Is Air or Water More Viscous for Slow Flow Around a Body?

[Clausius2]

The other day I met a mathematician and I told him I was considering air as the surrounding fluid for the slow viscous flow around a body. And he replied: "what?. This is viscous motion, so you cannot consider air.", implying that I should use water instead.

Who of us do you think was right?. What makes my motion more viscous, air or water? (for the same geometry and flow). Neglect compressibility.

I'd love to hear your opinions.

 

[AlephZero]

Your mathematican is talking nonsense.

The effect of viscosity on motion depends on the ratio of the viscous to inertia forces. Viscous forces tend to vary as size squared (e.g surface area) and inertia forces as size cubed (e.g. mass is proportional to volume). So viscosity tends to have relatively more effect on small objects than on large ones.

A simple demonstration that viscosity of air has a signicant effect on slow moving objects: drop a feather and a lump of lead, and see which hits the ground first. Maybe your mathematican read about a famous experiment at the Leaning Tower of Pisa, but forgot about common sense.

The dimensionless Reynolds number can be interpreted as a ratio of inertia to viscous forces. To a first approximation, the motion will be the same in air or water for the same value of Re.

 

[caslav.ilic]

What makes my motion more viscous, air or water?

It is not a clear matter what one defines as "more viscous".

For example, if you would like to test Stokes law, then you should use water, or another fluid with even greater dynamic viscosity. Dynamic viscosity of water is ~100 times greater than that of air.

However, you may want to examine influence of viscosity in cases which are frequently modeled with inviscid flow as base approximation (eg. flow around an airfoil). Then you are interested not in viscosity per se, but in ratio of viscous and inertial forces, which is measured by Reynolds numbers. Or, for the identical geometries and velocities, by the kinematic viscosity of the fluid. Since water also has ~1000 times greater density than air, its kinematic viscosity is ~10 times smaller than that of air, and so air should be used as "more viscous".

There are, for example, water tunnels for testing aeronautical applications, precisely because they allow for "less viscosity", ie. cheap tenfold increase of Reynolds number for the given model size. The caveat (at least one, that is) is that water is susceptible to cavitation as pressure gradients get too high, so these tunnels can be used only for low velocities.

 

[FredGarvin]

Depending on what you are actually doing, perhaps the person meant that an experimental set up with water as your fluid would be easier than dealing with air. Who knows. It could be an example of why engineers are smarter than mathematicians

 

[Cyrus]

Why would you ask a mathematician an engineering question?

 

[Clausius2]

Thanks guys. Amongst all I chose the answer of caslav.ilic:

However, you may want to examine influence of viscosity in cases which are frequently modeled with inviscid flow as base approximation (eg. flow around an airfoil). Then you are interested not in viscosity per se, but in ratio of viscous and inertial forces, which is measured by Reynolds numbers. Or, for the identical geometries and velocities, by the kinematic viscosity of the fluid. Since water also has ~1000 times greater density than air, its kinematic viscosity is ~10 times smaller than that of air, and so air should be used as "more viscous".
.

That's the keypoint. The viscous force is proportional to the dynamic viscosity $$\mu$$, thus it is clear it would be larger in water. But talking about viscous behavior given a geometry and a flow speed, the flow of air is more viscous than the flow of water by the reasons given above. It is the ratio of inertial to viscous terms which allows us to use a Stokesian approximation. For sure that for the same flow conditions the Stokesian approximation is more justified working with air rather than working with water because the Reynolds number is always smaller. For my it was not trivial when I realized, so I am happy if someone reads this and takes this stuff into account for his studies.