Introduction to Chemical Engineering Fluid Mechanics

INTRODUCTION

In a static fluid, gravity causes the pressure to increase with depth. Pressures at fluid–fluid interfaces may be influenced also by surface tension. Both kinds of pressure variation have numerous practical consequences, as will be discussed in this chapter. An advantage of beginning a more detailed analysis of fluid mechanics with statics is that viscous stresses, which are the most difficult conceptually and mathematically, are absent in fluids at rest. Thus, we can concentrate first on pressure, gravity, and surface tension, and return later to a more general and precise description of viscous stresses. Quantifying momentum changes within fluids also can be deferred. Viscous stresses and inertia are considered in detail in Chapter 6.

This chapter begins the use of vector notation, and it is suggested that the reader review certain parts of the appendix before proceeding. Needed particularly is familiarity with vector representation (Section A.2), vector dot products (Section A.3), the gradient operator (Section A.4), and cylindrical and spherical coordinates (last part of Section A.5).

PRESSURE IN STATIC FLUIDS

Properties of pressure

Pressure is a force per unit area, making it a type of stress. It has three properties, whether or not there is flow.

Properties (i) and (ii) provide a mechanical definition of pressure and establish the direction of pressure forces when P > 0. Property (iii) is a consequence of (i), as shown at the end of this section. If (i) and (iii) seem contradictory, keep in mind that the pressure itself is a scalar, whereas the pressure force acting on a surface is a vector. The orientation of the surface determines the direction of the force vector, but not the value of P.

Static pressure equation

A differential equation that describes pressure variations in a static fluid will be derived using a force balance on the small fluid cube in Fig. 4.1.