5.5-Differential Form of the Co, fluid mech

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Differential Form of the Continuity Equation
5.5 Differential Form of the Continuity Equation
In the analysis of fluid flows and the development of numerical models, one of the fundamental independent
equations needed is the differential form of the continuity equation. This equation is derived in this section. The
derivation is accomplished by applying the integral form of the continuity equation to a small control volume
and taking the limit as the volume approaches zero.
A small control volume defined by the
x
,
y
,
z
coordinate system is shown in Fig. 5.15. The integral form of the
continuity equation, Eq. (5.24), is
where
V
is the velocity measured with respect to the local control surface. Applying the Leibnetz theorem for
differentiation of an integral allows the unsteady term to be expressed as
where
V
s
is the local velocity of the control surface with respect to the reference frame. For a control volume
with stationary sides, as shown in Fig. 5.15,
V
s
= 0, so the continuity equation for the control volume can be
written as
Because the volume is very small (infinitesimal), one can assume that the velocity and densities are uniformly
distributed across each face (control surface), and the mass flux term becomes
and the continuity equation assumes the form
Figure 5.15
Elemental control volume.
Considering the flow rates through the six faces of the cubical element and applying those to the foregoing form
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Differential Form of the Continuity Equation
of the continuity equation, results in
(5.30)
Dividing Eq. (5.30) by the volume of the element (
x
y
z
) yields
Taking the limit as the volume approaches zero (that is, as
x
,
y
, and
z
uniformly approach zero) yields the
differential form of the continuity equation
(5.31)
If the flow is steady, the equation reduces to
(5.32)
EXAMPLE 5.10 APPLICATIO OF DIFFERETIAL FORM
OF COTIUITY EQUATIO
The expression
V
= 10
x
i
10
y
j
is said to represent the velocity for a twodimensional (planar)
incompressible flow. Check to see if the continuity equation is satisfied.
Problem Definition
Situation:
Velocity field is given.
Find:
Determine if continuity equation is satisfied.
Plan
Reduce Eq. (5.33) to twodimensional flow (
w
= 0 and substitute velocity components into equation).
Solution
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Differential Form of the Continuity Equation
Continuity equation for twodimensional flow
Continuity is satisfied.
And if the fluid is incompressible, the equation further simplifies to
(5.33)
for both steady and unsteady flow.
In vector notation, Eq. (5.33) is given as
(5.34)
where is the del operator, defined as
Copyright ¨ 2009 John Wiley & Sons, Inc. All rights reserved.
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