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Rheology, Math, and Coating Equipment Design

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Rheology again? The science of rheology is critical in so much that goes into fluid coating that we could spend a lifetime discussing the details regarding a particular flow regime and still forget to worry about viscoelastic functions that could screw up the entire system. Since this is such an important area of study, I thought we would go beyond what was discussed in the past (here:, here:, and here: and move into a path of understanding how rheology impacts us on a daily level.

Rheology is the study of flow. In our case we are interested in fluid flow of materials between 1-100,000 cP for most coated products in the roll-to-roll web industry. The math behind rheology is explaining the function of applied stress (shear) to a fluid to deform, move and alter the flow of fluid into a prescribed path. Coating equipment design is the application of the mathematical formulas of rheology to a fluid or set of fluids to create a precision engineered machine. That’s where all the theory becomes useful! As long as the theory matches the reality, everyone is happy. So the key is to constantly work towards an accurate design model based off a mathematical basis and empirical data to develop correction factors or rework the mathematical functions if they fail.

There are two main regimes of rheological fluids: Newtonian and non-Newtonian. Newtonian fluids behave according to more simplified mathematical functions when stresses are applied to them. Polymeric (non-Newtonian) fluids, however, behave more erratically when the same said stresses are applied. Even more so if the polymeric fluids act like rubber bands (are viscoelastic). The world of Newtonian fluids allows treatment in the mathematical formulas of viscosity as a constant. Polymeric fluids treat viscosity as a function with different behavior over the entire range of shear forces that act on the fluid. The range that needs to be studied can be as much as 1-10,000 Hz of shear stress. This complex study of rheological shear requires a more robust testing method than the standard one-point viscometer available in most coating locations. When designing equipment for a polymeric fluid, the mathematical formulas require a flow curve of viscosity versus shear stress instead of a single number.

To obtain the range of shear stress required in complex coating equipment, a rotational rheometer is required. The information that comes out a shear stress sweep of a particular fluid provides the shear stress versus viscosity data that helps equipment designers move forward. To be clear, the starting point for the rheological understanding of a fluid is viscosity:


Where h is viscosity, τ is the shear stress, and γ is shear rate. So if the viscosity varies with the stresses applied, the fluid is complex and cannot be treated as a constant. The main functions used for applying rheology to coating equipment design are the Modified Cross model:

h= ho/(1+(ho/τ*γ)^(1-n))

and the Carreau model:

h= ho/(1+ho/τ*γ)^(1-n)

With these basic equations, the world of rheology provides a path for product and process development to work in concert to develop successful fluid coating applications. The complex viscosity models allow for an understanding of how a fluid moves from an area of constriction (a feed pipe) to an area of relaxation (a distribution manifold) to a delivery area of restriction (slot die lip opening). Once the fluid distributes on the substrate the leveling effects, surface energies, and tensions of the fluid and substrate take over to develop the final coated product.

Success in the world of fluid coating depends on understanding the intricacies of complex rheological behavior, the associated mathematical formulas, and the application of this understanding to the development of coating equipment technology. The dance between the product research (fluid formulation) and the process development would move to the music more systematically with a third partner – a rheologist who can help get the rheology right.

If you are interested in discussing this concept further, contact Mark D. Miller, Founder and CEO of Coating Tech Service, LLC ( at This email address is being protected from spambots. You need JavaScript enabled to view it. or (612) 605-6019.


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