fluid_mechanics

6.2 Conservation of Mass 269
streamlines for a given velocity field is given in Sections 4.1.4
and 6.2.3.) It is noted that all of the streamlines (except for the one
through the origin) are curved. However, because the flow is irrotational,
there is no rotation of the fluid elements. That is, lines
OA and OB of Fig. 6.4 rotate with the same speed but in opposite
directions.
As shown by Eq. 6.17, the condition of irrotationality is equivalent
to the fact that the vorticity, Z,
is zero or the curl of the velocity is zero.
In addition to the rotation associated with the derivatives 0u0y and 0v0x, it is observed
from Fig. 6.4b that these derivatives can cause the fluid element to undergo an angular
deformation, which results in a change in shape of the element. The change in the original right
angle formed by the lines OA and OB is termed the shearing strain, dg, and from Fig. 6.4b
where dg is considered to be positive if the original right angle is decreasing. The rate of change
of dg is called the rate of shearing strain or the rate of angular deformation and is commonly
denoted with the symbol g. The angles da and db are related to the velocity gradients through Eqs.
6.10 and 6.11 so that
and, therefore,
dg
g lim
dtS0 dt lim
dtS0
dg da db
c 10v 0x2 dt 10u0y2 dt
d
dt
g 0v
0x 0u
0y
(6.18)
As we will learn in Section 6.8, the rate of angular deformation is related to a corresponding shearing
stress which causes the fluid element to change in shape. From Eq. 6.18 we note that if
0u0y 0v0x, the rate of angular deformation is zero, and this condition corresponds to the case
in which the element is simply rotating as an undeformed block 1Eq. 6.122. In the remainder of this
chapter we will see how the various kinematical relationships developed in this section play an important
role in the development and subsequent analysis of the differential equations that govern fluid motion.
6.2 Conservation of Mass
Conservation of
mass requires that
the mass of a
system remain
constant.
As is discussed in Section 5.1, conservation of mass requires that the mass, M, of a system remain
constant as the system moves through the flow field. In equation form this principle is expressed as
DM sys
Dt
We found it convenient to use the control volume approach for fluid flow problems, with the control
volume representation of the conservation of mass written as
0
0t cv
r dV cs
r V nˆ dA 0
(6.19)
where the equation 1commonly called the continuity equation2 can be applied to a finite control
volume 1cv2, which is bounded by a control surface 1cs2. The first integral on the left side of Eq. 6.19
represents the rate at which the mass within the control volume is changing, and the second integral
represents the net rate at which mass is flowing out through the control surface 1rate of mass
outflow rate of mass inflow2. To obtain the differential form of the continuity equation, Eq. 6.19
is applied to an infinitesimal control volume.
0
6.2.1 Differential Form of Continuity Equation
We will take as our control volume the small, stationary cubical element shown in Fig. 6.5a. At
the center of the element the fluid density is r and the velocity has components u, v, and w. Since
the element is small, the volume integral in Eq. 6.19 can be expressed as
0
(6.20)
0t r dV 0r
cv 0t dx dy dz