Ideal Fluids in Motion

1
Ideal Fluids in Motion

• Ideal Fluid

Steady Flow, Incompressible Flow, Non viscous
Flow, Irrotational Flow.

• The Equation of Continuity

If a fluid is incompressible, its density r is
constant throughout. Thus the volume of fluid
entering a tube at one end per unit of time must
be equal to the volume of fluid leaving the other
end per unit time. In the time Dt we have A1v1Dt
A2v2Dt.
A1v1 A2v2 (continuity equation) RV Av
volume flow rate constant Rm rRV rAv
mass flow rate constant

• Bernoullis Equation

Application of W DKE DU
F2 P2A2
M
Dx2 v2Dt
F1 P1A1
M
Dx1 v1Dt
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Bernoullis Equation Applications

• Bernoullis Equation

P ½rv2 rgy constant
(conservation of energy for a fluid)

• Constant Height (y1 y2)

P ½rv2 constant
If the speed of a fluid element increases as the
element travels along a horizontal streamline,
the pressure of the fluid must decrease, and
conversely.

• Example (velocity of efflux)

We can use Bernoullis equation to calculate the
speed of efflux, v2, from a horizontal orifice
(and area A2) located a depth h below the water
level of a large talk (with area A1).
(1)
(1?2)
(2)
v1 v2A2/A1
P1 P2 Patm
(Torricellis Law)
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Bernoullis Equation Application

• Venturi Meter

A Venturi meter is used to measure the flow of a
fluid in a pipe. The meter is constructed
between two sections of a pipe, the
cross-sectional area A of the entrance and exit
of the meter matches the pipes cross-sectional
area. Between the entrance and exit, the fluid
(with density r) flows from the pipe with speed V
and then through a narrow throat of
cross-sectional area a with speed v. A manometer
(with fluid of density rM) connects the wider
portion of the meter to the narrow portion. What
is V in terms of r, rM, h, a, and A?
d
C
(1?2)
(C?C)
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Bernoullis Equation Application

• Siphon

The figure shows a siphon, which is a device for
removing liquid from a container. Tube ABC must
initially be filled, but once this is done,
liquid will flow until the liquid surface of the
container is level with the tube opening A. With
what speed does the liquid emerge from the tube
at C? What is the greatest possible height h1
that a siphon can lift water?
(S?A)
A Area of container a area of tube
S
y0
V
(A?C)
v
(B?A)
v