fluid_mechanics

276 Chapter 6 ■ Differential Analysis of Fluid Flow
and the operator D1 2Dt is the material derivative 1see Section 4.2.122. In the last chapter it was
demonstrated how Eq. 6.43 in the form
a F contents of the 0
control volume 0t Vr dV cs
VrV nˆ dA
cv
(6.44)
could be applied to a finite control volume to solve a variety of flow problems. To obtain the
differential form of the linear momentum equation, we can either apply Eq. 6.43 to a differential
system, consisting of a mass, dm, or apply Eq. 6.44 to an infinitesimal control volume, dV, which
initially bounds the mass dm. It is probably simpler to use the system approach since application
of Eq. 6.43 to the differential mass, dm, yields
D1V dm2
dF
Dt
where dF is the resultant force acting on dm. Using this system approach dm can be treated as a
constant so that
dF dm DV
Dt
But DVDt is the acceleration, a, of the element. Thus,
dF dm a
(6.45)
which is simply Newton’s second law applied to the mass d m . This is the same result that would
be obtained by applying Eq. 6.44 to an infinitesimal control volume 1see Ref. 12. Before we can
proceed, it is necessary to examine how the force dF can be most conveniently expressed.
Both surface forces
and body forces
generally act on
fluid particles.
6.3.1 Description of Forces Acting on the Differential Element
In general, two types of forces need to be considered: surface forces, which act on the surface of the
differential element, and body forces, which are distributed throughout the element. For our purpose,
the only body force, dF b , of interest is the weight of the element, which can be expressed as
dF b dm g
(6.46)
where g is the vector representation of the acceleration of gravity. In component form
dF bx dm g x
(6.47a)
dF by dm g y
(6.47b)
dF bz dm g z
(6.47c)
where g x , g y , and g z are the components of the acceleration of gravity vector in the x, y, and z
directions, respectively.
Surface forces act on the element as a result of its interaction with its surroundings. At any
arbitrary location within a fluid mass, the force acting on a small area, dA, which lies in an arbitrary
surface, can be represented by dF s , as is shown in Fig. 6.9. In general, dF s will be inclined with
respect to the surface. The force dF s can be resolved into three components, dF n , dF 1 , and dF 2 ,
where dF n is normal to the area, dA, and dF 1 and dF 2 are parallel to the area and orthogonal to
each other. The normal stress, s n , is defined as
dF n
s n lim
dAS0 dA
δF n
δ F s
δA
δF 2
δF 1
Arbitrary
surface
F I G U R E 6.9 Components of force acting
on an arbitrary differential area.