Mechanics of Fluids

92 The principles governing fluids in motion
Fig. 3.3
interpretation of the pattern formed by streamlines in steady flow: in regions
where the streamlines are close together the velocity is high, but where the
same streamlines are more widely spaced the velocity is lower. This conclusion,
which applies to incompressible fluids, does not necessarily apply to
the flow of compressible fluids in which large density changes occur.
3.4 BERNOULLI’S EQUATION
The velocity of a fluid in general varies from one point to another even in the
direction of flow. Since, by Newton’s First Law, a change of velocity must
be associated with a force, it is to be expected that the pressure of the fluid
also changes from point to point.
The relation between these changes may be studied by applying Newton’s
Second Law to a small element of the fluid over which the changes of velocity
and pressure are very small. The element is so chosen that it occupies part of
a stream-tube of small cross-section (see Fig. 3.3). The ends of the element
are plane and perpendicular to the central streamline, but may be of any
geometrical shape.
The forces under investigation are those due to the pressure of the fluid all
round the element, and to gravity. Other forces, such as those due to viscosity,
surface tension, electricity, magnetism, chemical or nuclear reactions are
assumed negligibly small. Even the assumption of negligible viscosity is less
restrictive than it may at first seem. The fluids more frequently encountered
have small values of viscosity, and except when eddies are present viscous
forces are significant only very close to solid boundaries. The behaviour of