# Why viscosity depends on the scale of things?

Water feels like honey for bacteria and air is very viscous for small insects.

My question is why viscosity depends on the scale of things?

## 3 Answers

Actually, viscosity does not depend on the scale of things. It is an intrinsic property of fluids, and it's defined as the measure of a fluid's resistance to flow. It describes the internal friction of a moving fluid: a fluid with large viscosity resists motion because its molecular structure gives it a lot of internal friction, while a fluid with low viscosity flows easily because its molecular structure results in very little friction when it is in motion. This property isn't a constant, but it's a function of temperature.

What depends on the scale is the force that you need to move when you are in contact with the fluid. To visualize this, let's imagine a simple situation, where we have a layer of fluid trapped between two horizontal plates, one fixed and one moving horizontally at constant speed $u$: If the speed of the top plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to $u$ at the top. This is called laminar motion. Each layer of fluid will move faster than the one just below it, and friction between them will give rise to a force resisting their relative motion. An external force is therefore required in order to keep the top plate moving at constant speed.

For a Newtonian fluid, the shear stress $\tau$ is proportional to the strain rate in the fluid, which we can express as the velocity gradient along $y$, and the viscosity $\mu$ is the constant of proportionality:

$\tau=\mu \dfrac{\partial u}{\partial y}$

Also, for any given material, the shear stress $\tau$ is defined as the ratio between the force causing the deformation and the cross-sectional area of material with area parallel to the applied force vector:

$\tau=\dfrac{F}{A}$

Putting the two together we obtain:

$F=\mu A \dfrac{\partial u}{\partial y}$

Which means that, if we fix the velocity $u$ at which we want to move the plate, the force $F$ required to cause the motion will be proportional to the contact area $A$. Or, in other words: bigger the plate, harder it gets to move it. But the viscosity of the fluid doesn't change.

To add a more formal perspective to the answers already given, if we non-dimensionalize the fundamental equations for fluid flow, the Navier-Stokes equations, using the characteristic flow velocity $U$ and characteristic length scale $L$, we obtain these equations in a form which only depends on the flow geometry, and is independent of scale. We find that the only parameter remaining in this case is the Reynolds number,

$$Re=\frac{U L}{\nu},$$

where $\nu$ is the kinematic viscosity of the fluid. Intuitively the Reynolds number can be understood as the ratio of inertial over viscous forces. Thus flows at small Reynolds numbers are dominated by viscous effects, corresponding to the appearance of "feels like honey". Flows in small gaps, around bacteria, small insects, etc. are all characterized by low Reynolds numbers and thus appear "very viscous", whereas flows around cars and airplanes, say, correspond to high-Reynolds number flow and appear nearly "inviscid". Here is an instructive article by Purcell on "Life at Low Reynolds Numbers".

• +1 In fact $Re^{-1}$ may be interpreted as dimensionless viscosity. – Deep Feb 5 '17 at 6:51

As Francesco indicated, viscosity does not depend on the scale of things. But in bacteria and small insects, the flow passages are very small (i.e., the surface to volume ratio is very large), and the viscous drag on the flowing fluid occurs at the flow surfaces. So viscosity has a bigger relative effect when the fluid is flowing through small flow passages than through large flow passages. From the "Hagen_Poiseuille" pressure-drop/flow-rate relationship, for a given flow rate, the pressure drop varies inversely with the 4th power of the tube diameter.

## protected by Qmechanic♦Feb 4 '17 at 13:49

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