Next: The Gradient Operator,
Up: Introduction
Previous: Scalar and Vector Products
In one dimensional problems, if the applied force is , we solve
In three dimensional situations we have, in cartesians coordinates,
we have
This is written in concise form by using vector notation so that

(1.11) 
In a continuuous medium, a variation of pressure, , from
one location to another produces a pressure force. is the force
per unit area exerted on a surface by a fluid (liquid or gas) or
plasma. Thus, if the force on a face is . This is
illustrated in Figure 1.5.
Figure 1.5:
The pressure force acting on a face of area .

Consider a fluid element, lying between the points and . On the left hand side the pressure force is
and on the right hand side the force is
The minus sign arises because the force on the right hand side acts
towards the left. Therefore, the net force in the direction
PER UNIT VOLUME is
Thus, the equation of motion (1.11) per unit volume for a
liquid, gas or plasma of density (mass per unit volume) becomes

(1.12) 
Pressure gradients produce a force in the opposite direction to the
actual gradient. Thus, the pressure gradient force acts from areas of
high pressure to areas of low pressure and is equal in magnitude to
the gradient of the pressure. This is illustrated in Figure
1.6.
Figure 1.6:
A sketch of pressure as a function of and the direction
of the force in relationship to the isobars.

In three dimensional problems, where
, (1.12)
generalises to give three components as
or in vector form

(1.13) 
Next: The Gradient Operator,
Up: Introduction
Previous: Scalar and Vector Products
Prof. Alan Hood
20001106