The most economic particles model of a[n utterly] fake fluid—part 1

Real fluids are viscous.

Newton was the first to formulate a law of viscosity; his law forms an essential part of the engineering fluid mechanics even today.

The way the concept of viscosity is usually presented to undergraduates is in reference to a fluid moving over a horizontal solid surface, e.g., water flowing over a flat river-bed. The river-bed itself is, of course, stationary. The students are then asked to imagine a laminar flow in which the horizontal layers of fluid go slipping past each other with different velocities. The viscous forces between the fluid layers tend to retard their relative motion. Now, under the assumption that the layer adjacent to the stationary solid surface has zero velocity, and that the flow is laminar, a simple parabolic profile is obtained for the velocity profile. The velocity progressively increases from 0 at the solid surface to some finite mainstream value as you go up and away from the horizontal solid surface. Newton’s law is then introduced via the equation:

$\tau \propto \dfrac{dU}{dy}$

where $\tau$ is the shear stress between the fluid layers slipping past each other, and $\frac{dU}{dy}$ is the velocity gradient along the vertical direction. The constant of proportionality is viscosity, $\mu$:

$\mu \equiv \dfrac{\tau}{\left(\dfrac{dU}{dy}\right)}$

This picture of layers of fluids slipping past with progressively greater velocities, as in a deck of card given a gentle horizontal push, is easy to visualize; it helps people visualize what otherwise is not available to direct perception.

That’s quite fine, but then, as it happens, sometimes, concrete pictures also tend to over-concretize the abstract ideas. The above mentioned picture of viscosity is one of these.

You see, the trouble is, people tend to associate viscosity to be operative only in this shear mode. They can’t readily appreciate the fact that viscous forces also arise in the normal direction. The reason is, they can’t as easily imagine velocity gradients along the flow direction. No engineering (or physics) text-book ever shows a diagram illustrating the action of viscosity along the direction of the flow.

One reason for that, in turn, is that while in solids stress depends on the extent of deformation, in fluids, it depends on the rate of deformation. Indeed the extent of possible deformation, in fluids, is theoretically undefined (or infinite, if you wish). A fluid will continually go on changing its shape so long as a shear stress is applied to it… It’s easily possible to pour water from a tap onto a tilted plate, and then, from that plate onto the bottom of a kitchen sink, without any additional stress coming into picture as the water continuously goes on deforming in the act of pouring. The fact that water has already suffered deformation while being poured from the tap to the plate, does not hinder the additional deformation that it further suffers while falling off from the plate. And, all this deformation, inasmuch as it involves a change of the initial shape, involves only shear. And, as to the stress, when it comes to fluids, the extent of deformation does not matter; the rate of deformation—or the velocity gradient—does. Stress in fluids is related to the velocity gradient, not to the deformation gradient (as in solids).

Another, related, reason for the difficulty in visualizing viscosity appearing in the normal direction is that, in our usual imagination, we can’t visualize fluids being gripped from its ends and pulled apart, the way solids (e.g. rubber band) can be. The trouble is not in the stretching part of it; the trouble is in the “being held” part of it: you can’t grab of a piece of a fluid in exactly the same way as you can, say, a bite of food. There is no bite of water, only a gulp of it. But the practical impossibility of holding fast onto an end of a fluid also carries over when it comes to imagining fluids being stretched purely along the normal direction, i.e., without involving shear.

Of course, as far as exerting a normal force to a fluid is concerned, people have no difficulty imagining that. You can always exert a compressive normal force on a fluid, by applying a pressure. But then, that is only a compressive force, and, a static situation. You don’t have to have spatially varying velocities to arrive at the concept of pressure—indeed, you don’t make any reference to the very idea of velocity, in that concept. Pressure refers to static forces.

Now, when people try to visualize velocity gradients in the normal direction, they unwittingly tend to take the visualization on the lines parallel to the viscosity-defining picture. So, they take, say, a 10 m/s velocity vector at origin, an 8 m/s velocity vector at the point x = 1, a 6 m/s vector at x = 2, and so on. Soon, they end up imagining having a zero magnitude velocity vector.

But this is a poorly imagined situation because it can never be realized in one-dimension—quantitatively, it violates the mass conservation principle, i.e. the continuity equation (at least for the incompressible 1D flow without sources/sinks, it does).

Now, when pushed further, people do end up imagining an L’ kind of bend in a pipe (or a fluid bifurcating at aT’ joint), i.e., taking velocity vectors to be just x-components of a 2D/3D velocity field.

But, speaking in general terms, at least in my observation, people still can’t easily imagine viscosity being defined in reference to velocity gradients along the direction of the flow. Many engineers in fact express a definite surprise at such a definition of viscosity. The only picture ever presented to them refers to the shear deformation, and given the peculiar nature of fluids, velocity gradients in the normal direction (i.e. along the flow) are not as easy to visualize unless you are willing to break continuity.

Recently when Prof. Suo wrote an iMechanica post about viscosity (in reference to a course he is currently teaching at Harvard), the above-mentioned observations came rushing to my mind, and that’s how I had a bit of discussion with him on this topic, here [^].

As mentioned in that discussion, to help people visualize the normal viscosity, I then thought of introducing a particles model of fluid, specifically, the Lennard-Jones (LJ) fluid [^]. It also goes well with my research interests concerning the particles approaches to fluids.

But then, of course, I have been too busy just doing the class-room teaching this semester, and find absolutely no time to pursue anything other than that—class-room teaching, or preparation for the same, or follow-up activities concerning the same (e.g. designing assignments, unit tests, etc.). But no time at all is left for research, blogging, or why, even just building a few software toys at home. (As a matter of fact, I find myself hard-pressed to find time even for just grading of unit-test answer-books.)

Therefore, writing some quick-n-simple illustrative software (actually, completing writing this software—something which I had began last summer) was out of the question. Still, I wanted to steal some time, to think about this question.

I therefore decided to drastically simplify the matters. I would work on the problem, but only to the extent that I can work on it off my head (i.e. without using even paper and pencil, let alone a computer or a software)—that’s what I decided.

So, instead of taking the $(1/r)^{12} - (1/r)^6$ potential, I began wondering what if I take a simple $1/r$ attractive potential (as in Newtonian gravity). After all, most every one knows about the inverse-square law, and so, it would be easier for people to make the conceptual connections if a fluid could be built also out of the plain inverse-distance potential.

So, the question was: (Q1) if I take a few particles with (only attractive) gravitational interactions among them—would they create a fluid out of them, just the way the LJ potential does? And if the answer is yes, then would these particles also create a solid out of them, too, just the way the LJ potential does?

Before you rush into an affirmative answer, realize here that the LJ potential carries both attractive and repulsive terms, whereas the gravitational interaction is always only attractive.

But, still, suppose such a hypothetical fluid is possible, then, (Q2) what would distinguish this hypothetical fluid from its corresponding solid? How precisely would the phase transition between the solid and fluid occur? For instance, how would the fluid consisting of only gravity-interacting particles, melt or solidify?

And, (Q3) what is the minimum number of such particles that must be present before they can create a solid? a fluid? a liquid? a gas?

Of course, answering these questions is not a big deal (neither is thinking up these questions). The point is, I had some fun thinking along these lines, in whatever time I could still find.

However, since this post is already more than a thousand words-long, let me stop here, and ask you to think about the above mentioned questions. In my next post, I will give my answers to them. In the meanwhile, think about it, have fun, and if you think you have got an answer that you could share with me, feel free to drop a comment or an email.

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Music: Ashok Patki
Lyrics: Ashok Paranjape

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