Basics of Fluid Mechanics, 2014a

CHAPTER 4
Fluids Statics
4.1 Introduction
The simplest situation that can occur in the study of fluid is when the fluid is at rest or
quasi rest. This topic was introduced to most students in previous study of rigid body.
However, here this topic will be more vigorously examined. Furthermore, the student
will be exposed to stability analysis probably for the first time. Later, the methods
discussed here will be expanded to more complicated dynamics situations.
4.2 The Hydrostatic Equation
A fluid element with dimensions of DC,
dy, and dz is motionless in the accelerated
system, with acceleration, a as
shown in Figure 4.1. The system is in a
body force field, g G (x, y, z). The combination
of an acceleration and the body
force results in effective body force which
is
y
P
⎛
⎜
⎝P + ∂P
⎞
∂y dy ⎟
⎠ dxdz
dx
z
P
x
dz
⎛
⎜
⎝P + ∂P
dy
⎞
∂z dz ⎟
⎠ dxdy
⎛
⎜
⎝P + ∂P
⎞
∂x dx ⎟
⎠ dydz
Fig. -4.1. Description of a fluid element in accelerated
system under body forces.
g G − a = g eff (4.1)
Equation (4.1) can be reduced and simplified for the case of zero acceleration, a =0.
In these derivations, several assumptions must be made. The first assumption
is that the change in the pressure is a continuous function. There is no requirement
that the pressure has to be a monotonous function e.g. that pressure can increase
and later decrease. The changes of the second derivative pressure are not significant
compared to the first derivative (∂P/∂n × dl >> ∂ 2 P/∂n 2 ). where n is the steepest
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