An Introduction to Fluid Mechanics pp 273-377 | Cite as

# Incompressible Viscous Flows

## Abstract

Flows of viscous fluids are discussed in this chapter, in which the fluid viscosity is intrinsically important. For simplicity, fluid density is considered constant, and the focus is on the characteristics of incompressible viscous flows. First, a general formulation of the field equations for viscous flows is presented, and the vorticity equation is derived, which provides a useful perspective in describing viscous flows. The exact solutions to the full Navier-Stokes equation for selected problems are presented. The approximate solutions to the Navier-Stokes equation for low-Reynolds-number flows, in the context of Stokes’ approximation, are discussed for selected problems. Similarly, large-Reynolds-number flows are introduced in the context of boundary-layer theory and Prandtl’s boundary-layer equations. These are considered equally an approximation to the Navier-Stokes equation, and some exact solutions to the obtained boundary-layer equations are presented by using similarity methods. On the other hand, the momentum integral and the Kármán-Pohlhausen method are introduced as the approximate methods in solving the boundary-layer equations, with a discussion on the stability of boundary layer. Buoyancy-driven flows, which are induced essentially by density variation, are discussed in the context of the Boussinesq approximation to the Navier-Stokes and thermal energy equations. The solutions to the resulting equations are presented for some problems with simple geometric configurations. The stability of a horizontal fluid layer is explored to study the conditions of the onset of thermal convection.

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