Viscosity

If you pour water into a glass, the water deforms from a liquid stream into a fluid surface enclosed by the containment of your glass. If you poured syrup into that same cup, the syrup would take longer to pour out and deform into a liquid containment. Why's this? Why does one liquid go from being poured to flowing much, much faster than another? The answer lies in the fluids' viscosities.

The easiest way to think of a fluid's viscosity is its internal fluid friction, expressed in Newtons per meters. However, if we go more in-depth, we realize the absolute viscosity is essentially how resistive a given fluid is to flow. If we take a fluid with a laminar flow, then its fluid layers will slide against each other, generating what we normally know as friction. While knowing this important distinction is great, we'd want to know how to apply it to hydraulics and other disciplines. We'd want to know, specifically, how much force is required to push the fluid such that the layer keeps a constant speed even with viscous forces against it. This is the general definition of the absolute viscosity of a fluid but that's not the only type of viscosity we might want to use.

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There are two types of viscosities that we mainly use: dynamic(absolute) and kinetic viscosity. They're both related simply through density.

Nu(the greek letter on the left) stands for the kinematic viscosity while mu(the greek letter on the numerator of the right side) stands for dynamic viscosity and rho(the greek letter on the denominator of the right side) stands for the fluid's density. Kinematic viscosity has units of stokes, where 1 stoke = 0.1 m2/s. Dynamic viscosity, as given by the equation relating the viscosities, has units of kg/meters second, also called Poises. 1 Poise = 1 kg/meters seconds.

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If this distinction is so easy to configure mathematically, why even make the distinction at all?

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First off, if you're in an academic context and you're given one of these and need to use the other, you can always make the conversion to the other but let's make a case for each given viscosity with an example.

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The big generalization we can make between kinematic and dynamic viscosities is their namesake topics in physics. Kinematics talks about how motion works without its underlying causes and forces while dynamics does talk about forces, energy, and the underlying causes behind motion. This tells us exactly what we need to know: dynamic viscosity deals with the force required for a fluid to move through viscous conditions and kinematic viscosity gives information on how a fluid will flow in viscous conditions, specifically with the area it covers. If you look at the units for kinematic viscosity, it can tell us how much area a viscous fluid covers in a given time, hence referring to its kinematic properties. Kinematic viscosity also tells us about what changes in a fluid's momentum and flow as its temperature changes.

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For example, pretend we have two cups, one with water and one with vegetable oil. If we wanted to pour the vegetable oil into an empty cup, it would fill the empty cup slower than the water would, signifying that vegetable oil resists deformation much more than water, which you'd expect. Quantifying why and also explaining why this happens would require kinematic viscosity since we're talking about the differences in flow rates of the fluids. Now, let's say we try and stir our cup with water and our cup with vegetable oil. Stirring the water at rest is much easier to do than stirring the oil at rest, signifying the force which the oil needs to overcome to resist deformation is much greater. As hinted from the discussion of forces here, it's apparent that we are explaining the differences caused by the dynamic viscosities of both the water and vegetable oil.

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Bottom line: kinematic viscosity tells us how much area or how fast a fluid can move given viscous conditions while dynamic viscosity tells us how much force we need to apply to a fluid to keep it moving in viscous conditions.

Stokes' Law is just a special case of viscosity when two conditions are met. First, the fluid flow itself is extremely laminar, as it is characterized by a very low Reynolds Number. Second, the object moving through the given viscous fluid is a perfect sphere. Many times, this law is applied to not exactly perfect spheres but it's best to ensure that the object in question is as close as possible to a perfect sphere. The law mathematically states that the viscous force on a sphere moving through a viscous fluid is:

where R is the radius of the sphere, v is the flow velocity, and mu is the dynamic viscosity of the fluid.

If we want to know what will happen to the pressure across a certain length of a cylindrical pipe in which fluid is flowing viscidly, we can employ Poiseuille's Law, also known as the Hagen-Poiseuille Equation. To derive this, we will need calculus and so we can't assume that for quantities volumetric flow rate, variables like the area are constant anymore. This gives us the following equation: