# Viscosity

Written by Jerry Ratzlaff on . Posted in Thermodynamics

Viscosity ( $$\eta$$ (Greek symbol eta) ) is the measure of the internal friction/resistance to the flow of a liquid. Lower viscosity fluids flow easily in pipes where high viscosity fluids have a have a higher pressure drop.  Viscosity of fluids is typically temperature dependent and is not affected as dramatically by pressure as gas viscosity. In fact, typically as a liquid temperature increases, the velocity decreases. When a gas temperature increases, the viscosity increases!

There are two different types of viscosity, dynamic viscosity and kinematic viscosity. Higher the viscosity the more resistance to flow. Lower the viscosity the less resistance to flow. Temperature also plays a part, lower the temperature, higher the resistance. Higher the temperature, lower the resistance. A Viscometer meter can be used to measure viscosity.

## Coefficient of Viscosity

Coefficient of viscosity ( $$\eta$$ (Greek symbol eta) ) (also called viscosity coefficient) is the tangential friction force required to preserve a unit velocity gradient between two parallel layers of liquid of unit area.  The greater the coefficient of viscosity the greater the force required to move the layers at a velocity.

### Coefficient of Viscosity Formula

$$\large{ \eta = \frac { Fl } { Av } }$$

Where:

$$\large{ \eta }$$  (Greek symbol eta) = coefficient of viscosity

$$\large{ A }$$ = area

$$\large{ F }$$ = tangential force

$$\large{ l }$$ = distance between the layers

$$\large{ v }$$ = velocity

Solve for:

$$\large{ F = \eta \frac { Av } { l } }$$

## Dynamic Viscosity

Dynamic viscosity ( $$\mu$$ ) (also called absolute viscosity of simple viscosity) is the force required to move adjacent layers that are parallel to each other at different speeds.  The velocity and shear stress are combined to determine the dynamic viscosity.

### Dynamic Viscosity Formula

$$\large{ \mu = \frac {\tau} {\dot {\gamma}} }$$

Where:

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity

$$\large{ \tau }$$  (Greek symbol tau) = shear stress

$$\large{ \dot {\gamma} }$$  (Greek symbol gamma) = shear rate

## Kinematic Viscosity

Kinematic viscosity ( $$\nu$$ ) is the ratio of dynamic viscosity to density or the resistive flow of a fluid under the influance of gravity.

### Kinematic Viscosity Formula

$$\large{ \nu = \frac {\mu} {\rho} }$$

Where:

$$\large{ \nu }$$  (Greek symbol nu) = kinematic viscosity

$$\large{ \rho }$$  (Greek symbol rho) = density

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity

## Newton's Law of Viscosity

Newton's law of viscosity states that shear stress between adjacent fluid layers is porportional to the velocity gradients between the two layers.  The ratio of shear stress to shear rate is a constant for a given temperature and pressure.

### Newton's Law of Viscosity Formula

$$\large{ \tau = \mu \frac {d \nu}{dy} }$$

Where:

$$\large{ \tau }$$  (Greek symbol tau) = shear stress

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity

$$\large{ \frac {d \nu}{dy} }$$ = rate of shear deformation

## Reduced Viscosity

### Reduced Viscosity Formula

$$\large{ \mu_r = \frac {\mu_i} {c} }$$

Where:

$$\large{ \mu_r }$$  (Greek symbol mu) = reduced viscosity

$$\large{ c }$$ = mass concentration

$$\large{ \mu_i }$$  (Greek symbol mu) = relative viscosity increment

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity

$$\large{ \mu_s }$$  (Greek symbol mu) = viscosity of a solution used

Where:

$$\large{ \mu_i = \frac { \mu \;-\; \mu_s } { \mu_s } }$$

## Relative Viscosity

Relative viscosity ( $$\eta_r$$ ) is the ratio of the viscosity of a solution to the viscosity of a solution used.

### Relative Viscosity Formula

$$\large{ \eta_r = \frac {\eta} {\eta_s} }$$

Where:

$$\large{ \eta_r }$$  (Greek symbol eta) = relative viscosity

$$\large{ \eta }$$  (Greek symbol eta) = viscosity of a solution

$$\large{ \eta_s }$$  (Greek symbol eta) = viscosity of a solution used

## Viscosity Index

Viscosity index ( $$VI$$ ) is a measure of a fluid's sensitivity to change in viscosity with change in temperature.

### Viscosity Index Standards

• ASTM Standards
• ASTM D2116 - Standard Method for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity
• ASTM D2270 - Standard Practice for Calculating Viscosity Index from Kinematic Viscosity at 40 and 100°C
• ASTM STP168 - ASTM Viscosity Index Calculated from Kinematic Viscosity
• ASTM STP532 - Viscosity Testing of Asphalt and Experience with Viscosity Graded Specifications

## Viscosity of Slurry

The viscosity of slurry ( $$\eta_s$$ ), slurry being a mixture of liquids and solids, is comparitive to the viscosity of the liquid phase.

### Viscosity of Slurry Formula

$$\large{ \eta_s = \frac {\mu} {\eta_r} }$$

Where:

$$\large{ \eta_s }$$  (Greek symbol mu) = slurry viscosity

$$\large{ \eta_r }$$  (Greek symbol eta) = relative viscosity

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity

## Viscosity of a Mixture

This calculates the effective viscosity of gasses.

### Viscosity of a Mixture formula

$$\large{ \mu = \frac {\sum ({\mu_i y_i \sqrt{M_i}})} {\sum ({y_i \sqrt{M_i}})} }$$

Where:

$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity of the gas mixture

$$\large{ M_i }$$ = molecular weight of gas component

$$\large{ \mu_i }$$  (Greek symbol mu) = dynamic viscosity of a gas component

$$\large{ y_i }$$ = mole fraction or percent of gas component