Fluid Mechanics-Fluid Statics

In this chapter we will consider an important class of problems in which the fluid is either
at rest or moving in such a manner that there is no relative motion between adjacent particles.
In both instances there will be no shearing stresses in the fluid, and the only forces that
develop on the surfaces of the particles will be due to the pressure. Thus, our principal concern
is to investigate pressure and its variation throughout a fluid and the effect of pressure
on submerged surfaces. The absence of shearing stresses greatly simplifies the analysis and,
as we will see, allows us to obtain relatively simple solutions to many important practical
problems.
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2
Fluid Statics
2.1 Pressure at a Point
As we briefly discussed in Chapter 1, the term pressure is used to indicate the normal force
per unit area at a given point acting on a given plane within the fluid mass of interest. A
question that immediately arises is how the pressure at a point varies with the orientation of
the plane passing through the point. To answer this question, consider the free-body diagram,
illustrated in Fig. 2.1, that was obtained by removing a small triangular wedge of fluid from
some arbitrary location within a fluid mass. Since we are considering the situation in which
there are no shearing stresses, the only external forces acting on the wedge are due to the
pressure and the weight. For simplicity the forces in the x direction are not shown, and the
z axis is taken as the vertical axis so the weight acts in the negative z direction. Although we
are primarily interested in fluids at rest, to make the analysis as general as possible, we will
allow the fluid element to have accelerated motion. The assumption of zero shearing stresses
will still be valid so long as the fluid element moves as a rigid body; that is, there is no relative
motion between adjacent elements.

Although we have answered the question of how the pressure at a point varies with direction,
we are now faced with an equally important question—how does the pressure in a fluid
in which there are no shearing stresses vary from point to point? To answer this question
consider a small rectangular element of fluid removed from some arbitrary position within
the mass of fluid of interest as illustrated in Fig. 2.2. There are two types of forces acting
on this element: surface forces due to the pressure, and a body force equal to the weight of
the element. Other possible types of body forces, such as those due to magnetic fields, will
not be considered in this text.
If we let the pressure at the center of the element be designated as p, then the average
pressure on the various faces can be expressed in terms of p and its derivatives as shown in
Fig. 2.2. We are actually using a Taylor series expansion of the pressure at the element center
to approximate the pressures a short distance away and neglecting higher order terms that
will vanish as we let and approach zero. For simplicity the surface forces in the x
direction are not shown. The resultant surface force in the y direction is