Another guest post this week. Morris Flynn from the ept. of Mech. Eng., at U. Alberta writes about a couple of viscous flow demonstrations. This demonstration serves to highlight two important features of (immiscible) viscous flows, namely (i) their capacity for supporting large compressive loads, and, (ii) the Taylor-Saffman instability that develops when a less viscous fluid pushes on a more viscous fluid.

**Equipment**

- Two pieces of acrylic plastic. Each piece should be at least 5 mm thick and have a cross sectional area about the size of two business cards.
- Dish soap. Lubricating oil can also be used, but requires a more careful cleanup.
- (Optional) An overhead projector so that you can project the demonstration onto an overhead screen.

**Demonstration**

- Place one of the pieces of acrylic flat on the overhead projector stage.
- Add several drops of dish soap to the top surface of the acrylic piece.
- Gently place the second acrylic piece on top of the first. The soap drops should expand into a pancake shape.
- Gently pull on the upper acrylic piece while holding the other fixed to the stage. Notice the interesting instability pattern that appears.

**Analysis, part i (squeezing the acrylic pieces together)**

Reynolds’ lubrication equation describes flow that is primarily or exclusively in one particular direction, say the horizontal or *x*-direction. Regarding the *x*-component of the Navier-Stokes equation, the inertial terms are smaller than the viscous terms by a factor of (*L*/*h*)/Re. Here, *L* and *h* represent, respectively, the characteristic length scales in the horizontal and vertical directions. Also Re=ρ*Uh*/μ is the Reynolds number where ρ is the fluid density, μ is the fluid dynamic viscosity and *U* is a characteristic flow speed. For a lubricating flow where Re is less than 10 say, (*L*/*h*)/Re is expected to be small so that fluid inertia is negligible.

If the upper acrylic piece is pushed downwards at a constant speed, Reynolds’ lubrication equation can be written as

As the gap width *h* becomes indefinitely small, the only way this equation can be satisfied is if the pressure gradient and therefore *p*, the fluid pressure, becomes indefinitely large. Thus solid-solid contact is inhibited because an infinite force is necessary to squeeze out all of the interstitial fluid. This is the reason why lubricating oils play such a critical role in reducing wear in engines and related mechanical equipment.

**Analysis, part ii (pulling the acrylic pieces apart)**

By analogy with the Darcy’s Law equations, which consider flow in a porous medium, flow in the horizontal direction is described by

*U* now represents the (constant) speed of the interface between the soap, having a dynamic viscosity μ_{1}, and the intruding air, having a dynamic viscosity μ_{2}. Also *k*, the analogue of the porous medium permeability, is related to the gap width between the pieces of acrylic.

For whatever reason, let’s suppose that the otherwise smooth interface develops a “kink.” We can apply the above equation to understand whether this perturbation is likely to grow or decay in time. The force on the soap displaced by the air due to the kink is proportional to *p*_{2}–*p*_{1} where

Here *r*_{1} and *r*_{2} denote, respectively, the radial position of the interface in the unperturbed and perturbed states. If the force in question is positive, there is a positive feedback leading to an amplification of the perturbation or, in other words, a further distortion of the interface. From the last equation, *p*_{2}>*p*_{1} whenever μ_{1}>μ_{2}. The interface is therefore unstable when the displacing fluid (air in this case) has a smaller dynamic viscosity than the displaced fluid (dish soap in this case). This result has long been the bane of petroleum engineers, who frequently inject water into hydrocarbon formations to maintain reservoir pressure and to sweep oil towards a producing well. The fact that the interface between the injected water and the oil is very often unstable leads to phenomena such as fingering and water breakthrough. These cause a commensurate drop in the rate of oil production.

**Reference:**

Homsy, G. M., 1987: Viscous fingering in porous media. Annu. Rev. Fluid Mech., **19**, 271—311.

Description compiled by Morris R. Flynn, Dept. of Mech. Eng., U. Alberta (mrflynn”at”ualberta”dot”ca, 780-492-5593)

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.