Laminar Flow Reversibility: Why does the Blob Rewind?

Irreversibility in thermodynamics is usually explained with a movie example. A movie strip when rewound, wounds back in time, the events it evolved forward in time. Events in life do not usually reverse completely - unless, reality is what is depicted as in The Matrix. Movies are completely reversible. But reality is not so.

But here is a simple and elegant real-life fluid mechanics experiment that is reversible. Take two concentric cylindrical containers made of transparent glass or perspex. The cylinders are of different diameters and placed one inside the other. The inner cylinder is fixed to a motor from the top and it can be made to rotate at fixed speeds. A small gap separates the bottom surface of the inner cylinder from that of the outer cylinder. The annular space is filled with a highly viscous fluid. Say honey or Golden Syrup. Put a blob of ink or colouring agent using an ink filler somewhere inside the fluid, away from all of the cylinder walls. This configuration is shown as the first diagram in the accompanying schematic.

Rotate the inner cylinder slowly. The blob shears and spreads as shown in the second diagram. After a few turns, the blob is stretched and thinned all around the annulus as shown in the third diagram. A snapshot taken at this stage would not have evidence about which cylinder was rotated and its initial direction of rotation.

Interestingly, if we now rotate the inner cylinder in the reverse, the ink-strip begins to rewind. If the sequence is rewound with care, rotating the inner cylinder slowly as before, the strip would gain back its concentration and eventually becomes the blob. Well, almost exactly, the same blob.

A live experiment where the blob remembers its shape history in time like a movie event.

Here is an excellent video of this experiment filmed at the University of New Mexico Physics Department.

[YouTube Video Link]

Let us explore the technical explanation. What is observed is surprising to us because, it is a low Reynolds number phenomenon. Most of our daily life fluid mechanics - like water running from a tap or river flow - are high Reynolds number phenomena. What is Reynolds number? For most circumstances, it can be understood as a ratio of the acceleration terms (forces) and viscous terms (forces) in the momentum conservation statement that is believed to hold true for any fluid flow.

Reynolds number, named after Osborne Reynolds a pioneer in fluid mechanics understanding, is a way to quantify what forces dominate in a flow. Read my Pipe Turbulence for more on this number and its significance in fluid flow. If it is a large number (say, 1000 or 10000 when compared to 1), the acceleration terms dominate over the viscous terms (the brakes). Stirring your tea vigorously with a spoon and the air flow over a jet plane wing respectively generate Reynolds numbers of order 1000 and 10000. But the wading of our hands through rice patter or wheat dough generates only a low Reynolds number flow. Like the one observed in the above experiment.

Low Reynolds number flows are almost reversible.

But why? For this we need to appreciate how the momentum conservation reads for such flows.

The momentum conservation statement for fluid flows is the Navier-Stokes equations, which for incompressible flows in Cartesian coordinates can be written in the form

The first equation is for momentum conservation and the second one is for mass conservation and in tandem they represent the equations of motion for any fluid flow. Here p denotes the pressure in the fluid and V with an arrowhead represents the velocity vector for the fluid flow. ? and µ are the density and viscosity of the fluid. In the first equation, the left most terms are the acceleration terms while the rightmost are the viscous terms.

Reynolds number is the ratio of the acceleration and viscous terms and after simplification can be written as

In our reversibility experiment the slow rotation that shears and stretches the blob is a low Re flow. This means the LHS of equation (1) can be dropped in comparison to the viscous terms in the RHS. The resulting equations in 2 dimension (a good approximation for our experiment) can be written as

Observe in the above equations there is no term changing in time (no time derivative). Perhaps enough to convince us the phenomenon explained by any solution of this equation is reversible. More rigour can also be mustered. Assuming the rotation is slow, the blob stretches into a thin annular strip eventually. In other words, this final state is a steady state solution of the above equation. So, there should be finite values for u, v and p at this state that satisfy the above equations. Let them be u₁, v₁ and p₁, satisfying also the boundary conditions at the cylinder walls, where the fluid has to move with the velocity of the wall, say, u₁ = u_W and v₁ = v_W .

This solution is valid at state (1) or (3) in the diagram above.

If we now reverse the experiment, the boundary condition at the wall is reversed. That is, u₁ = -u_W and v₁ = -v_W . Likewise, if we now replace all u, v and p in the above equations with -u, -v and -p respectively, each term would have changed sign. Hence, the solution remains the same (as before).

That is the explanation for the almost 100 percent reversibility of highly viscous flows or low Re flows.

To round off, if we try to pull the same trick with the original Navier-Stokes equation in (1) above, it won’t work. the acceleration terms on the LHS (being quadratic in u and v) won’t change sign as they are product terms.

We kept saying almost reversible. There is a reason. Diffusion. The blob is made of a separate constituent from the base fluid. The blob medium diffuses in time in the fluid and lose its concentration. This works even if the cylinders remain stationary. So it works even when the cylinder is rotated. Hence, when we rewind, the blob doesn’t exactly regroup into the same volume, as some of it continually is diffusing. This phenomenon (governed separately by a Fickian diffusion term) is not captured in the Navier-Stoeks equation. It should not be confused with the viscous diffusion that is present in the RHS of equations (1) and (3). Viscous diffusion diffuses momentum not concentration.

We kept saying very high viscous flows and low Reynolds number flows in the same breadth as if one implies the other. Not necessarily. For instance, if we rotate the cylinders very fast, the acceleration terms (and forces) cannot be neglected in our experiment even though the fluid is highly viscous and its flow can be identified as highly viscous flow. The correct way to classify our phenomenon is as a low Reynolds number phenomenon.

Reference

1. From Calculus to Chaos, An Introduction to Dynamics by David Acheson, Oxford Uty. Press, 1987. [Amazon Link] [Google Books]

Related posts:

1. Turbulence in Flow around Bodies
2. Flow Transition in Porous Media
3. Predicting Flow Transition in Porous Media
4. Blood: Clot, Flow and Slip
5. Pipe Turbulence