Introduction to the Variational Formulation in Mechanics

11Incompressible Fluid Flow

11.1 Introduction

In this chapter we will tackle the construction of a model for incompressible fluid flow phenomena exploiting the variational framework presented earlier. This implies that we will make use of the Principle of Virtual Power (PVP) presented in Chapter 3 to find the variational equation that governs the problem of mechanical equilibrium. It is worth mentioning that, notwithstanding that the fluid flow problem is addressed here as a separated chapter, the variational structure comprises the very same ingredients already discussed in the mechanics of solid continua also seen in Chapter 3. In fact, from the kinematical perspective, the same kinematical descriptors will be utilized to characterize the motion actions that can be executed over particles composing the (fluid) continuum. A peculiarity to be discussed in detail here is that of the incompressibility constraint, which characteristically emerges in fluid mechanics problems at low Mach numbers.¹ We will underline the way in which this constraint enters into the variational formulation and the consequences of such a constraint.

Thus, this chapter differs from most of the literature in the fluid mechanics domain in that the derivation of the governing equations is realized through a purely variational formalism, instead of the standard approach based on (mass and momentum) conservation laws. As said before, the PVP provides a unified framework to model this problem and, unlike the ...