However, not all fluids obey a Newtonian stress-strain relationship. An enormous variety of ``visco-elastic'' fluids exist that obey more complicated constitutive relationships, such as nonlinear relationships, similar to plastic deformation of solids, or history effects, where the stress history needs to be known before the deformation can be predicted. Such fluids are commonly encountered in the plastics and chemical industries.
Flows inside ducts, channels and pipes are very important because they occur in many practical applications (water pipes, air conditioning ducts, gas lines, ventilation shafts, heat exchanger tubes, etc.). Friction is usually important in these flows because there is a resistance to relative motion: when one layer of fluid is moving with respect to an adjacent layer, there exists friction between the layers. The amount of friction depends on the fluid viscosity and the velocity gradient (that is, the relative velocity between fluid layers). The velocity gradients are set up by the no-slip condition at the wall. When a fluid is in contact with a solid surface, there can be no relative motion between the fluid in contact with the solid surface and the surface itself: if the wall has zero velocity, then the fluid in contact with the wall has zero velocity also.
To see how the no-slip condition arises, and how the no-slip condition and the fluid viscosity lead to frictional stresses, we can examine the conditions at a solid surface on a molecular scale. When a fluid is stationary, its molecules are in a constant state of motion with a random velocity v. For a gas, v is equal to the speed of sound. When a fluid is in motion, there is superimposed on this random velocity a mean velocity V, sometimes called the bulk velocity, which is the velocity at which fluid from one place to another. At the interface between the fluid and the surface, there exists an attraction between the molecules or atoms that make up the fluid and those that make up the solid. This attractive force is strong enough to reduce the bulk velocity of the fluid to zero. So the bulk velocity of the fluid must change from whatever its value is far away from the wall to a value of zero at the wall (figure 7). This is called the no-slip condition.
|Figure 7. Velocity profile in a boundary layer.|
|Figure 8. Flow in a channel, showing boundary layers near the walls.|
|Figure 9. Flow over an airfoil, showing boundary layers near the surface, and the formation of a wake.|
Since friction is always present when there is relative motion between fluid layers, and because boundary layers are always formed near solid surfaces, a certain amount of energy is continually given up as heat when a body moves through a fluid. The force acting on a body due to the viscous resistance of the surrounding fluid is called the drag force and it acts in the direction opposite to the direction of motion. As a consequence of the fluid friction, the entropy always rises, and the flow is not reversible.