system
Given the following post, respond as if you're commenting on Reddit:
human
Is Mayonnaise a compressible fluid?

I know it's technically a colloid.  I'm working on a question from an eve friend who wants to make a mayonnaise jet pack.
gpt
First some clarifications that are somewhat relevant to your question.

* A fluid exhibits *incompressible flow* if the density of fluid parcels is constant. This implies that div(**u**) = 0, which is roughly a statement that infinitesimal volumes are conserved. 

* Something is an (homogeneous) *incompressible material* if the density is constant in time and throughout the entire material.

All incompressible fluids exhibit incompressible flow, but the converse is not true. The density of a fluid can vary throughout the fluid with the flow itself still being incompressible. We very often *assume* the density is constant throughout to make some problems easier or to get a first approximation, but it's important to understand the difference between an incompressible flow and an incompressible material.

(The phrase "this fluid is incompressible" technically means that the fluid is an incompressible material, but it is often interpreted as "this fluid exhibits incompressible flow". So to be absolutely clear, we often either put in the specific word "flow" or just say that the flow is *isochoric*. The word "isochoric" literally means "same volume" and emphasizes the fact that div(**u**) = 0 is an expression of conservation of volume.)

---

That being said, you are really asking whether mayonnaise exhibits incompressible flow. Mayonnaise is essentially an emulsion of eggs and oil. So it is a liquid (oil) with solid particles (egg) suspended throughout. Generally speaking, all liquids exhibit incompressible flow. This is not strictly true (e.g., seawater is about 5% more dense at the bottom of the ocean than at the surface), but it is true to a very good approximation for many, many applications. Compressibility becomes important if the flow velocity is comparable with the speed of sound in the material or if you are interested in pressure waves (sound), density waves, or shock waves. For everyday problems with mayonnaise (squeezing through a bottleneck, spreading with a knife), the flow of the mayonnaise can very well be approximated as incompressible.

However, you are definitely right to be suspicious of mayonnaise, as it is very clearly not like other liquids like water or even honey. Suppose you have a jar of water or honey and you tilt the jar to pour out the fluid. The gravitational force is said to be applying a shear stress to the fluid. The water or honey immediately begins to pour out as soon as you tilt the jar, no matter how little you have tilted the jar. Yes, the honey flows much more slowly than the water, but it *does* flow right away. (Honey is said to be more viscous than water.)

Fluids like water and honey are called *Newtonian fluids* and a defining feature is that the shear rate is a linear function of the shear stress. This just means that the fluid flows as soon as you apply any shear stress and the rate at which it flows increases in proportion to the applied shear stress. That proportionality constant is related to the viscosity of the fluid.

Mayonnaise is a *non-Newtonian fluid*, which is simply any fluid that does not exhibit the defining feature just described. There are many types of non-Newtonian fluids. Mayonnaise is an example of a *Bingham fluid* (or Bingham plastic, or yield-stress fluid). A Bingham fluid will flow in response to an applied stress, but only once *enough* stress has been applied. Think of sliding a block on a table. The block doesn't move until you apply a large enough force to overcome static friction. It's pretty much the same idea with Bingham fluids. If you tilt a jar of mayonnaise just a little bit, it will not flow at all. At some point, you will have tilted the jar enough so that the gravitational force is large enough to cause the mayonnaise to flow. (If the mayonnaise is especially thick, it may not even flow if the jar is turned upside down.)

So for a Newtonian fluid, the shear rate and shear stress are proportional. The graph of shear rate versus shear stress is a line through the origin, whose slope is related to the viscosity of the fluid. For a Binghman fluid (e.g., mayonnaise), the graph of shear rate versus shear stress is a line *not* through the origin. It has some positive *y*-intercept. So you need to give two numbers to characterize the fluid: the slope (related to the viscosity once it starts to flow) and the y-intercept (called the yield stress, the minimum required stress for the fluid to flow).

So think of squeezing mayonnaise through a bottleneck. Once you squeeze hard enough to overcome the yield stress, the mayonnaise flows through the bottleneck just as any other fluid with some viscosity. Then it falls under gravity and finally lands on your sandwich. As soon as the shear stress is no longer applied (i.e., once it lands on your sandwich), the mayonnaise stops flowing. This is why when you squeeze mayonnaise out of a bottle with, say, a circular opening, you end up just getting long tubes of mayonnaise on your sandwich, which you then have to spread with a knife to get a nice even coating. (This happens for pretty much all emulsions.)

**Summary:** Mayonnaise is a liquid that exhibits incompressible flow. But it is a non-Newtonian fluid, specifically a Bingham fluid. So it requires a certain minimum shear stress before it begins to flow.