Gingerbread Mould Pressing

What is modelled

Pressing a disc of dough between two plates pushes the material outward in a thin layer. The resistance of such a thin layer is huge and grows fast as the layer thins, so the dough thins quickly at first and then more and more slowly. Dough is shear-thinning (it flows more easily when pushed harder) and has a yield stress, so it eventually stops squeezing. The pressure is highest in the centre and zero at the edge, so a carving near the centre fills better than one at the rim.

Equations

• Power-law squeeze under a force F: F = (2π K Rn+3/(n+3)) ((2n+1)/n)n (−dh/dt)n / h2n+1; for n = 1 this is Stefan's law.
• Pressure p(r) = p0 [1 − (r/R)n+1], centre value p0 = F (n+3) / (π R2 (n+1)).
• Yield stop: h = 2π τy R3 / (3F).
• Dough entering a slot of width w: stops at depth p w / (2 τy); the speed follows the power-law slot flow.

Checks run in the page

• The Newtonian Stefan law against a numerical integration.
• The power-law solution against a numerical integration.
• The pressure profile integrates to the pressing force.
• The cavity fill time by direct stepping against the integral of dx/u.

Honest simplifications

The dough is treated as a homogeneous power-law material with a yield stress added in an approximate way (a force offset); the plate is a flat disc with no slip, dough that leaves the disc is ignored, the dough does not stick, shrink or rise, and the carved cut is a straight slot with plug-like flow. The numbers are typical of a stiff paste, not a measurement of any recipe.

• Press Stiff dough and then Heavy press: the cut fills deeper only when the force is large enough.
• Press Deep narrow cut: a deep and thin cut needs far more pressure.
• Move the cavity position toward the edge and watch the fill drop with the pressure.