Viscosity measurement

Instruments based on several principles are used to measure viscosity, i.e. measuring quantities whose value in a given physical system is related to the viscosity of the measured liquid.

Rotary viscometers

The principle common to all types of rotary viscometers is the measurement of the moment of force that must be overcome by a rotating body immersed in a liquid. Under ideal conditions for the magnitude of this moment:

$M=k\omega \eta$

, where M is the moment of force, ω is the angular frequency of the rotating body, η is the dynamic viscosity and k is the constant of the device including mainly its geometry. Of course, the rotation of the body must be so slow that there is no turbulent flow. The usual geometry is a cylindrical vessel in which a rotating cylinder is immersed. A great advantage of rotary viscometers is that they are in principle able to measure the viscosity of non-Newtonian liquids as well.

Capillary viscometers

Capillary or outflow viscometers are based on measuring the volume flow of the measured liquid through a tube of defined dimensions. The starting principle is the Hagen-Poiseuille equation for a vertically placed capillary of circular cross-section with radius r and length l through which a liquid of volume V flows in time τ:

$\eta ={\frac {\pi \,r^{4}\,\Delta p\,\tau }{8\,V\,l}}$

Furthermore, for the pressure difference Δp, the formula known from primary school applies in slow flow:

$\Delta p=h\rho g$

The measurement is usually applied by comparing the seepage time of a reference (known) liquid and the liquid being measured. For the ratio of their dynamic viscosities, after substituting into the above equations:

${\frac {\eta }{\eta _{ref}}}={\frac {\tau \rho }{\tau _{ref}\rho _{ref}}}$

With a simple modification we get:

${\frac {\frac {\eta }{\rho }}{\;\;{\frac {\eta _{ref}}{\rho _{ref}}}\;\;}}={\frac {\tau }{\tau _{ref}}}$

The fraction on the right is actually the ratio of kinematic viscosities, i.e.:

${\frac {\nu }{\nu _{ref}}}={\frac {\tau }{\tau _{ref}}}$

Therefore, with most capillary viscometers, we measure kinematic viscosity, because the driving force of the flow through the capillary is the force of gravity. This can be corrected to some extent by applying an external force that will be significantly greater than the gravitational force.

There are a number of specific technical implementations, e.g.:

• Ford's cup is the simplest type used for approximate measurement of usually technical oils. Actually, it is just a standardized model of the original "can with a hole in the bottom". This viscometer measures kinematic viscosity.
• The Ostwald viscometer is actually a U-shaped glass tube in which the volume is precisely marked with lines. This viscometer measures kinematic viscosity.
• The Ubbelohde viscometer is a somewhat more complicated design than the Ostwald viscometer. This viscometer measures kinematic viscosity.
• A Mariotte bottle is actually a closed container with a horizontal capillary at the bottom. The dominant driving force is the pressure applied above the liquid surface. So the Mariotte bottle measures dynamic viscosity.

Capillary viscometers cannot be used to measure non-Newtonian fluids because a non-Newtonian fluid flowing through a capillary generally does not have a parabolic velocity profile. This means that the Hagen-Poiseuille relation will not apply in the given form.

Solid viscometers

Solid-body viscometers are based on measuring the rate of fall or, conversely, ascent to the surface of the test body. The drag force acting on a body surrounded by a liquid is described by Stokes' law, which for a commonly used spherical body of radius r moving at speed v has the form:

$F=6\pi \eta rv$

In general, the equation of motion of the test body is drawn up for the device in question, in which, in addition to the resistance force, gravity and buoyancy also figure. For a test ball of radius r and density ρ, and after adding the assumption that the ball already falls at a constant speed, the resulting relationship will take the form:

$\eta ={\frac {2}{9}}\,{\frac {r^{2}g(\rho -\rho _{liquid})}{v}}$

The measurement of the time t during which the body covers the given distance is usually used for the actual measurement. After considering this and after all the fixed parameters into a single constant k can be written:

$\eta =K(\rho -\rho _{liquid})t$

An example of a technical implementation is the Höppler viscometer.

Solid-body viscometers can only be used to measure Newtonian fluids. Another limitation is that the liquid needs to be transparent.

Other types of viscometers

• Vibration viscometers are based on the study of the propagation of low-frequency waves with a variable high amplitude in a liquid.
• Ultrasonic viscometers are roughly similar to vibration viscometers, they differ primarily in that they are clearly acoustic measurements at high frequencies and low amplitudes.
• Float viscometers are based on the study of float entrainment in a flowing liquid.

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Categories: Biophysics