Begin2.DVI
ben green
(Ben Green)
#1
∆M. It is assumed that the velocity is the same at all points over the tiny element
of surface area. In a time interval ∆t, the amount of fluid which crosses the element
dS is given by ∆M=ρV ·ˆendS ∆t. The total mass of fluid flowing out of the volume
Vbounded by the surface Sis given by
∆M= ∆ t
∫∫
S
ρV ·dS= ∆ t
∫∫∫
V
div (ρV)dV.
Also the total mass of the fluid enclosed within the volume Vbounded by Scan be
represented as the integral
M=
∫∫∫
V
ρdV. (9 .87)
The rate of change of the mass with time is
∂M
∂t
=
∫∫∫
V
∂ρ
∂t
dV. (9 .88)
Hence, in a time interval ∆t, the amount of fluid in the volume Vdiminishes by the
amount
∆M=−∆t
∫∫∫
V
∂ρ
∂t
dV. (9 .89)
The amount of fluid flowing out of the arbitrary volume is equated to the amount
of fluid decreasing within the volume to obtain
∆t
∫∫∫
V
div (ρV)dV =−∆t
∫∫∫
V
∂ρ
∂t dV
or
∫∫∫
V
[
div (ρV) +
∂ρ
∂t
]
dV = 0.
(9 .90)
For an arbitrary volume V within the fluid, the relation (2.42) must hold and con-
sequently
∂ρ
∂t
+div (ρV) = 0. (9 .91)
This equation is called the continuity equation of hydrodynamics which can also be
expressed in the form
∂ρ
∂t
+∇ρ·V +ρ∇V = 0. (9 .92)
The first two terms on the left-hand side of this last equation represents the time
rate of change of the density ρ, that is,
dρ
dt =
∂ρ
∂t +∇ρ·
V. (9 .93)
