitsmesser.jpg

𝜈 = 𝑡𝑖𝑚𝑒 × 𝐶

Operation:

Fluid is first drawn into the upper bulb via suction

The fluid is then allowed to flow back into the lower bulb, and the time it takes to move from one marking to the next (labelled ‘c’ and ‘d’) is observed

This time is then used to calculate kinematic viscosity, either via a manufacturer-provided equation or by comparison with a standard.

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"Ostwaldscher Zähigkeitsmesser" by Wilhelm Ostwaldhttps://commons.wikimedia.org/wiki/File:Ostwaldscher_Z%C3%A4higkeitsmesser.jpg#/media/File:Ostwaldscher_Z%C3%A4higkeitsmesser.jpg

• Patented in 1932 by Fritz Hoppler

• First viscometer to measure dynamic viscosity!

• Based on Stoke’s Law• A sphere of known weight and size is

dropped into fluid, and is allowed to reach terminal velocity

• The terminal velocity is measured, and the frictional force (Stoke’s Drag) is calculated. From there, dynamic viscosity can be determined.

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𝜂 =2

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𝑟2𝑔(𝜌𝑠 − 𝜌𝑓)

𝑉𝑠

• All of the preceding viscometers have several similar disadvantages

• They all rely on gravity-driven flow• This means that highly viscous samples will have long

measurement times

• In addition, this means that shear rates cannot be controlled or changed• For Newtonian fluids, this does not matter, but it makes

accurate characterization and comparison of Non-Newtonian fluids impossible.

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• Also similar to cup-and-bob or Coutte viscometers

• Mentions of similar types of rotational viscometers date back to at least 1931

• These viscometers feature a cylindrical element (the spindle, or bob) suspended in a cup of fluid.

• The spindle is rotated inside the fluid by a motor, and the torque required to rotate the spindle at a certain speed is measured

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𝜂 = 𝐾𝑇𝑜𝑟𝑞𝑢𝑒

𝜔

• This torque value can then be converted to dynamic viscosity values using manufacturer-provided calculations

• Compared to cone and plate viscometers, spindle-type viscometers typically require greater sample volumes, but have improved resolution for low viscosity fluids

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𝜂 = 𝐾𝑇𝑜𝑟𝑞𝑢𝑒

𝜔

Fluid

• Similar in mechanism to spindle-type viscometers.

• Instead of a cylindrical element suspended in the fluid, these feature a conical element on the surface of the fluid.

• This design allows for accurate characterization of shear rates.

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Cone

Plate

𝜂 =𝜏 (𝑡𝑜𝑟𝑞𝑢𝑒, 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦)

𝛾 (𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦)

Fluid

• This makes a cone and plate design superior to a spindle-type design for Non-Newtonian fluids

• The design allows for:• smaller sample volumes than

spindle type• increases complexity of setup

(‘setting the gap’, cone must be carefully aligned with surface of the fluid)

• sensitivity to particulates, and reduced resolution at low viscosity (surface area is lower, so friction will be lower for same viscosity)

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Cone

Plate

𝜂 =𝜏 (𝑡𝑜𝑟𝑞𝑢𝑒, 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦)

𝛾 (𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦)

• Combination of old principles and new technology

• Old principle: Hagen-Poiseuille flow for incompressible fluids

• New technology: fabrication of microfluidics and MEMS technology

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Microfluidic Viscometers

• As fluid passes through the flow channel at a fixed flow rate, it experiences a drop in pressure that is measured with multiple sensors

• This pressure drop is directly related to shear stress, which in turn can be used to calculate fluid viscosity

• This technique allows for accurate determination of viscosity and control of shear rate

• Allows for characterization of Non-Newtonian samples, small sample volume measurements, and a broad shear rate range (no turbulence concerns)

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𝜏 ~ Δ𝑃

𝛾 ~ 𝑄𝜂 =

𝜏

𝛾

Other Modern Viscometry methods

• In the last several decades, the assortment of instruments used for viscometry has exploded into a vast arrangement. We’ve mentioned the most common viscometry methods above, but here is a handful of others:

• Oscillating Piston viscometer• Analyzes travel time of piston oscillating in fluid to calculate shear stress

• Stabinger viscometer• Modified version of a Couette viscometer (inverted version of standard spindle

viscometer, where the sample container rotates and the spindle is stationary

• Bubble viscometer• Measures rise time of bubbles in low viscosity fluid to calculate the viscosity

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