### Summary

Students are introduced to the similarities and differences in the behaviors of elastic solids and viscous fluids. Several types of fluid behaviors are described—Bingham plastic, Newtonian, shear thinning and shear thickening—along with their respective shear stress vs. rate of shearing strain diagrams. In addition, fluid material properties such as viscosity are introduced, along with the methods that engineers use to determine those physical properties.

### Engineering Connection

### Educational Standards

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### Pre-Req Knowledge

### Learning Objectives

- Describe the similarities and differences between elastic solids and viscous fluids.
- Explain four different types of fluid behavior: Newtonian, shear thinning, shear thickening, and Bingham plastic.
- Demonstrate an understanding of how engineers measure and calculate fluid material properties such as viscosity.
- Communicate scientific information about why the molecular-level structure is important in the functioning of designed materials.

### Introduction/Motivation

### Lesson Background and Concepts for Teachers

*τ*(tao) is the shear stress in the fluid,

*μ*(nu) is the viscosity, and

*du/dy*is the shear velocity of the fluid. The shear stress of a fluid is defined in a similar manner as stress in a solid: force divided by area. The above equation is very similar to the Hooke's law equation (discussed in the Mechanics of Elastic Solids lesson):

*σ*(sigma) is the stress in the solid,

*E*is Young's modulus, and

*ε*is the strain that the solid experiences. In each equation, the stress in the material (caused by a force on the material) is equal to a material property (Young's modulus or viscosity) multiplied by either the strain or velocity of the material, which tells something about the response of the material to the force (either moving the material or deforming it). Therefore, the Young's modulus and viscosity are similar in that they both measure a material's resistance to deformation (or movement).

- For example, at your neighborhood gas station, the pumps are designed to measure the volume of gasoline being purchased. By knowing the viscosity of the fluid and the force being applied to it from the gas pump, engineers can calculate the velocity that the gas will move. Using this information, along with the dimensions of the gas nozzle, the amount of gas being purchased can be calculated.
- For example, if engineers know the viscosity of printer ink and what velocity they want the ink to move, they can design a printer so that the correct amount of force is applied to the ink.
- For example, for the mass production and packaging required in the food and beverage industry, knowing the viscosity of the fluids to be packaged (think milk vs. molasses) gives engineers the information they need to design factory equipment that regulates how fast a fluid can be packaged based on the tolerable forces that can be applied to the fluid.

Measuring Viscosity

- For Newtonian fluids, engineers place the fluid in a container and drop a ball of known mass and volume into the container. By measuring the amount of time it takes the ball to travel through a specified distance of the fluid, they can calculate the resistance the ball experiences through the fluid. (This is similar to the forces a skydiver experiences when jumping out of an airplane. At some point, the force of air resistance matches the force of gravity and the skydiver reaches terminal velocity—the point at which the skydiver no longer accelerates and reaches a constant velocity.) For the ball with a known mass and shape, calculating the force of gravity on the ball is easy. This force must balance with the force of shear resistance (viscosity) and dictates the ball's speed (velocity). So if an engineer can measure the speed of the ball, s/he can directly predict the viscosity of the fluid! (Students investigate this method in more detail during the associated Measuring Viscosity activity.)
- The second method for determining viscosity, rheometry, is very expensive and typically used only on fluids that are not Newtonian. A rheometer (see Figure 2) can either control the velocity of a fluid and measure the force it takes to apply that velocity, or apply a force and measure the resulting velocity. Using either method, engineers acquire the force and velocity data needed to use the viscosity equation and calculate the viscosity of the fluid. In the testing machine, the fluid is placed either in a cylinder or on a plate, and different probes are used to apply force to the fluid (Figure 2-right). The probe can vary in geometry, depending on the fluid viscosity. High-viscosity fluids are placed on plates with either a cone or another flat plate used to apply the force (Figures 2b, 2c). All other fluids, especially low-viscosity fluids, require a cylinder configuration (Figure 2a).

Fluid Behavior

*τ*) vs. rate of shearing strain diagrams (

*du/dy*). The shear stress is calculated using the force data, and the rate of shearing strain is calculated using the deformation data. This is similar to a stress-strain diagram with solids. When engineers test solids and generate stress-strain diagrams, they calculate the slope of the initial line (covered in more detail in the Mechanics of Elastic Solids lesson), which is equal to the Young's modulus or stiffness of the material. With fluids, engineers also calculate the slope of the line formed on the shear stress-rate of shearing strain diagram. This value is equal to the viscosity of the fluid.

**Bingham plastic**materials behave as solids at low stresses, but flow as viscous fluids at high stresses. Because the particles in these materials have weak bonds, at high stresses they break, causing them to flow and be characterized as fluids. When the stress is relieved, the bonds form again, characterizing the materials as solids. Two material properties are needed to describe this material: viscosity and yield stress. The slope of the shear stress-rate of shearing strain diagram is the viscosity (as described above) and the intersection of the y-axis (shear stress axis) is the yield stress. The yield stress defines the transition point between solid and liquid.

- A common example of this fluid type is toothpaste.

**Newtonian fluids**are identified by linear plots in the diagram, which means that these fluids have constant viscosities that are independent of velocity (rate of shear). Regardless of how fast or slow you stir these liquids, they always require the same proportional forces.

- Everyday examples of this fluid type include: water, gasoline and most gases (remember gases are fluids as well!).

**shear thinning**materials, viscosity decreases as velocity (rate of shear) increases. As you stir this type of fluid faster, it becomes much easier to stir. While scientists do not fully understand the cause of this phenomenon, engineers have used fluids with this behavior to their advantage.

- For example, paint is a shear thinning fluid. It is easy to adhere on a roller because of the increase in velocity the roller imposes on the fluid. However, once the paint is applied to the wall and the force on the fluid is reduced, the viscosity increases to its original state and the paint stays on the wall without dripping.
- Another example is whipped cream. Engineers used its characteristics to their advantage when designing pressurized can containers for easy dispensing of whipped cream. When a force is applied to this fluid, its viscosity decreases and it flows smoothly like a liquid out of the nozzle. Once it comes to a rest on your tasty treat, it becomes rigid again (increased viscosity), like a solid.
- Additional common examples include ketchup, blood and motor oil.

**shear thickening**materials, viscosity increases as velocity (rate of shear) increases. As you stir this type of fluid faster, it becomes much harder to stir. This is due to closely packed particles combined with just enough fluid to fill the spaces between them. At low velocities, the fluid dominates the behavior and is able to continue to adequately fill the spaces between the particles because they are not moving fast. At high velocities, the fluid cannot keep up with the particle movement and is unable to fill the spaces between them, so the particles to rub against each other creating friction between them. Engineers have also used this phenomenon to improve our lives.

- One example is body armor. The fluid in body armor reacts to sudden forces (increases in velocity, such as bullets) and immediately increases its viscosity, which in turn stops the blow (the bullets). The only caveat to this is that slow velocities (like a knife) do not produce this change in viscosity. To address this vulnerability, an additional material (Kevlar fabric) is added to body armor to protect against these types of attacks. The combination of Kevlar and a shear thickening fluid performs better at protecting than Kevlar alone. The fluid-Kevlar combination body armor is also one-third the thickness of body armor containing only Kevlar, so it is more lightweight and comfortable to wear.
- Another innovative design using shear thickening fluids is found in vehicle traction control, which is a system used for all-wheel drive vehicles that reacts to the differences in motion between the front and rear wheels. When the vehicle has sufficient traction, the front and rear wheels have similar motion, so no shear force is applied to the fluid. However, when the primary drive wheels begin to slip, the difference in motion between the front and rear increases, applying a shear force to the fluid and resulting in an increase in viscosity. This viscosity increase applies torque to the secondary drive wheels, creating a system in which all four wheels become engaged only when needed.
- Another example is cornstarch in water; see the Additional Multimedia Support section for the link to a fun online video that demonstrates its behavior in response to different forces.

### Vocabulary/Definitions

Newtonian fluid: |
A fluid with a viscosity that is independent of its velocity (rate of shear). |

strain: |
Deformation per unit length. |

stress: |
Force per unit area, or intensity of forces distributed over a given section. |

torque: |
A force that causes an object to rotate. |

velocity: |
Speed (and direction) of an object. |

viscosity: |
A measure of the resistance of a fluid to shear stress. |

Young's modulus: |
A measure of the stiffness of a material. |

### Associated Activities

- Measuring Viscosity - Students calculate the viscosity of various household fluids by measuring the amount of time it takes marble or steel balls to fall given distances through the liquids. They experience what viscosity means, and also practice using algebra and unit conversions.

### Lesson Closure

### Attachments

### Assessment

*Worksheet:*After presentation of the lesson content, have students complete the attached Viscosity Worksheet. Review their answers to gauge their mastery of the subject matter.

### Contributors

Brandi N. Briggs, Michael A. Soltys, Marissa H. Forbes

### Copyright

© 2011 by Regents of the University of Colorado.

### Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

### Acknowledgements

Last modified: March 2, 2015