Page 33 - Mechanical Engineers Reference Book

1/22 Mechanical engineering principles

1.5.3 Fluid flow
1.5.3.1 Definitions
(a) Continuity For almost all analysis, a fluid is considered to
be a continuum, that is, with non-discontinuities or cavities in
the flow stream. Cavitation, two-phase flow, ‘bubbly’ flow,
etc. are special cases with non-standard relationships. av I
av 1
Therefore for a continuum, by considering the flow through v- - - 6x Vf- - 6x
an elemental cuboid the continuity equation in three dimen- ax 2 ax 2
sions may be shown to be
(1.22)
where v, is the fluid velocity in the x direction, etc. For a fluid 6X
of constant density aU I
u-- -6y
aY 2
(1.23) (a) Vorticity
That is, the velocity of an incompressible fluid flow cannot
increase in all three directions at the same time without
producing discontinuity or cavitation.
For two-dimensional flow: 6Y I au
Uf- 6y
(1.24) av

For one-dimensional flow the continuity equation may be
linked with the conservation of mass, which states that for
steady flow conditions mass flow rate, h, is constant through-
out a flow system:
m = pAv (1.25)
where A is the cross-sectional area normal to the direction of
flow.
6X
(b) Circulation r Circulation is defined as the line integral
of the tangential velocity around a closed contour: (b) Rotation
r = f v,ds (1.26) Figure 1.38
r is positive if the closed contour is on the left.
is a tube of infinitely small cross section with a stream line as
(c) Vorticity i Vorticity is defined as the circulation per unit its axis.
area, and by considering the circulation around the element in
Figure 1.38(a) it can be shown that (g) Energy Energy is the stored form of heat and work. The
basic concepts applied in fluid mechanics are:
(1.27)
The conservation of energy
That energy is transferred only as heat or work
(d) Rotation w Rotation is defined as the instantaneous mean That energy in a fluid flow system is stored only as internal
angular velocity of two mutually perpendicular lines in a plane energy, kinetic energy or potential energy.
of the flow field. By considering the angular velocities of the Other forms of energy (electrical, magnetic, chemical, etc.)
w=- :i2 :I may have to be taken into account in some circumstances, but
two lines OA and OB in Figure 1.38(b) it can be shown that
are not usually included in general fluid mechanics relation-
(1.28)
ships.
Enthalpy and entropy need to be considered for gas flow
or the rotation is equal to half the vorticity. analysis (see Section 1.5.8). The basic energy-flow equation is
the steady-flow energy equation:
(e) Stream lines The stream line is a line drawn in a flow
stream which is everywhere tangential to the direction of flow. V2
&A
A family of stream lines may be described mathematically by a Q + *= (h + + gz) (1.29)
stream function I), where I,+ = fn(x,y). Each stream line has
the same function with a value of I) peculiar to that line. where Q is the rate of heat transfer,
W is the rate of work transfer (power),
(f) Stream tubes Since a line has no thickness, there can be h is the specific enthalpy (if e is the specific internal
no flow along a stream line. The stream tube is a concept energy, p the pressure and p the fluid density, then
introduced to enable flow along a stream line to be studied. It h = e + WP)),