Model reveals why slicing a tomato works better than dicing

If you’re anything like me, you have a set of kitchen knives so dull that you could swap the handle and the blade with no noticeable effect on your fingers. And yet these knives will still cut—you just have to saw away at that soft tomato with less downward pressure. Surprisingly, although physicists had a general idea of why sawing with a knife helps cut soft material, no one had really looked at the details. Luckily, this oversight has been corrected and you can now optimize your cutting technique using—as certain radio personalities put it—the power of science.

Too squishy to crack, too solid to flow

Gels—a tomato is a gel with a skin—fall into a funny category. A gel is a liquid (usually water), at least if you decide by volume. However, a gel doesn’t really flow, because the liquid is contained within a polymer network. You could say that a gel is a bit like a sponge: if you squeeze a gel hard enough, water will be expelled, and when you relax your grip, water will be sucked back into the network. But not all gels hold their shape if they dry out, while a dry sponge still retains its structure.

Although gels are flexible, they are still breakable. If you stretch a gel, the gel becomes strained. As long as the strain is below a certain limit, it will always return to its original shape when you stop stretching it. The maximum strain for which this is true is called the elastic limit. To cut a gel, the strain has to be above the elastic limit.

When the strain is above the elastic limit, the polymer network melts to form tiny particles embedded in the water. When the strain drops below the limit again, the polymer network may reform, but it will not return to its original shape. A cut is basically a boundary across which the polymer network remains broken.

To cut a polymer, a blade must press down into it, which strains the polymer. This results in water being expelled from the gel to form a liquid layer between the knife edge and the polymer network. This means that cutting a gel is a fluid dynamics problem.

It’s all in the boundary conditions

The key to understanding fluid dynamics problems is to figure out the boundary conditions. Essentially, all problems have edges, and if you know what is happening at the edges, you can use that knowledge to figure out what is happening everywhere else—though you might need a supercomputer to do the figuring.

In this case, we have a solid blade that is sliding along one axis while slowly moving downward to slice. This is equivalent to the blade being held fixed and the gel being moved, which is how the researchers set up their solution. The extremely thin layer of liquid in contact with the solid blade moves with the blade (this is called the no-slip condition). Elsewhere, the gel starts to behave elastically again exactly where the strain equals the critical strain. At that location, the fluid is moving with the gel rather than the blade.

The solution will show a smoothly increasing fluid/gel velocity from the blade to the strain boundary. Likewise, the strain, carried by the viscosity of the water, will reduce from the blade out to the critical strain boundary. So where exactly is that location? That depends on how fast the gel is moving (or in reality, how fast the blade is moving horizontally). Furthermore, the cutting velocity (the speed at which the blade moves vertically into the gel) depends on the speed at which the gel parts.

That makes the boundary condition, in one sense, well-defined. But in another sense, it’s rather fluid.

The researchers were able to draw an immediate conclusion based on these boundary conditions: a gel requires that the blade is drawn (e.g., slicing), and purely downward pressure (dicing) is generally not sufficient to initiate a cut (witness exploding tomatoes after they’re given to a kid to cut). Now, the skin of a tomato changes that, but the model does not include the skin, so don’t take the tomato example too literally.

The researchers’ numerical solution revealed that the latter boundary condition (a smooth interface where the strain is on the threshold of exceeding the elastic maximum) is only nicely defined when the knife cuts downward at less than 24 percent of the speed at which it saws horizontally. At these cutting speeds, the cut boundary is a constant distance from the edge of the knife, and both propagate smoothly into the gel. At higher speeds, the boundary still exists, but it fluctuates continuously, sometimes ballooning out from the knife edge and sometimes collapsing in toward the knife edge. In other words, when the downward speed is too high in comparison with the draw speed, the cut result becomes unpredictable—the tomato explodes.

Simplicity at its finest

The most amazing part of this work is its simplicity. OK, fluid dynamics is in no way simple, but the alternative is understanding crack propagation. This is the idea that at some random location—usually a microscopic defect—a material begins to fracture. The fracture propagates in a random way, using up the strain energy acquired from the knife. Where will that crack go, how long will the crack be, how many cracks will form? These are all random processes that are enough to drive anyone to distraction.

This solution doesn’t rely on the randomness of crack formation and propagation, so it’s much easier to test quantitatively. It should be possible to visualize the formation of the fluid layer, the melting of the polymer, and to do all of this at a range of different cutting speeds.