5-Summary, fluid mech
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Summary
SUMMARY
There are two ways to describe a flow field, the Lagrangian and Eulerian approaches. The Eulerian, in which
flow properties are available at points in the field, is the most appropriate for fluid mechanics.
Flow rate refers to either the volume per unit time or the mass per unit time passing through a surface. The
volume flow rate, or discharge, is given by
where
D
A
is the vector normal to the surface with magnitude equal the differential surface area and
V
is the
velocity vector. If the area vector and velocity vector are aligned, then
where
V
is the average velocity. The corresponding mass flow rate is
If the density is uniformly distributed across the area,
A fluid system is a given quantity of matter consisting always of the same matter. A control volume (cv) is a
geometric volume defined in space and enclosed by a control surface (cs). Mass can cross the control surface.
The Reynolds transport theorem relates the time rate of change of an extensive property of a system to the rate of
change of the property in the control volume plus the net outflow of the property across the control surface. It
provides a link between the Lagrangian and Eulerian forms for the rate of property changes in a fluid.
The continuity equation derives from the application of the Reynolds transport theorem to the conservation of
mass principle and is expressed as
where
V
is the velocity with respect to the control surface and
D
A
is the differential area directed outward from
the control volume. An alternative form of the continuity equation is
where
M
cv
is the mass in the control volume and and are the mass flow rates of flows entering and
leaving the control volume, respectively.
For steady, onedimensional flow in a pipe, the continuity equation reduces to
where the subscripts 1 and 2 refer to the inlet and outlet of the pipe. If, in addition, the flow is incompressible,
then
The differential form of the continuity equation for incompressible flow is ·
V
= 0
Cavitation occurs when the pressure drops to the local vapor pressure of the liquid and bubbles appear due to
liquid boiling. The presence of cavitation can cause serious equipment failures.
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Summary
Copyright ¨ 2009 John Wiley & Sons, Inc. All rights reserved.
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