Viscosity

Introduction and Application in Paint Flow & Pigment

Dispersion

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Background

Viscosity is a measure of a fluid’s resistance to flow. Think of viscosity as a coefficient that relates the stress applied to a fluid and the fluids response. For example:

where is the stress (Pa), is the

resulting velocity gradient in the fluid, and is the viscosity (sometimes referred to as shear viscosity in this context).

This basaltic lava flow emanating from a very small vent on Mt. Etna is about as viscous as a thick salsa.

Photo by L. Connor

What is viscosity?

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Background

Consider a fluid trapped between two plates. When a stress is applied to the upper plate while the lower plate is held still, a vertical velocity gradient is created in the fluid. This velocity gradient is equivalent to a strain rate. Since

the units of viscosity are Pascal seconds (Pa s).

In order to explain shear rate, first there are a few items that need to be discussed:

• Reynolds Number• Turbulent and Laminar Flow• Newtonian and non-Newtonian fluids• Fountain Flow.

What are the units of viscosity?

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2Figure from Mader, 2006, Volcanic processes as a source of statistical data, In: Mader et al., (eds) Statistics in Volcanology, Geological Society of London, 1-14.

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The units of Viscosity are:

Pascal.second [Pa.s] in SI, Poise in CGS

1 poise = 100 centipoise (cps)1 poise = 0.1 Pa·sec1 poise = 0.0671969 lb/(ft·sec)1 poise = 4.031814 lb/(ft·min)

Pa = Pascal = N·m pressure, stress

stoke = ɳ (poise)/ρ(g/cm)

Viscosity Units

Reynolds Number

When discussing fluid flow, there are two distinct types of flow:•Turbulent and Laminar (there is also transitional flow which occurs between the two distinct types)

Turbulent flow is very random and there is a lot of mixing that takes place within the molecules.

Laminar flow is very ordered, the molecules move in layers.

The Reynolds number is a dimensionless number that is used to describe the type of flow that is occurring.

Reynolds Number

The Reynolds Number (Re) of a fluid moving through a round channel is equal to;

Re = density x velocity x diameter = ρ .v . d viscosity µ

If a fluid has a Re of:2300 > »Laminar Flow2300-4000 »Transitional Flow> 4000 »Turbulent Flow

Turbulent flow provides a higher degree of mixing and a more consistent temperature across the flow channel than laminar flow.

In applications where we use water flow to cool tooling, we want turbulent flow in order to get better heat transfer.

Faster Flow – less drag

Frictional Drag

Frictional Drag

Laminar Flow

While Turbulent flow is very chaotic an there is a lot of mixing that takes place, Laminar flow is much smoother and there is a difference in velocity across the flow channel.

The molecules at the channel wall are flowing more slowly than the molecules in the middle due to frictional drag.

Laminar flow is sometimes envisioned as a series of plates or layers that flow across each other.

The shear rate is determined by the velocity divided by the thickness of the laminate or:

γ = v/h

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Types of viscosity

Newton's law of viscosity, is a constitutive equation (like Hooke's law, Fick's law, Ohm's law). It is not a fundamental law of nature but an approximation that holds in some materials and fails in others.

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Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. Thus there exist a number of forms of viscosity:Newtonian: fluids, such as water and most gases which have a constant viscosity.Shear thickening: viscosity increases with the rate of shear.Shear thinning: viscosity decreases with the rate of shear. Shear thinning liquids are very commonly, but misleadingly, described as thixotropic.Thixotropic: materials which become less viscous over time when shaken, agitated, or otherwise stressed.Rheopectic: materials which become more viscous over time when shaken, agitated, or otherwise stressed.A Bingham plastic is a material that behaves as a solid at low stresses but flows as a viscous fluid at high stresses.A magnetorheological fluid is a type of "smart fluid" which, when subjected to a magnetic field, greatly increases its apparent viscosity, to the point of becoming a viscoelastic solid.

Forms of Viscosity

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Viscosity Coefficients

Dynamic viscosity, also absolute viscosity, the more usual one (typical units Pa·s, Poise, P);Kinematic viscosity is the dynamic viscosity divided by the density (typical units cm2/s, Stokes, St).Viscosity is a tensorial quantity that can be decomposed in different ways into two independent components. The most usual decomposition yields the following viscosity coefficients:Shear viscosity, the most important one, often referred to as simply viscosity, describing the reaction to applied shear stress; simply put, it is the ratio between the pressure exerted on the surface of a fluid, in the lateral or horizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referred to as a velocity gradient).Volume viscosity (also called bulk viscosity or second viscosity) becomes important only for such effects where fluid compressibility is essential. Examples would include shock waves and sound propagation. It appears in the Stokes' law (sound attenuation) that describes propagation of sound in Newtonian liquid.Alternatively,Extensional viscosity, a linear combination of shear and bulk viscosity, describes the reaction to elongation, widely used for characterizing polymers.

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Newton’s Law of Viscous Flow

Examples of non–Newtonian fluids are:· polymer solutions and melts· slurries of ore, sand, coal· cement, asphalt· paint, ketchup, peanut butter· blood, saliva

Newtonian and non-Newtonian Fluids

If ɳ is constant the fluid is called Newtonian. If ɳ is not constant and depends on the gradient then the fluid is non-Newtonian.

Water is a Newtonian fluid.

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Background

Any fluid, including the solvent in paint industry is considered Newtonian if there is a linear relationship between stress and strain. That is, if:

Newtonian and non-Newtonian viscosity

In a non-Newtonian fluid, the above linear relationship does not hold true. For example, in a Bingham fluid:

where o is a yield stress (also called yield strength) required to “get the fluid moving”. Although still linear, a new term is added to the linear equation.

high slope, high viscosity

low slope, low viscosity

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Background

Viscosity is highly dependent on temperature. Think of a pool of molten glass. As the temperature drops, it will take more and more applied stress to make the glass flow.

Temperature and Viscosity

An Arrhenian model of viscosity is one in which viscosity is exponentially dependent on temperature:

where o is viscosity under standard temperature conditions, E is the activation energy, R is the universal gas constant, and T is temperature.

( ) expTE

RTo

This glass is an alkaline silicate melt with low viscosity at 1000 °C and very “workable” into shapes at about 800 to 900 °C. At about 700 °C, the viscosity is high enough for the shape to not deform under its own weight.

Learn More about the Arrhenian model

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More about the Arrhenian Model

The Arrhenian model is named for Svante Arrhenius, who developed a method of predicting the increased speed of a chemical reaction with increased temperature. His equation has the form:

Arrhenian models are also common in statistics and used to predict the higher rate of failure of just about anything at higher temperatures. For more about the statistical application of the model, see:

k AE

RT

exp

where k is the rate coefficient that describes how much faster the reaction will proceed, A is the Arrhenius coefficient, which varies with the specific chemical reaction, E is the activation energy, R is the gas constant, and T is temperature.

Note that this is the exact form of the Arrhenian viscosity model, with A replaced by the viscosity of the fluid at some standard temperature condition.

More about E,R,T

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Pseudoplastic and Dilatant Flow

Pseudoplastic and Dilatant • Coating that decrease in viscosity with

increase of shear stress (shear-thinning).• Coating that increase in velocity with

increase of shear stress is called dilatant.• If h decreases as the shear force on the

fluid increases then the fluid is called thixotropic.

• paint, tomato ketchup, toothpaste, peanut butter

• This is the more common behaviour because the shear force destroys the weak secondary (van der Waals) forces which hold the molecules together.

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Why is Rheology Important for Paint Industries

Mixing•Pigment Dispersion•PumpingStorage•SettlingApplication•Spray•Dip•Flow coat•Roller coat•BrushFilm formation•Flow and levelling•Coalescence

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Paint Application

Brush Application•Shear thinning-easy brushing•low resistance•Flow and Leveling - recovery•Sagging - high low shear viscosity•Open time - lapping•Settling - thixotropic

What Effects Spray Performance•Paint Viscosity (Elongational)•Surface tension•Shear thinning•Thixotropy•Pseudo plastic flow•Solvent evaporation•Thixotropy•Sagging•Flow and Leveling

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Common Viscosity Measurement