Kinematics is the study of motion without reference to the forces or stresses that produce
the motion. In this chapter, fluid kinematics is presented in two and three dimensions starting
with simple fluid-particle-path concepts and then proceeding to topics of greater complexity. These include: particle- and field-based descriptions for the time-dependent position, veloc-
ity, and acceleration of fluid particles; the relationship between the fluid velocity gradient
tensor and the deformation and rotation of fluid elements; and the general mathematical
relationships that govern arbitrary volumes that move and deform within flow fields. The
forces and stresses that cause fluid motion are considered in subsequent chapters covering
the dynamics or kinetics of fluid motion.
In general, three independent spatial dimensions and time are needed to fully describe
fluid motion. When a flow does not depend on time, it is called steady; when it does depend
on time it is called unsteady. In addition, fluid motion is studied in fewer than three dimen-
sions whenever possible because the necessary analysis is usually simpler and relevant
phenomena are more easily understood and visualized.
A truly one-dimensional flow is one in which the flow’s characteristics can be entirely
described with one independent spatial variable. Few real flows are strictly one dimensional,
although flows in long, straight constant-cross-section conduits come close. Here, the inde-
pendent coordinate may be aligned with the flow direction, as in the case of
low-frequency pulsations in a pipe as shown in Figure 3.1a, where z is the independent
coordinate and darker gray indicates higher gas density. Alternatively, the independent
coordinate may be aligned in the cross-stream direction, as in the case of viscous flow in
a round tube where the radial distance, R, from the tube’s centerline is the independent
coordinate (Figure 3.1b). In addition, higher dimensional flows are sometimes analyzed
in one dimension by averaging the properties of the higher dimensional flow over an appro-
priate distance or area (Figure 3.1c and d).
A two-dimensional, or plane, flow is one in which the variation of flow characteristics can be
described by two spatial coordinates. The flow of an ideal fluid past a circular cylinder of
infinite length having its axis perpendicular to the primary flow direction is an example of
a plane flow (see Figure 3.2a). (Here we should note that the word “cylinder” may also be
used in this context for any body having a cross-sectional shape that is invariant along its
length even if this shape is not circular.) This definition of two-dimensional flow officially
includes the flow around bodies of revolution where flow characteristics are identical in
any plane that contains the body’s axis (see Figure 7.24). However, such flows are custom-
arily called three-dimensional axisymmetric flows.
A three-dimensional flow is one that can only be properly described with three indepen-
dent spatial coordinates and is the most general case considered in this text. Sometimes
curvilinear coordinates that match flow-field boundaries or symmetries greatly simplify
the analysis and description of flow fields. Thus, several different coordinate systems are
used in this text (see Figure 3.3). Two-dimensional (plane) Cartesian and polar coordinates
for an arbitrary point P (Figure 3.3a) may be denoted by the coordinate pairs (x, y), (x1 , x2 ),
or (r, q) with the corresponding velocity components (u, v), (u1 , u2 ), or ðu r; uqÞ. Here the axis
perpendicular to the plane of interest will be the z-axis or x3 -axis. In three dimensions, Car-
tesian coordinates (Figures 2.1 and 3.3b) may be used to locate a point P via the coordinate
triplets (x, y, z) or (x1 , x2 , x3 ) with corresponding velocity components (u, v, w) or (u1 , u2 , u3 ).
Cylindrical polar coordinates for P (Figure 3.3c) are denoted by (R, 4, z) with corresponding
velocity components ðuR; u4; u zÞ. Spherical polar coordinates for P (Figure 3.3d) are denoted
by (r, q, 4) with the corresponding velocity components ðu r; uq; u4Þ. In all cases, unit vectors
are denoted by e with an appropriate subscript as in (2.1) and Figure 2.1. More information
about these coordinate systems is provided in Appendix B.

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