So far in this course, you have been considering inviscid flows, that is flows in which the fluid viscosity is not important. In this chapter we introduce the effect of viscosity into the dynamics. Fluid viscosity is a friction effect internal to the fluid, and it arises due to layers of fluid sliding over each other during fluid motion. Including viscosity in our fluid model will lead to the famous Navier–Stokes equations, which are the governing equations for many fluid flows of practical interest. In many real life applications, the effect of viscosity completely changes the nature of the fluid flow from that which would be observed with an inviscid fluid, and this effect can be confined to a region of the fluid or can be throughout the entire flow.
Figure7.0.1.Steady, laminar flow of a viscous fluid through a uniform, circular pipe. The velocity profile is parabolic. A similar profile is found for flow between two parallel plates. Credit Fu, Liang, Zhang (Symmetry, 2020).
Figure7.0.3.Thin film flow / lubrication theory. If the thickness of a fluid layer is much less than the lengthscale in the other two dimensions, we may considerably simplify the governing equations. Credit J Mo Painting.
Even in some of these flows, the effect of viscosity is important, for example in the flow past an aerodynamic body there is a thin layer of fluid close to the body surface, called a boundary layer, in which viscous effects are important. Flows that are completely dominated by viscosity are called viscous flows, and these tend to include flows that are on a small spatial scale, have a low velocity scale, or are of a particularly viscous fluid, for example:
Many flows in internal physiology, for example the flow of blood through blood vessels, especially capillaries (the smallest vessels), the flow of mucus in the respiratory system, fluid flows within tissues, the flow of synovial fluid in joints. These are often on a small spatial scale and have a low velocity scale.
Several manufacturing processes, for example paint, adhesives or food products, since the spatial scales involved are not large and the fluids are often quite viscous.
