Assumption of local thermodynamic equilibrium in a fluid

[Shivam Sinha]

Moving fluids are generally in a state of non-equilibrium. However, in fluid dynamics, people generally assume a state of local thermodynamic equilibrium and argue that in such a condition, equilibrium thermodynamic concepts such as pressure, temperature, entropy, internal energy etc. can be defined for a local fluid element. As such, the moving fluid can be considered as a continuum of local thermodynamic states and relations from equilibrium thermodynamics can be applied locally.

I am having trouble understanding this concept. For example, for a system in global equilibrium, pressure is defined as normal stress acting at a point, and it is equal in all directions.

Now consider a moving fluid. If we assume local equilibrium for this fluid, we can define a local property called pressure in the same way it was defined for a system in global equilibrium. This means that in a moving fluid, pressure at a point should be the normal stress at that point, and it should be equal in all directions (which is clearly not the case).

Perhaps my idea of local equilibrium or my conception of pressure is flawed. Can someone clarify my doubts? Thanks!

 

[boneh3ad]

Why do you think pressure is not equal in all directions in a moving fluid?

 

[Shivam Sinha]

Why do you think pressure is not equal in all directions in a moving fluid?

Pardon me, what I meant to say was that the normal stress at a point in a moving fluid is not equal in all directions i.e it depends on the orientation of the surface on which the stress is acting.

 

[256bits]

Perhaps my idea of local equilibrium or my conception of pressure is flawed

and it should be equal in all directions (which is clearly not the case).

So far up the that point ( in bold ), everything seems OK.

 

[Shivam Sinha]

So far up the that point ( in bold ), everything seems OK.

Please explain further.

Are you suggesting that the "local pressure" in a moving fluid is the normal stress at a point (which is not equal in all directions)? This is the basis of my confusion. In equilibrium thermodynamics, pressure is defined to be the normal stress at a point which is equal in all directions. I was expecting the same definition for a system with local equilibrium.

 

[256bits]

Please explain further.

Take a look at something different such as a static column of water. The pressure increases with depth by the value mgh. Taking a cubic element of water with sides z, one can see that in horizontal directions x and y perpendicular to the vertical z-axis, the pressure forces are equal on vertical faces of the cube, On the top and bottom faces, the pressures are different by a factor of mgz. ( the difference in pressure being accounted for by the weight of the cube )

When the cubes' side length is decresed further to Δz, or even smaller to dz, what happens to the pressure difference of top and bottom faces?

And, then to a point?

 

[boneh3ad]

Pardon me, what I meant to say was that the normal stress at a point in a moving fluid is not equal in all directions i.e it depends on the orientation of the surface on which the stress is acting.

That's a very important distinction, though. Regardless of fluid motion, the pressure is always equal in every direction. The difference is that, for a flowing fluid, the normal stress has more components than just the pressure. There is a normal viscous component as well, though it is almost always incredibly tiny compared to the pressure component. Either way, the existence of viscous normal forces doesn't invalidate the assumption of local equilibrium. Pressure can still be defined as being equal in all directions at any point.

 

[Chestermiller]

That's a very important distinction, though. Regardless of fluid motion, the pressure is always equal in every direction. The difference is that, for a flowing fluid, the normal stress has more components than just the pressure. There is a normal viscous component as well, though it is almost always incredibly tiny compared to the pressure component. Either way, the existence of viscous normal forces doesn't invalidate the assumption of local equilibrium. Pressure can still be defined as being equal in all directions at any point.

In high viscosity flows, the normal component is far from incredibly tiny compared to the pressure component. What you consider"almost always" depends on the kind of fluids you are personally used to dealing with. Or, to put it another way, "almost always in the eye of the beholder."

 

[boneh3ad]

You've got to have a very viscous fluid (borderline viscoelastic, I'd think, which I know you've dealt with in your career) or else a more exotic fluid phenomenon (e.g. a shock wave) for the viscous normal stresses to be meaningfully large. Otherwise, in the majority of fluid flows encountered by the majority of people, those stresses are generally quite small.

Either way, I'm not sure that is germane to the discussion, as my main point was that pressure is always the same in all directions and deviation from isotropy of the normal stresses is going to be viscous in nature rather than thermodynamic.

 

[Chestermiller]

You've got to have a very viscous fluid (borderline viscoelastic, I'd think, which I know you've dealt with in your career) or else a more exotic fluid phenomenon (e.g. a shock wave) for the viscous normal stresses to be meaningfully large. Otherwise, in the majority of fluid flows encountered by the majority of people, those stresses are generally quite small.

Either way, I'm not sure that is germane to the discussion, as my main point was that pressure is always the same in all directions and deviation from isotropy of the normal stresses is going to be viscous in nature rather than thermodynamic.

It's even important for shock waves in compressible flow nozzles. See solved example 11.4-7 in Bird, Stewart, and Lightfoot, Transport Phenomena, Chapter 11, p 350. It is also going to be important in sharply converging and diverging flow of incompressible fluids in ducts. And it is going to be important in external flow near the leading and trailing edges of bodies.

 

[boneh3ad]

That's why I specifically cited shock waves in my previous post.