## Fighting Viscous Forces to Improve Fluid Coating Success

- Published: September 18, 2014, By Mark Miller

When it comes to slot die fluid coating, the mathematical equations that dictate flow are fundamental, but the details are complicated. The goal with understanding the math should be not to make things more complicated, but to simplify the understanding and lead engineers and operators towards common controls for adjusting at the coating line. So let’s do just that…

The basic equations that control fluid flow are Newton’s second law of motion (equation of change of velocity) and conservation of mass (equation of change of density). These equations present the relationship of viscous forces to internal fluid criteria (rheology, density, etc.). If you understand the interaction between the fluid you have chosen to coat and the forces involved, you will be able to predict behavior in the coating process. **Competing forces are the main ingredient to understanding fluid flow! **

For the equations that we are going to discuss, the nomenclature is-

• Viscosity = μ

• Substrate speed = U

• Density = ρ

• Wet film thickness = H

• Capillary pressure = σ

• Gravity = g

• Bulk modulus = K

• Relaxation time = λ

The ratio of inertial forces to viscous force indicates the magnitude of force of flow versus the resistant force of viscosity – this is referred to as the Reynolds number:

**Re = ρUH/μ **

When the Reynolds number is below 2300, the flow is laminar. When the Reynolds number is above 4000, the flow is turbulent. This is important because laminar flow allows for consistent two dimensional flow analysis and less opportunity for coating defects.

The ratio of viscous force to surface tension of a substrate indicates the force of flow versus the resistant adherence of the fluid to the substrate – this is referred to as the Capillary number:

**Ca = μU/σ **

The viscosity induced pressure gradient works opposite capillary pressure to determine the controlling force. Capillary flow is dominant when the Capillary number is ~10^{-5}. Viscous flow is dominant for high Capillary numbers. This is important because viscous flow is understood and predictable.

The ratio of gravitational forces to the net viscous force defines the Stokes number:

**St = ρgH ^{2}/μU **

When the Stokes number is much less than 1, particles follow fluid flow. If the Stokes number is much greater than 1, particles fall out of suspension and do not follow the fluid flow path. This is important for reasons of streak defects and agglomeration after fluid deposition onto the substrate.

The ratio of viscous stress to elastic stress in a bounded area defines the Elasticity number:

**El = μU/KH **

When the elasticity number is greater than 1, the viscoelastic component is more important than the Newtonian flow. Conversely, when the elasticity number is less than 1, the Newtonian flow component drives flow. This is important because the regime of rheological force is critical to the behavior of the fluid flow.

Given enough time, even a solid will flow. This phenomena is described by the Deborah number:

**De = λU/H **

Low numbers represent liquid flow that occurs more easily, while higher numbers show resistance to flow. This concept provides understanding of the stress relaxation time and the effect of flow resistance to defect formation.

With an understanding of the forces that control fluid flow at the exit of a slot die, you are better prepared to setup, analyze and implement quality fluid flow parameters. Happy coating!