Chapter 20, Problem 55P

It is a common practice when examining a fluid's viscous behavior to plot the shear strain rate (velocity gradient)

$\frac{dv}{dy}=\stackrel{\dot{}}{\gamma }$

on the abscissa versus shear stress $\left(\tau \right)$ on the ordinate. When a fluid has a straight-line behavior between these two variables it is called a Newtonian fluid, and the resulting relationship is

$\tau =\mu \stackrel{\dot{}}{\gamma }$

where $\mu$ is the fluid viscosity. Many common fluids follow this behavior such as water, milk, and oil. Fluids that do not behave in this way are called non-Newtonian. Some examples of non-Newtonian fluids are shown in Fig. P20.55.

For Bing hamplastics, the reis a yields tress ${\tau }_y$ that must be overcome before flow will begin,

$\tau ={\tau }_y+\mu \stackrel{\dot{}}{\gamma }$

FIGURE P20.55

A common example is toothpaste.

For pseudoplastics, or “shear thinning” fluids, the shear stress is raised to a power n less than one,

$\tau =\mu {\stackrel{\dot{}}{\gamma }}^n$

Such fluids, such as yogurt, mayonnaise, and shampoo, exhibit a decrease in viscosity with increasing stress. Note that for cases where $n>1$, called dilatant (or “shear thickening”) fluids, viscosity actually increases with shear stress. Examples include starch in water and wet beach sand.

The following data show the relationship be tween the shear stress $\tau$ and the shear strain rate $\stackrel{\dot{}}{\gamma }$. The yield stress ${\tau }_y$ is the amount of stress that must be exceeded before flow begins. Find the viscosity $\mu$ (slope), ${\tau }_y$, and the $r^2$ value using a regression method. What is the type of fluid?

Stress $\tau ,{\text{N/m}}^{\text{2}}$	3.25	4.25	4.65	5.65	6.05
Shear strain rate $\stackrel{\dot{}}{\gamma },\text{1/s}$	0.9	2.1	2.9	4.1	4.9