Capillary Number (Viscous Stress against Surface Tension)

The capillary number is the odd one out on these pages, and the reason is that it has no length in it. $Ca = \mu v/\sigma$ compares a viscous shear stress scale directly against a surface tension stress scale, and the two lengths that would appear — the one setting the shear rate and the one setting the interface curvature — cancel if you assume they are the same. That assumption is doing real work, and it is why $Ca$ can be quoted without the endless "which diameter?" argument that follows Reynolds and Weber around. It is a genuine convenience and it deserves to be recognised as one.

The physics is a contest at a moving contact line or a moving interface. Viscosity drags the interface out of the shape surface tension wants. At very low $Ca$ the interface wins completely: contact angles sit at their static values, an interface in a pore holds its equilibrium curvature, and a plate withdrawn slowly from a bath comes out essentially dry. As $Ca$ rises, dynamic contact angles depart from static ones, films thicken, and eventually the flow simply drags the interface wherever it likes.

The best-known result is Landau and Levich's: a plate withdrawn from a bath carries a film of thickness proportional to $\ell_c\,Ca^{2/3}$, where $\ell_c$ is the capillary length. That two-thirds power is why a coating line's thickness is so much less sensitive to speed than intuition suggests — doubling the line speed thickens the film by only about 60 %. It also caps how thin you can go by slowing down, which is why real coaters meter rather than rely on withdrawal.

In porous media the same group governs whether trapped oil moves. A blob of oil sitting in a pore is held by the capillary pressure of the constrictions at either end; the viscous pressure gradient of the passing water tries to push it through. The ratio is $Ca$, and the residual saturation curve — the famous capillary desaturation curve — is flat until $Ca$ reaches somewhere around $10^{-5}$ to $10^{-4}$ and then falls steeply. Because you cannot raise the water viscosity or the velocity by four orders of magnitude, the only practical lever is $\sigma$, and that is precisely what surfactant flooding does: drive the oil–brine interfacial tension from a few mN/m toward $10^{-3}$ mN/m and $Ca$ rises by decades without changing anything else.

Two conventions to pin down before comparing to anyone's chart. Which viscosity — in a two-phase displacement it is conventionally the displacing phase, with the mobility ratio between the phases treated as a separate parameter, and getting this backwards can be a factor of a hundred. Which tension — the tension at the interface actually in question, which for oil against brine is nothing like the 72 mN/m of water against air.

And a caution that bites hardest here. Most coating and displacement liquids are not Newtonian, so $\mu$ is a reading at one shear rate rather than a property. A shear-thinning coating is far less viscous in the metering nip than in the pan. Take $\mu$ at the shear rate the process actually imposes, and treat the answer as an estimate of an estimate — which, on these pages, it always was.