Trying to better understand what viscosity really is

[caltrop]

I'm trying to get a better handle on the actual physical phenomena underlying viscosity (for Newtonian fluids). Something I could word in the format of "this happens (and this and this), and so the fluid resists flow." What I've found online is that when gasses are at higher temperatures, they have more intermolecular collisions, and when liquids are at lower temperatures, the molecules are closer together and thus interact more, but there's something missing here. The molecules interact more, and so... what?

I understand that viscosity provides resistance to flow distinct from inertial effects (i.e. I could have one bottle with a given mass of water, another bottle with an equal mass of molasses, and I should be able to throw them both the same distance). My best guess (and this is only a guess) is that viscosity relates to a liquid's tendency to transform the kinetic energy of its molecules into internal energy (i.e. the tendency to go from coherent motion to waste heat). I don't know if this is right, and even if it is, I'm don't understand how increased intermolecular interaction increases this rate of conversion.

Any help or different ways of looking at viscosity would be greatly appreciated. Thanks! :-)

 

[A.T.]

The molecules interact more, and so... what?

...and so they resist more to deformation of the fluid.

 

[caltrop]

...and so they resist more to deformation of the fluid.

Indeed, but resist how? That's the mechanism I'm trying to understand better.

Perhaps it would be better if I asked in terms of force or energy.

If I apply shear stress to a solid (if I correctly understand elastic deformation), it generates a force counter to the applied force because some atoms are moved closer together by the deformation and repel one another, while others are forced further apart and attract one another. In terms energy, we are doing work on the material as we deform it, and the energy transferred into the material through this work is being stored as potential energy due to these interatomic attractions/repulsions. This energy is released if we remove the force and allow the material to "spring back."

However, I don't know the corollary for fluids: If I apply a shear stress to a fluid, what is happening between the atoms/molecules to cause it to create an opposing force? What happens to the energy I expend in order to deform the fluid?

 

[CWatters]

Cohesive forces between the molecules in a fluid means that a moving object/molecule tries to drag nearby molecules along with it. I guess this effectively makes objects bigger increasing drag.

 

[Wledig]

In the case of liquids it's useful to think of viscosity arising from internal friction. Consider the picture below representing the flow of a fluid between two parallel plates:

It's an experimental fact that if you were to drag the upper plate with a constant force, the liquid right below it would move with the same velocity. The same however does not hold for further down layers of the fluid, experience again tells us that the velocities in x follow the linear relationship: $$v_x(y) = \frac{v_0}{d}y $$ Which you can take as a statement of proportionality, the further down it is the slower it moves. This might be easier to visualize if you consider the fluid as a deck of cards. It's fair then to conclude that the shear stress will be proportional to the variation of velocity and have the form: $$ \frac{F}{A} = \eta \frac{dv}{dy} $$ This relation altough quite simple, is rather general in the context of transport theory. Now for the gas, viscosity is better understood as being caused by the diffusion of momentum. You can make statistical considerations and say for instance that there will be 1/3 of total molecules moving in the y direction, half of these(1/6), in the +y, and the other half in the opposite -y. If you consider a fixed position at y, molecules crossing it from below will have smaller momentum in x and molecules crossing it from the top will have bigger momentum in x. That's due to the assumption that momentum in x is a function of y and of the mean free path. Now as molecules move around you get momentum exchange, the resultant force then will be the difference between contributions from top and bottom and as it turns out, after doing a bit of Taylor expansion, you reach the exact formula as before but this time $\eta$ is explicitly dependant on the mean free path. And since the mean free path itself is dependant on temperature, the logical conclusion is that you get more viscosity as the temperature increases.

 

[caltrop]

Let's start by seeing if I understand the mechanism as discussed for liquids.

I'll first discuss the solid analogy (the deck of cards) to see if I get that.
Let's suppose we do the above sliding plate experiment with thin sheets of solid material (we can consider our sheets and plates to be infinite planes for simplicity's sake).
Once we are at steady state (every layer is moving at a constant velocity), we will still be applying a force to the top plate in the direction of motion, so we are still doing work on the top plate, but since there is no acceleration of the plates or sheets, we have no net gain of kinetic energy. We're inputting energy, but there's no gain in kinetic energy, so there must be energy going into some other form.
This is possible because there are frictional losses: i.e. there is heat (internal energy) generated in the process. The plates and sheets will exhibit rising temperature as the process goes on. That's the only way we can continue to do work on the system without any of the plates/sheets accelerating.
If we then wanted to run the same process (same height y and same relative speeds of plates, thus same velocity gradient) using sheets having a higher coefficient of friction, we would need to apply a larger force F, i.e. do more work for a given distance, i.e. have a higher rate of energy input.
To maintain steady state, this would mean that we'd also have to have the conversion of energy to heat to go up by the same amount. We'd have a higher rate of heat generation and (assuming the same specific heat as before), the temperature of the system would go up more quickly.
Thus, the coefficient of kinetic friction could be seen as a proportionality constant for the rate at which heat is generated for a given relative velocity.

Did I get the above correct? If so, is the below correct as well?

For the case of the sliding plates (and laminar liquid flow), we are again doing work on the system at steady state, and again no layer of liquid is accelerating, so once again, the we must have heat generation at the same rate as our input power.
For a higher viscosity fluid, to maintain the same velocity gradient will require a higher power input and will result in an equally higher rate of heat generation.
Viscosity could thus be seen as a proportionality constant for the rate of heat generation for a given relative velocity.
So we can say: "Molasses is more viscous than water, which means that if we want to deform them both at the same rate, the molasses will generate more heat and, thus, require more energy input."

 

[Wledig]

This is possible because there are frictional losses: i.e. there is heat (internal energy) generated in the process. The plates and sheets will exhibit rising temperature as the process goes on. That's the only way we can continue to do work on the system without any of the plates/sheets accelerating.

Yes, taking this to the context of fluids you'd explain it by saying that the viscous resistance of the fluid is equal and opposite to the force applied on the plate.

We'd have a higher rate of heat generation and (assuming the same specific heat as before), the temperature of the system would go up more quickly.

I believe so, these are interesting considerations you're making.

Did I get the above correct? If so, is the below correct as well?

Yes, I think you understood the analogy quite well.

 

[LURCH]

The molecules interact more, and so... what?

Would it be correct to say that each interaction uses some energy?

 

[A.T.]

Would it be correct to say that each interaction uses some energy?

The interactions dissipate energy, by converting bulk motion into random motion (heat).

 

[Chestermiller]

See Transport Phenomena by Bird, Stewart, and Lightfoot, Chapter 1, for detailed molecular explanations and derivations related to viscosity of both liquids and gases.