# Is there an intuitive explanation for the Kinematic Viscosity unit of mm^2/s?

I've learned that a high viscosity means that the fluid is more thick. Dynamic Viscosity thus represents the resistance to shear forces. Presumably, higher resistances are represented by higher numbers in the relevant ISO spec.

Thus I'm puzzled about the unit of Kinematic Viscosity (which is basically Dynamic Viscosity modulated by a measure of the density of the fluid). The unit is:

m^2/s

But how can this unit be understood intuitively? The higher the viscosity, the lesser the flow-rate and thus less m^2 fluid should flow per second in whatever measurement setup they are using. Looking at the unit for kinematic viscosity, it seems that a higher value means that the fluid has a lower viscosity (i.e. is less thich), because, well, more square meters of fluid seems to flow per second.

How can the unit of kinematic viscosity be understood intuitively, when seen in relation to the definition of viscosity?

Dynamic viscosity represents the resistance of fluid to shear forces as you said. This is what lay people think of when they think viscosity. High dynamic viscosity = more resistance to flow. E.g. honey has much higher dynamic viscosity than water, or cold motor oil has higher dynamic viscosity than warm motor oil.

Kinematic viscosity is something totally different. Don't try to think of it as resistance of fluid to flowing, because that's not what it is. Thinking of it as so many $m^2$ of fluid flowing per second is not what it is about.

To understand what it is, you have to understand that there are many different types of fluid flow, and they behave very differently. One of the fundamental distinctions has to do with the ratio of inertia forces to viscous forces, which is called the Reynold's number. Wikipedia has a pretty good page on it. Flows with high reynolds number will behave completely differently than flows with low reynolds number.

The Reynold's number is defined as $Re = \rho s L / \mu$, where $\rho$ is density, $s$ is velocity, L is a length, and $\mu$ is dynamic viscosity. Of these four quantities, 2 are intrinsic to a particular fluid (density and dynamic viscosity) and 2 are more to do with the situation (the length and the velocity). The reynold's number is super important and it comes up all the time in fluid mechanics. Because it comes up so often, and because both density and viscosity are intrinsic to the type of fluid, we simplify the Reynold's number equation to just $Re = s L / \nu$, where $\nu$ is kinematic viscosity. Now it's 3 equations, exactly 1 of which has to do with the type of fluid, and 2 that have to do with the particular situation. This is just for convenience. We could have just kept the original two quantities.

As for the units, the other two quantities in the reynold's number equation are length and velocity. Length * velocity will give you m^2/s. Note that is the same units as kinematic viscosity. So when you calculate $Re = s L / \nu$, you come out with a pure number that has no units. Unitless numbers occur very frequently in fluid mechanics.

• Wow, nice answer! Goes slightly above my head, but the central point for me will be: "Thinking of it as so many m2 of fluid flowing per second is not what it is about." Thanks :) Jan 17 '18 at 1:28
• I can thus still asume that a higher number in <whatever> viscosity measure (be it dynamic or kinematic) still means that the fluid is more thick-ish... Jan 17 '18 at 1:30
• For dynamic viscosity, yes. higher number is more "thick-ish". For kinematic viscosity, not really. For kinematic viscosity, higher number means more thickish relative to the density. E.g. a liquid with a higher kinematic viscosity might be more "thick-ish", but it also might just be less dense. Kinematic viscosity is the ratio of "thick-ish" to density. If all you care about is "thickish", then ignore kinematic viscosity completely and only look at dynamic viscosity. Jan 17 '18 at 2:01
• You "buried the lede" :-) -- the units of constants, aka "fudge factors," are chosen to make the answer come out in the right units. Jan 17 '18 at 13:43
• Looking at the Reynolds equation, you could look at it like this: s*L represents inertial forces, kinematic viscosity represents viscous forces and that is viscosity in relation to density
– mart
Apr 19 '18 at 7:16

Suppose you want to compare the performance of a lifting surface like an airfoil when operating in two different fluids.

The lift equals some f1(geometry) * density * velocity squared.

One of the major drag components equals some f2(geometry) * dynamic viscosity * velocity squared.

In order to have a convenient lift to drag ratio expression, we take the drag formula and multiply it by density over density and factor (density * velocity squared). Now we have L/D = f1/f2 * density/viscosity. This is very convenient when density is constant as in liquids or incompressible flows. For instance, it tells us that a foil in water can operate at a higher L/D than one in air at the same lift force, because the kinematic viscosity of water is about 18 times less than that of air.

That's all there is to it. It's handy when you want to cancel density out of other terms in the equation; it has a simpler set of units to keep track of; and it facilitates more compact notations.

Let's start from the definition of dynamic viscosity, which is the proportionality constant in Newton's linear relationship between shear and velocity gradient: shear = (dynamic) viscosity x velocity gradient. Thus, dynamic viscosity has dimensions of stress per velocity gradient. Physically, it is a measure of how much stress a fluid experiences for a certain velocity gradient. Kinematic viscosity is defined as the dynamic viscosity divided by fluid density. Thus, kinematic viscosity is a measure of the stress a fluid experiences for a certain velocity gradient per unit of its density.

• whats the unit of shear in this equation? shear = viscosity x velocity gradient Jan 21 '18 at 20:06
• Force per unit area. Jan 23 '18 at 4:33