Basics of Fluid Mechanics, 2014a

4.4. FLUID IN A ACCELERATED SYSTEM 93
4.4 Fluid in a Accelerated System
Up to this stage, body forces were considered as one-dimensional. In general, the
linear acceleration have three components as opposed to the previous case of only
one. However, the previous derivations can be easily extended. Equation (4.8) can
be transformed into a different coordinate system where the main coordinate is in the
direction of the effective gravity. Thus, the previous method can be used and there
is no need to solve new three (or two) different equations. As before, the constant
pressure plane is perpendicular to the direction of the effective gravity. Generally the
acceleration is divided into two categories: linear and angular and they will be discussed
in this order.
4.4.1 Fluid in a Linearly Accelerated System
For example, in a two dimensional system, for the effective gravity
where the magnitude of the effective gravity is
and the angle/direction can be obtained from
g eff = a î + g ˆk (4.81)
|g eff | = √ g 2 + a 2 (4.82)
tanβ = a g
(4.83)
Perhaps the best way to explain the linear acceleration is by examples. Consider
the following example to illustrate the situation.
Example 4.9:
A tank filled with liquid is accelerated at
a constant acceleration. When the acceleration
is changing from the right to the
left, what happened to the liquid surface?
What is the relative angle of the liquid
surface for a container in an accelerated
system of a =5[m/sec]?
Solution
27.1
g
a
⎡
m
⎣
5
sec
g eff
Fig. -4.13. The effective gravity is
for accelerated cart.
⎤
⎦
This question is one of the traditional question of the fluid static and is straight forward.
The solution is obtained by finding the effective angle body force. The effective angle
is obtained by adding vectors. The change of the acceleration from the right to left is