We now turn our attention to flows in silicates, including mantle convection, volcanic eruptions and sheet flow of lava on the surface. Like flows in the atmosphere and oceans, the star of these flows is the Navier–Stokes equation, with the continuity equation remaining a co-star, but we now have a much different cast of supporting characters. Rotation, inertia and turbulence, which affect most atmospheric and oceanic flows, are completely negligible in the force balances for silicates. This is good news, because much of the difficulty in analyzing those flows stems from the nonlinear inertia terms; on the face of it the laminar, linearized Navier–Stokes equation would seem to be much easier to solve. However, two new characters that intrude on this idyllic scene are

Each of these brings nonlinearity into the formulation; the first introduces a velocity nonlinearity while second couples the Navier–Stokes and energy equations. The energy equation contains, among other factors, a nonlinear advective term, presenting us once again with a set of challenging equations to analyze and solve.

The set of equations governing silicate flows is summarized in the following chapter. Following this, the equations are applied to two rather disparate flow problems: mantle convection and volcanic flows. Earth's mantle convects in order to cool both itself and core. The mantle is cooled by the introduction of cold lithospheric slabs, as explained in Chapter 31, while heat from the core is conveyed upward by discrete plumes, as explained in Chapter 32. Changing gears, Chapter 33 presents an overview of volcanic flows. Then flow in a volcanic conduit is briefly investigated in Chapter 34 and lava sheet flow on the surface is investigated in Chapter 35.