Mechanics of Fluids

30 Fundamental concepts
Uniform flow
tubes of other materials. Equation 1.12 therefore overestimates the actual
capillary rise. Mercury, which has an angle of contact with clean glass of
about 130 ◦ in air, and therefore a negative value of cos θ, experiences a
capillary depression.
Surface tension becomes important when solid boundaries of a liquid
surface are close together or when the surface separating two immiscible
fluids has a very small radius of curvature. The forces due to surface tension
then become comparable with other forces and so may appreciably affect
the behaviour of the liquid. Such conditions may occur, for example, in
small-scale models of rivers or harbours. The surface tension forces may be
relatively much more significant in the model than in the full-size structure;
consequently a simple scaling-up of measurements made on the model may
not yield results accurately corresponding to the full-size situation.
In apparatus of small size the forces due to surface tension can completely
stop the motion of a liquid if they exceed the other forces acting on it. It is
well known, for example, that a cup or tumbler may be carefully filled until
the liquid surface is perhaps 3 mm above the rim before overflowing begins.
Among other instances in which surface tension plays an important role are
the formation of bubbles, the break-up of liquid jets and the formation of
drops, the rise of water in soil above the level of the water table, and the
flow of thin sheets of liquid over a solid surface.
In most engineering problems, the distances between boundaries are
sufficient for surface tension forces to be negligible compared with the
other forces involved. The consequent customary neglect of the surface
tension forces should not, however, lead us to forget their importance in
small-scale work.
1.8 BASIC CHARACTERISTICS OF FLUIDS IN MOTION
1.8.1 Variation of flow parameters in space and time
In general, quantities such as velocity, pressure and density, which describe
the behaviour or state of a fluid, vary with respect to both space and time.
The location of a point in space can be defined using a coordinate system,
examples of which are the Cartesian system, with the variables x, y, z or the
cylindrical polar coordinates, with variables x, r, θ. Defining time by t and
using Cartesian coordinates x, y, z to define positions in space then, as an
example, the velocity u is a function of x, y, z and t. Mathematically, we
write u = u(x, y, z, t), and this is called the velocity field of the flow. Similarly,
p = p(x, y, z, t) is the pressure field. If we wish to discuss the flow
within a region or domain in a general way, without reference to any particular
flow or fluid properties, then we refer to the flow field.
If, at a particular instant, the various quantities do not change from point
to point over a specified region, then the flow is said to be uniform over that
region. If however, changes do occur from one point to another, the flow is
said to be non-uniform. These changes with position may be found in the
direction of the flow or in directions perpendicular to it. This latter kind of