# What causes viscosity of a fluid?

Consider a fluid like water. Intuitively I would say that its viscosity is caused by intermolecular interactions among its molecules. But the Einstein-Smoluchowski relation (and the Fluctuation-Dissipation Theorem in general) says that viscosity is caused by the erratic motion of fluid particles. What I'm missing?

• It is caused by both the factors you listed. See Fluid Dynamics by Batchelor, Chapter 1.
– Deep
Commented May 26, 2018 at 4:54
• there is a story told about Heisenberg on his deathbed. He says, "I have a couple of questions for The Old One (God) when I get to meet him: why viscosity, and why relativity. And I think he'll have an answer for the second one!" Commented May 26, 2018 at 6:22
• @LoScrondo That's right.
– Deep
Commented May 26, 2018 at 15:39
• @nielsnielsen I believe it is "Lamb" instead of "Heisenberg", "turbulence" instead of "viscosity", and "quantum" instead of "relativity"! This is recounted I think in Chapter 2 of Turbulence by Davidson. But hey, it's probably apocryphal, so no harm done :-)
– Deep
Commented May 26, 2018 at 15:44
• @deep, thanks for your observation, I suspect that this particular literary device has probably been used in a variety of contexts- your challenge being, can you devise a new deathbed anecdote using another famous physicist? the world awaits your reply! Commented May 26, 2018 at 15:58

I think that the origin of viscosity is easy to understand with a help from analogy.

Imagine multiple parallel rail tracks. Trains consisting of flatbed wagons are moving along the same direction on all tracks but with slightly varying velocities. Workers standing on wagons are throwing sacks of sand in random directions onto nearby trains.

If we assume that in the reference frame of a given train the distribution of thrown sacks' momentum is isotropic it would not be so in the frame of another train. For example, sacks thrown from the fastest train would carry on average more momentum in the direction of motion than sacks thrown onto it and so this train would be slowing down while neighboring trains would be speeding up. We have a mechanism of diffusion for a momentum component along the direction of the tracks.

By formalising this analogy I believe it is possible to derive Einstein–Smoluchowski relation (at least up to a constant multiplier). So while indeed intermolecular interactions among (the fluid) molecules is an immediate mechanism for momentum exchange, the kinetic theory origin of viscosity is random motion of molecules that causes diffusion of net average of momentum.

• A very interesting analogy, thank you! Indeed, are you saying that the diffusion of net average of momentum accounts also for the intermolecular interactions, or it is possible you are talking about the kinetic theory of viscosity, but not the phenomenological one? In short, in the trains analogy, where is the role of the intermolecular interactions? Commented May 28, 2018 at 19:14
• @LoScrondo: Intermolecular interactions ensure that molecules have distribution of momentum corresponding to a given temperature (plus a net shift which gives collective motion of fluid element) and ensure that molecule would be interacting with other molecules (so that its free path is small). In my analogy this role is played by workers who throw sacks of sand and catch them. Commented May 28, 2018 at 19:58

From what I've found in textbooks, SE, and Wikipedia the viscosity theorization could be addressed considering (at least) two momentum transfer principles. The kinetic model based on brownian motion is dominant in gases, whereas for liquids the intermolecular forces play a major role.

• Interesting paradox: In a gas, viscosity is proportional to the mean free path and speed of the molecules. It goes up with sqrt(T) at constant density. In a liquid, having larger aggregates of molecules decreases the MFP, but it increases the viscosity, and the trend with increasing temperature is the opposite. Commented Jun 2, 2018 at 15:16